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Eur. Phys. J. C (2014) 74:3110

DOI 10.1140/epjc/s10052-014-3110-1

Regular Article - Experimental Physics

The HERMES Collaboration

A. Airapetian13,16, N. Akopov27, Z. Akopov6, W. Augustyniak26, A. Avetissian27, H. P. Blok18,25, A. Borissov6, V. Bryzgalov20, M. Capiluppi10a,10b, G. P. Capitani11, E. Cisbani22a,22b, G. Ciullo10a,10b, M. Contalbrigo10a, P. F. Dalpiaz10a,10b, W. Deconinck6, R. De Leo2, E. De Sanctis11, M. Diefenthaler9,15, P. Di Nezza11, M. Dren13, M. Ehrenfried13, G. Elbakian27, F. Ellinghaus5, E. Etzelmller13, R. Fabbri7, L. Felawka23, S. Frullani22a,22b, D. Gabbert7, G. Gapienko20, V. Gapienko20, F. Garibaldi22a,22b, G. Gavrilov6,19,23, V. Gharibyan27, M. Hartig6, D. Hasch11, Y. Holler6, I. Hristova7, A. Ivanilov20, H. E. Jackson1, S. Joosten12,15, R. Kaiser14, G. Karyan27, T. Keri13,E. Kinney5, A. Kisselev19, V. Korotkov20, V. Kozlov17, P. Kravchenko19, V. G. Krivokhijine8, L. Lagamba2, L. Lapiks18, I. Lehmann14, P. Lenisa10a,10b, W. Lorenzon16, B.-Q. Ma3, D. Mahon14, S. I. Manaenkov19, Y. Mao3, B. Marianski26, H. Marukyan27, A. Movsisyan10a,27, M. Murray14, Y. Naryshkin19, A. Nass9, W.-D. Nowak7, L. L. Pappalardo10a,10b, R. Perez-Benito13, A. Petrosyan27, P. E. Reimer1, A. R. Reolon11, C. Riedl7,15, K. Rith9, A. Rostomyan6, D. Ryckbosch12, A. Schfer21, G. Schnell4a,4b,12,a, K. P. Schler6, B. Seitz14, T.-A. Shibata24, M. Stahl13, M. Stancari10a,10b, M. Statera10a,10b, E. Steffens9, J. J. M. Steijger18, S. Taroian27, A. Terkulov17, R. Truty15,A. Trzcinski26, M. Tytgat12, Y. Van Haarlem12, C. Van Hulse4a,12, V. Vikhrov19, I. Vilardi2, S. Wang3, S. Yaschenko6,9, S. Yen23, D. Zeiler9, B. Zihlmann6, P. Zupranski26

1 Physics Division, Argonne National Laboratory, Argonne, IL 60439-4843, USA

2 Sezione di Bari, Istituto Nazionale di Fisica Nucleare, 70124 Bari, Italy

3 School of Physics, Peking University, Beijing 100871, China

4 (a)Department of Theoretical Physics, University of the Basque Country UPV/EHU, 48080, Bilbao, Spain, (b)IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain

5 Nuclear Physics Laboratory, University of Colorado, Boulder, CO 80309-0390, USA

6 DESY, 22603 Hamburg, Germany

7 DESY, 15738 Zeuthen, Germany

8 Joint Institute for Nuclear Research, 141980 Dubna, Russia

9 Physikalisches Institut, Universitt Erlangen-Nrnberg, 91058 Erlangen, Germany

10 (a)Sezione di Ferrara, Istituto Nazionale di Fisica Nucleare, 44122 Ferrara, Italy, (b)Dipartimento di Fisica e Scienze della Terra, Universit diFerrara, 44122 Ferrara, Italy

11 Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, 00044 Frascati, Italy

12 Department of Physics and Astronomy, Ghent University, 9000 Gent, Belgium

13 II. Physikalisches Institut, Justus-Liebig Universitt Gieen, 35392 Gieen, Germany

14 SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK

15 Department of Physics, University of Illinois, Urbana, IL 61801-3080, USA

16 Randall Laboratory of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA

17 Lebedev Physical Institute, 117924 Moscow, Russia

18 National Institute for Subatomic Physics (Nikhef), 1009 DBAmsterdam, The Netherlands

19 B.P. Konstantinov Petersburg Nuclear Physics Institute, Gatchina 188300, Leningrad Region, Russia

20 Institute for High Energy Physics, Protvino 142281, Moscow Region, Russia

21 Institut fr Theoretische Physik, Universitt Regensburg, 93040 Regensburg, Germany

22 (a)Sezione di Roma, Gruppo Collegato Sanit, Istituto Nazionale di Fisica Nucleare, 00161 Rome, Italy, (b)Istituto Superiore di Sanit, 00161Rome, Italy

23 TRIUMF, Vancouver, BC V6T 2A3, Canada

24 Department of Physics, Tokyo Institute of Technology, Tokyo 152, Japan

25 Department of Physics and Astronomy, VU University, 1081 HV Amsterdam, The Netherlands

26 National Centre for Nuclear Research, 00-689 Warsaw, Poland

27 Yerevan Physics Institute, 375036 Yerevan, Armenia

Received: 8 July 2014 / Accepted: 30 September 2014 / Published online: 15 November 2014 The Author(s) 2014. This article is published with open access at Springerlink.com

Spin density matrix elements in exclusive electroproduction on 1H and 2H targets at 27.5 GeV beam energy

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Abstract Exclusive electroproduction of mesons on unpolarized hydrogen and deuterium targets is studied in the kinematic region of Q2 > 1.0 GeV2, 3.0 GeV < W < 6.3 GeV, and t < 0.2 GeV2. Results on the angular

distribution of the meson, including its decay products, are presented. The data were accumulated with the HERMES forward spectrometer during the 19962007 running period using the 27.6 GeV longitudinally polarized electron or positron beam of HERA. The determination of the virtual-photon longitudinal-to-transverse cross-section ratio reveals that a considerable part of the cross section arises from transversely polarized photons. Spin density matrix elements are presented in projections of Q2 or t . Violation

of s-channel helicity conservation is observed for some of these elements. A sizable contribution from unnatural-parity-exchange amplitudes is found and the phase shift between those amplitudes that describe transverse production by longitudinal and transverse virtual photons, L T and

T T , is determined for the rst time. A hierarchy

of helicity amplitudes is established, which mainly means that the unnatural-parity-exchange amplitude describing the T T transition dominates over the two natural-parity-

exchange amplitudes describing the L L and T T

transitions, with the latter two being of similar magnitude.

Good agreement is found between the HERMES proton data and results of a pQCD-inspired phenomenological model that includes pion-pole contributions, which are of unnatural parity.

1 Introduction

Exclusive electroproduction of vector 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 , 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., , , , ...). 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 Unnatu

ral 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

a e-mail: [email protected]

exchange of a pomeron and the latter to the exchange of a (combination of) reggeon(s).

Spin density matrix elements (SDMEs) describe the nal 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 transitions from the initial helicity states of virtual photon and incoming nucleon to the nal 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-to-transverse cross-section ratio.

In the framework of pQCD, the nucleon structure can also be studied through hard exclusive meson production as the process amplitude contains Generalized Parton Distributions (GPDs) [57]. For longitudinal virtual photons, 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 sufcient 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 avor f or a gluon. These GPDs are of special interest as they are related to the total angular momentum carried by quarks or gluons in the nucleon [10].

Although there is no such rigorous proof for transverse virtual photons, phenomenological models use the modied perturbative approach [11] instead, which takes into account parton transverse momenta. The latter are included at subleading twist in the subprocess f M f , 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 pQCD-inspired phenomenological GK model can describe existing data on cross sections, SDMEs and spin asymmetries in exclusive vector-meson production for values of Bjorken-x, xB, below about 0.2 [1214]. 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 at 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 leptoproduction [17], will be compared in this paper to the HERMES

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proton data in terms of SDMEs and certain combinations of them.

Early papers on exclusive electroproduction are summarized in Ref. [18], which particularly contains results on SDMEs obtained at DESY for 0.3 GeV2 < Q2 < 1.4 GeV2 and 0.3 GeV < W < 2.8 GeV. The symbol Q2 represents the negative square of the virtual-photon four-momentum and W is the invariant mass of the photon-nucleon system. Recently, SDMEs in exclusive electroproduction were studied for1.6 GeV2 < Q2 < 5.2 GeV2 by CLAS [19] and it was found that the exchange of the pion Regge trajectory dominates exclusive production, even for Q2 values as large as5 GeV2.

2 Formalism

2.1 Spin density matrix elements

The meson is produced in the following reaction:

e + p e + p + , (1) with a branching ratio Br = 89.1 % for the decay: + + + 0, 0 2. (2)

The angular distribution of the three nal-state pions depends on SDMEs. The rst subprocess of vector-meson production, the emission of a virtual photon (e e + ), is described

by the photon spin density matrix [3],

U+L = U + Pb L , (3)

where U and L denote unpolarized and longitudinally polarized beam, respectively, and Pb is the value of the beam polarization. The photon spin density matrix can be calculated in quantum electrodynamics.

The vector-meson spin density matrix V V is expressed

through helicity amplitudes FV N N . These amplitudes describe the transition of a virtual photon with helicity to a vector meson with helicity V , while N and N are the helicities of the nucleon in the initial and nal states, respectively.

Helicity amplitudes depend on W, Q2, and t = t tmin,

where t is the Mandelstam variable and tmin represents the

smallest kinematically allowed value of t at xed virtual-

photon energy and Q2. 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 N centre-of-mass (CM) system. In this system, the spin density matrix of the vector meson is given by the von Neumann equation [3],

V V =

where N is a normalization factor, see Refs. [3,20].

After the decomposition of U+L into the standard set of 3 3 Hermitian matrices , the vector-meson spin den

sity matrix is expressed in terms of a set of nine matrices V V related to various photon polarization states: trans

versely polarized photon ( = 0, ..., 3), longitudinally polar

ized photon ( = 4), and terms describing their interference

( = 5, ..., 8) [3]. When contributions of transverse and lon

gitudinal photons cannot be separated, the SDMEs are customarily dened as

r04V V = (0V V + R4V V )(1 + R)1,

rV V =

V V (1 + R)1, = 1, 2, 3, R

V V (1 + R)1, = 5, 6, 7, 8.

(5)

The quantity R = dL/dT is the longitudinal-to-transverse

virtual-photon differential cross-section ratio and is the ratio of uxes of longitudinal and transverse virtual photons.

2.2 Helicity amplitudes

A helicity amplitude can be decomposed into a sum of a NPE amplitude T and a UPE amplitude U,

FV N N = TV N N + UV N N , (6)

for details see Refs. [3,20]. The relations between the amplitudes F, T , and U are the following [3]:

TV N N =

2[FV N N + (1)V FV N N ],(7)

UV N N =

2[FV N N (1)V FV N N ].

(8)

The asymptotic behaviour of amplitudes F at small t [4],

FV N N

N )( N )|, (9)

follows from angular-momentum conservation. Equations (7)(9) show that the double-helicity-ip amplitudes with

|V | = 2 are suppressed at least by a factor of t /M,

and the contributions of these double-helicity-ip amplitudes to the SDMEs are suppressed by t /M2. Therefore they will

be neglected throughout the paper.

For an unpolarized target, there exists no interference between NPE and UPE amplitudes and there is no linear contribution from nucleon-helicity-ip amplitudes to SDMEs. For brevity, the following notations will be used:

M |(V

N N

FV N N U+L F V N N , (4)

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TV T V 12

N N

TV N N T V N N . (10)

Using the symmetry properties [3,20] of the amplitudes T , Eq. (10) can be rewritten as

TV T

V = TV

Denitions of angles and reference frames are shown in Fig. 1. The directions of the axes of the hadronic CM system and of the -meson rest frame follow the directions of the axes of the helicity frame [3,20,21].

The angle between the production and the lepton scattering plane in the hadronic CM system is given by

cos =

(q v) (k k )

|q v| |k k |

2 . (11)

Here, the rst and second product on the right-hand side gives the contribution of NPE amplitudes without and with nucleon-helicity ip, respectively. Analogous relations hold for UPE amplitudes. An additional abbreviated notation in the text will be the omission of the nucleon-helicity indices when discussing the amplitudes with N = N , i.e.,

TV TV

2 12 = TV 12 12

UV UV

2 12 T V

1 2 + TV 12 12 T V 12

, (16)

sin = [

(q v) (k k )] q |q v| |k k | |q|

. (17)

Here k, k , q = kk , and v are the three-momenta of the

incoming and outgoing leptons, virtual photon, and meson respectively.

The unit vector normal to the decay plane in the rest frame is dened by

n =

p+ p

| p+ p |

2 12 = UV 12 12 . (12)

The dominance of diagonal V transitions (V = )

is called s-channel helicity conservation (SCHC).

2.3 Angular distribution

The SDMEs in exclusive electroproduction of mesons are determined using the process in Eq. (1). They are tted as parameters of WU+L( , , cos ), which is the three-

dimensional angular distribution, to the corresponding experimental distribution of the three pions originating from the -meson decay. The angular distribution WU+L( , , cos )

is decomposed into WU and WL, see Eq. (13), which are the

respective distributions for unpolarized and longitudinally polarized beams. From the t, 15 unpolarized SDMEs (see Eq. 14) are extracted and additionally 8 polarized SDMEs (see Eq. 15) from data collected with a longitudinally polarized beam.

WU+L( , , cos ) = WU ( , , cos ) + PbWL( , , cos ), (13)

WU ( , , cos ) =

3 82

, (18)

where p+ and p are the three-momenta of the positive

and negative decay pions in the rest frame.

The polar angle of the unit vector n in the -meson rest frame, with the z-axis aligned opposite to the outgoing nucleon momentum p and the y-axis directed along p q,

is dened by

cos =

p n

| p |

, (19)

while the azimuthal angle of the unit vector n is given by

cos =

(q p ) ( p n)

|q p | | p n|

, (20)

sin = [

(q p ) p ] (n p ) |(q p ) p | |n p |

. (21)

2(3r0400 1) cos2 2Re{r

2(1 r0400) +

} sin 2 cos r0411 sin2 cos 2

cos 2 (r111 sin2 + r100 cos2 2Re{r

1 10

} sin 2 cos r111 sin2 cos 2)

sin 2 (2Im{r

} sin 2 sin + Im{r211} sin2 sin 2)

+ 2 (1 + ) cos (r511 sin2 + r500 cos2 2Re{r

5 10

} sin 2 cos r511 sin2 cos 2)

+ 2 (1 + ) sin (2Im{r

2 10

6 10

} sin 2 sin + Im{r611} sin2 sin 2)

, (14)

WL( , , cos ) =

382 [ 1 2(2Im{r

3 10

} sin 2 sin + Im{r311} sin2 sin 2)

+ 2 (1 ) cos (2Im{r

} sin 2 sin + Im{r711} sin2 sin 2)

+ 2 (1 ) sin (r811 sin2 + r800 cos2 2Re{r

8 10

7 10

} sin 2 cos r811 sin2 cos 2)]. (15)

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Events/3MeV

250

200

150

100

Fig. 1 Denition of angles in the process eN eN, where

+0. Here, is the angle between the production plane and the lepton scattering plane in the center-of-mass system of the virtual photon and the target nucleon. The variables and are respectively the polar and azimuthal angles of the unit vector normal to the decay plane in the -meson rest frame

3 Data analysis

3.1 HERMES experiment

The data analyzed in this paper were accumulated with the HERMES spectrometer during the running period of 1996 to 2007 using the 27.6 GeV longitudinally polarized electron or positron beam of HERA, and gaseous hydrogen or deuterium targets. The HERMES forward spectrometer, which is described in detail in Ref. [22], was built of two identical halves situated above and below the lepton beam pipe. It consisted of a dipole magnet in conjunction with tracking and particle identication detectors. Particles were accepted when their polar angles were in the range 170 mrad in

the horizontal direction and (40140) 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 efciency of 98 % and a hadron contamination below 1 %.

3.2 Selection of exclusively produced mesons

The following requirements were applied to select exclusively produced 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 identied through the analysis of the combined responses of the four particle-identication detectors [22].

(ii) A 0 meson that is reconstructed from two calorimeter clusters as explained in Ref. [23] is selected requiring the two-photon invariant mass to be in the interval 0.11 GeV < M( ) < 0.16 GeV. The distribution of M( ) is shown in Fig. 2. This distribution is centered

100 110 120 130 140 150 160 170

M (

) [MeV]

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

at m0 = 134.69 19.94 MeV, which agrees well with

the PDG [24] value of the 0 mass.(iii) The three-pion invariant mass is required to obey0.71 GeV M(+0) 0.87 GeV.

(iv) The kinematic requirements for exclusive production of mesons are the following:

(a) The scattered-lepton momentum lies above 3.5 GeV.(b) The constraint t < 0.2 GeV2 is used.

both for proton and deuteron as E = M

2 X

M2p2Mp , with

Mp being the proton mass and M2X = (p+q p+

p p

0 )2 the missing mass squared, where p, q, p+ , p , and p0 are the four-momenta of target nucleon, virtual photon, and each of the three pions respectively. In this analysis, taking into account the 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 Q2 > 1.0 GeV2 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 dene 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 mesons. These data samples are referred to in the following as data in the entire kinematic region. The invariant-mass distributions for exclusively pro-

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entire kinematic region

0.0 GeV2 < -t/< 0.044 GeV2

Events/11MeV

Events/0.2GeV

350

400

150

300

250

100

200

150

100

0 0 5 10 15 20 0 5 10 15 20

E [GeV]

720 740 760

780 800 820 840 860

Events/0.2GeV

150

+ 0 ) [MeV]

0.044 GeV2 < -t/< 0.105 GeV2

150

0.105 GeV2 < -t/< 0.2 GeV2

M (

Events/11MeV

100

Events/0.2GeV

deuteron

100

200

150

100

0 0 5 10 15 20 0 5 10 15 20

E [GeV]

720 740 760

780 800 820 840 860

Fig. 4 The E distributions of mesons produced in the entire kinematic region and in three kinematic 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

angular distribution using an unbinned maximum likelihood method. The probability distribution function is

WU+L(R; , , cos ), where R represents the set of 23

SDMEs, i.e., the coefcients of the trigonometric functions in Eqs. (14, 15). The negative log-likelihood function to be minimized reads

ln L(R) =

i=1

M (

+ 0 ) [MeV]

Fig. 3 BreitWigner t (solid line) of +0 invariant mass distributions after application of all criteria to select mesons produced exclusively from proton (top) and from deuteron (bottom). The dashed line represents the PDG value of the mass

duced mesons are shown in Fig. 3. Note the reasonable agreement of the t result, m = 784.8 55.8 MeV for pro

ton data and m = 784.658.2 MeV for deuteron data, with

the PDG [24] value of the mass. The distributions 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 from a PYTHIA [25] 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 .

3.3 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 spin density matrix elements

4.1 The unbinned maximum likelihood method

The SDMEs are extracted from data by tting the angular distribution WU+L( , , cos ) to the experimental

ln WU+L(R; i, i, cos i)

N(R)

, (22)

where the normalization factor

N(R) =

NMC

j=1

WU+L(R; j , j , cos j ) (23)

is calculated numerically using events from a PYTHIA Monte Carlo generated according to an isotropic three-dimensional 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 NMC, respectively.

4.2 Background treatment

In order to account for the SIDIS background in the t, rst SIDIS-background SDMEs are obtained using Eqs. (22, 23) for the PYTHIA SIDIS sample in the exclusive region. Then, SDMEs corrected for SIDIS background are obtained

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400

Events/0.08GeV

Events/0.14GeV2

200

300

150

200

100

0 4 6

W [GeV]

0 0 2.5 5

Q2 [GeV2]

Events/0.012GeV2

Events/0.30GeV

200

150

100

0 0 0.2 0.4 0.6

-t/ [GeV2]

0 0 10

P+ [GeV]

Events/8.0MeV

Events/1.4MeV

400

150

300

100

200

100

0 0.6 0.8

M(+0) [GeV]

0 0.1 0.125 0.15

M() [GeV]

Fig. 5 Distributions of several kinematic variables from experimental data on exclusive -meson leptoproduction (black squares) in comparison with simulated exclusive events from the PYTHIA generator (dashed areas). Simulated events are normalized to the experimental data

as follows [26]:

ln L(R)

i=1

ln (1 fbg) WU+L(R; i, i, cos i)

N(R, )

fbg WU+L( ; i, i, cos i)

N(R, )

. (24)

From now on, R denotes the set of SDMEs corrected for

background, the set of the SIDIS-background SDMEs, and fbg is the fraction of SIDIS background. The normalization factor reads correspondingly

N(R, ) =

NMC

j=1

(1 fbg) WU+L(R; j , j , cos j )

+ fbg WU+L( ; j , j , cos j ) .(25)

4.3 Systematic uncertainties

The total systematic uncertainty on a given extracted SDMEr is obtained by adding in quadrature the uncertainty from the background subtraction procedure, rbgsys, and the one due

to the extraction method, r MCsys. The former uncertainty is assigned to be the difference between the SDME obtained with and without background correction. This conservative approach also covers the small uncertainty on the fraction of SIDIS background, fbg. The uncertainty r MCsys is estimated using the Monte Carlo data that were generated with an angular distribution determined by the set of SDMEs R.

The 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] and were then reconstructed and analyzed in the same way as experimental data. These Monte Carlo data were used to extract the SDME set RMC. In

this way, effects from detector acceptance, efciency, 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 MCsys)2 + ( r MCstat)2, (26)

where r MCstat is the statistical uncertainty of r MC as obtained in the tting procedure that uses MINUIT [28]. From

Eq. (26), r MCsys is determined, using the convention that r MCsys is set to zero if [(r r MC)2 ( r MCstat)2] is negative.

5 Results

The results on SDMEs in the SchillingWolf [3] representation are given in Tables 1, 2, 3, 4 and 5 in Appendix B and in the Diehl [4] representation in Table 6 in the same Appendix B. The SDMEs for the entire kinematic region are discussed in Sect. 5.1, while their dependences on Q2 and

t are discussed in Sect. 5.3.

5.1 SDMEs for the entire kinematic region

The SDMEs of the meson for the entire kinematic region ( Q2 = 2.42 GeV2, W = 4.8 GeV, and t =

0.080 GeV2) are presented in Fig. 6. These SDMEs are divided into ve classes corresponding to different helicity transitions. The main terms in the expressions of class-A SDMEs correspond to the transitions from longitudinal virtual photons to longitudinal vector mesons, L VL, and

from transverse virtual photons to transverse vector mesons, T VT . The dominant terms of class B correspond to the

interference of these two transitions. The main terms of class-C, class-D, and class-E SDMEs are proportional to small amplitudes describing T VL, L VT , and T VT

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 2 per degrees of free-

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Eur. Phys. J. C (2014) 74:3110

Fig. 6 The 23 SDMEs for exclusive electroproduction extracted in the entire HERMES kinematic region with

Q2 = 2.42 GeV2, W = 4.8 GeV, t = 0.080 GeV2. 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

r0400

r1 1-1

Im r2 1-1

Re r5 10

Im r6 10

Im r7 10

Re r8 10

r0410

r1 10

r2 10

r5 00

r1 00

Im r3 10

r8 00

r5 11

r5 1-1

Im r6 1-1

Im r7 1-1

r8 11

r8 1-1

r041-1

r1 11

Im r3 1-1

0 0.1 0.2 0.3 0.4 0.5 0.6

SDME values

-0.5 -0.4 -0.3 -0.2 -0.1

dom of 28/23 1.2. In Fig. 6, the eight polarized SDMEs

are presented in shaded areas. Their experimental uncertainties are larger in comparison to those of the unpolarized SDMEs because the lepton beam polarization is smaller than unity (|Pb| 40 %) and in the equation for the angular dis

tribution they are multiplied by the small kinematic factor

|Pb|1 0.2, cf. Eq. (14) vs. Eq. (15).

5.2 Test of the SCHC hypothesis

In the case of SCHC, the seven SDMEs of class A and class B (r0400, r111, Im{r211}, Re{r510}, Im{r610}, Im{r710}, Re{r810})

are not restricted to be zero, but six of them have to obey the following relations [3]:

r111 = Im{r211},

Re{r510} = Im{r610},

Im{r710} = Re{r810}.

The proton data yield

r111 + Im{r211} = 0.004 0.038 0.015,

Re{r510} + Im{r610} = 0.024 0.013 0.004,

Im{r710} Re{r810} = 0.060 0.100 0.018, and the deuteron data yield

r111 + Im{r211} = 0.033 0.049 0.016,

Re{r510} + Im{r610} = 0.001 0.016 0.005,

Im{r710} Re{r810} = 0.104 0.110 0.023.

Here and in the following, the rst uncertainty is statistical and the second systematic. In the calculation of the statistical uncertainty, the correlations between the different SDMEs are taken into account, see correlation matrices in Tables 8 and 9. It can be concluded that the above SCHC relations are fullled for class A and B. The SCHC relations for the class-A SDMEs r111 and Im{r211} can be violated only by the

quadratic contributions of the double-helicity-ip amplitudes T1

1 12 with |V | = 2. The observed

validity of SCHC means that their possible contributions are smaller than the experimental uncertainties. Also for class-B

1 12 and U1

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Eur. Phys. J. C (2014) 74:3110 Page 9 of 25 3110

Fig. 7 Q2 and t dependences

of class-A SDMEs. Proton data are denoted by squares and deuteron data by circles. Data points for deuteron data are slightly shifted horizontally for legibility. The representation of the uncertainties follows that of Fig. 6. The proton data are compared to calculations of a phenomenological model [17], where solid (dashed) lines denote results with (without) pion-pole contributions

0.2

r04

00 0.5

1-1

1-1 0.4

0.4

proton

Im r2

0.3

0.2

-0.2

0.2

0.1

-0.4

-0.2

0 1 2 3 4 1 2 3 4 1 2 3 4

Q2 [GeV2]

r04

00 0.5

1-1

0.2

1-1 0.4

0.4

Im r2

0.3

0.2

-0.2

0.1

-0.4

-0.2

0 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3

-t' [GeV2]

SDMEs, to which the same small double-helicity-ip amplitudes contribute linearly, no SCHC violation is observed. In addition, class-B SDMEs contain the contribution of the two small products T0

2 1 12 T 1

2 0 12 (U0

2 1 12 U1

2 0 12 ). As the

SCHC hypothesis is fullled, 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-ip amplitudes can be neglected.

All SDMEs of class C to E have to be zero in the case of SCHC. The class-C SDME r500 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 r500 [20],

r500 =

Re T0

2 1 12 T 0

2 0 12 + T0

2 1 12 T 01

2 0 12

, (27)

contains two amplitude products, at least one product is nonzero. However, without an amplitude analysis of the presented data it cannot be concluded which contribution to r500 dominates. Both amplitudes T0

2 1 12 and T0

12 1 12 have to be zero if the SCHC hypothesis holds.

Figure 6 shows that out of the six SDMEs of class D three,i.e., r511, r511, and Im{r611}, slightly differ from zero (see

Table 1). As will be discussed in Sects. 5.4 and 5.8, the largest UPE amplitudes in production are U11 and U10, and |U11| |U10|. The main term of the rst two SDMEs is

proportional to Re[U10U11], while Im{r611} is proportional

to Re[U10U11]. The calculated linear combination of these

three SDMEs, r511+r511Im{r611}, 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 uncertainty 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 production.

5.3 Dependences of SDMEs on Q2 and t and

comparison to a phenomenological model

In the following sections, kinematic dependences of the measured SDMEs and certain combinations 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/Q2 corrections, model calculations are only shown for Q2 > 2 GeV2, which leaves for the Q2 dependence only two data points that can be compared to the model calculation. This paucity of comparable points makes it sometimes difcult to draw useful conclusions about the data-model comparison.

The kinematic dependences of SDMEs on Q2 and t

are presented in three bins of Q2 with Q2 = 1.28 GeV2, Q2 = 2.00 GeV2, Q2 = 4.00 GeV2, and t with t =

0.021 GeV2, t = 0.072 GeV2, t = 0.137 GeV2.

Table 7 shows the average value of Q2 and t for bins in t and Q2, respectively.

The Q2 and t dependences of class-A SDMEs are

shown and compared to the model calculations in Fig. 7. All three SDMEs clearly show the need for the unnatural-parity contribution of the pion pole and the measured t

dependence is well reproduced both in shape and magnitude. The same holds for the two unpolarized class-B SDMEs that are shown in Fig. 8. For the polarized class-B SDMEs as well as for all class-C SDMEs, which are shown in Fig. 9, the pion-pole contribution has little or no effect, and the model describes the magnitude of the data reasonably well. The class-D and E SDMEs are shown in Figs. 10 and 11, respectively. These SDMEs are expected to be zero if the pion-pole contribution is not included. When comparing the

t dependences of the three unpolarized class-D SDMEs to

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3110 Page 10 of 25 Eur. Phys. J. C (2014) 74:3110

0.2

Im r6 10

0.05

0.6

10 0.4

Re r5 10

0.15

Im r7 10

Re r8

proton teron

0.4

0.2

0.1

-0.05

0.2

0.05

-0.1

-0.15

-0.2

-0.4

2 4 2 4 2 4

Q2 [GeV2]

2 4

Q2 [GeV2]

0.2

0.05

Re r5 10

0.6

10 0.5

0.15

Im r6 10

Im r7 10

Re r8

0.4

0.25

0.1

-0.05

0.2

0.05

-0.1

-0.25

-0.15

-0.2

-0.5

-0.4

0 0.2 0 0.2 0 0.2 0 0.2

-t' [GeV2]

Fig. 8 Q2 and t dependences of class-B SDMEs. Otherwise as for Fig. 7

the model calculation, also here the unnatural-parity pion-exchange contribution seems to be required. The two unpolarized class-E SDMEs are measured with reasonable precision, and agreement with the model calculation can be seen.

Within experimental uncertainties, the SDMEs measured on the proton are seen to be very similar to those measured on the deuteron. This can be understood by considering the different contributions to exclusive omega production. The pion-pole contribution is seen to be substantial [17]. For the NPE amplitudes, the dominant contribution comes from gluons and sea quarks, which are the same for protons and neutrons, while the valence-quark contribution is different. Thus altogether, only small differences between the proton and deuteron SDMEs are expected for incoherent scattering. As coherence effects are difcult to estimate, one can not exclude that they are of the size of the valence-quark effects. Therefore, the deuteron SDMEs are presently difcult to calculate reliably.

5.4 UPE in -meson production

In Fig. 12, the comparison of and 0 [20] SDMEs is shown. One can see that the SDMEs r111 and Im{r211} of class A

have opposite sign for and 0. The SDME r111 is negative

for the meson and positive for 0, while Im{r211} is posi

tive for and negative for 0. In terms of helicity amplitudes, these two SDMEs are written [20] as

r111 =

(|T11|2 + |T11|2 |U11|2 |U11|2),

(28)

The difference between Eqs. (29) and (28) reads

Im{r211} r111 =

1 (|T11|2 + |U11|2). (30)

For the entire kinematic region, this difference is clearly positive for the meson, hence

|U11|2 >

|T11|2, while for

|U11|2 [20]. This suggests a large

UPE contribution in exclusive -meson production. Applying Eq. (11) to relation (30), the latter can be rewritten as

Im{r211} r111 =

the 0 meson

|T11|2 >

1 (|T1

2 1 12 |2 |T1

2 1 12 |2 + |U1

2 1 12 |2

+ |U1

2 1 12 |2). (31)

The amplitudes with nucleon helicity ip, T1

2 1 12 and

2 1 12 , should be zero at t = 0 and are proportional to

t at small t (see Eq. (9) and Ref. [4]). The small con

tribution of |T1

2 1 12 |2 will be neglected from now on. As it

was established above, the UPE contribution is larger than the NPE one. This means that if the dominant UPE helicityip amplitude is U1

12 1 12 , expression (31) would increase proportionally to t . However, the experimental values of

(Im{r211} r111) (see Tables 3 and 5) do not demonstrate

such an increase; the values for the proton data even decrease smoothly with t . Hence the dominant UPE amplitude is

U11

2 1 12 , and it holds |U11|2 > |T11|2.

The existence of UPE in production on the proton and deuteron can also be tested with linear combinations of SDMEs such as

u1 = 1 r0400 + 2r0411 2r111 2r111, (32)

u2 = r511 + r511, (33)

u3 = r811 + r811. (34)

Im{r211} =

(|T11|2 + |T11|2

+|U11|2 |U11|2). (29)

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Eur. Phys. J. C (2014) 74:3110 Page 11 of 25 3110

Re r04

Re r1 10

Im r2 10

0.1

r5 00

0.15

0.1

proton deuteron

0.05

-0.05

-0.1

-0.05

-0.1

2 4 2 4

Q2 [GeV2]

r1 00

Im r3 10

0.3

0.2

00 0.6

0.2

0.4

0.1

0.2

0.1

-0.2

-0.1

-0.4

2 4 2

4 2 4

Q2 [GeV2]

0.1

0.2

0.15

Re r04

Re r1 10

Im r2 10

0.1

0.05

0.1

0.05

-0.05

-0.1

0 0.2 0 0.2

-t' [GeV2]

r1 00

00 0.5

0.2

Im r3 10

0.3

0.25

0.1

0.2

0.1

-0.25

-0.1

-0.5

-0.2

-0.1

-0.75

0 0.2 0

0.2 0 0.2

-t' [GeV2]

Fig. 9 Q2 and t dependences of class-C SDMEs. Otherwise as for Fig. 7

The quantity u1 can be expressed in terms of helicity amplitudes as

u1 =

1 (4 |U10|2 + 2|U11 + U11|2). (35)

A non-zero result for u1, implying that at least one of the four amplitudes U1

2 0 12 or (U1

(U11 + U11)U10, (36)

showing that these quantities are nonzero if at least one of

the products U11

2 0 12 (U1

2 1 12 +U1

12 1 12 ) is nonzero, indicates the existence of UPE contributions. In the entire kinematic region, u1 is 1.15 0.09 0.12 and 1.47 0.12 0.18 for proton and deuteron data, respectively. In Fig. 13,

the Q2 and t dependences of u1 for proton and deuteron

data are presented. It can be seen that u1 is larger than unity, which implies the existence of large contributions from UPE transitions.

The expression for the quantities u2 and u3 in terms of helicity amplitudes is

u2 + iu3 =

2 1 12 ) or U1

2 0 12 (U1

2 1 12 +

12 1 12 ) is nonzero. Therefore u2 and u3 provide information complementary to that given by u1. In Fig. 13, also the quantities u2 and u3 versus Q2 and t are presented both for

proton and deuteron data. As seen from this gure, there are no clear dependences on Q2 and t , but u2 for the proton

data is denitely nonzero and there is some evidence that it

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3110 Page 12 of 25 Eur. Phys. J. C (2014) 74:3110

0.05

0.1

Im r6

r5 11

1-1

0.05

0.1

0.05

teron

-0.05

-0.1

-0.05

1 2 1 2

4 4

1 2 3

Q2 [GeV2]

0.5

1-1

Im r7

11 0.4

0.25

0.5

0.2

-0.25

-0.2

-0.5

-0.4

-0.6

-0.75

3 4

1 2 3 4

Q2 [GeV2]

0.05

r5 11

1-1

1-1 0.15

0.05

Im r6

0.1

-0.05

0.05

-0.1

-0.05

-0.15

-0.1

-0.05

0 0.1 0.2 0.3 0

0.1 0.2 0.3 0 0.1 0.2 0.3

-t' [GeV2]

1-1

Im r7

0.2

1-1 0.2

0.5

-0.2

-0.5

-0.4

-0.6

-1 0 0.1 0.2 0.3 0

-t' [GeV2]

0.1 0.2 0.3 0 0.1 0.2 0.3

-t' [GeV2]

Fig. 10 Q2 and t dependences of class-D SDMEs. Otherwise as for Fig. 7

is also nonzero for the deuteron data. Note that u2 and u3 are compatible with zero in 0-meson electroproduction [20].

Figure 13 also demonstrates good agreement between proton data and the model calculation. It appears that including the pion-pole into the model fully accounts for the unnatural-parity contribution measured through u1 and u2, both in t

shape and magnitude. Conclusions on u3 are prevented by

the considerable experimental uncertainties.

5.5 Phase difference between amplitudes

Taking the amplitude without helicity ip,U112 1 12 , as the dominant UPE one, Eq. (36) can be simplied as

u2 + iu3 =

U11

2 1 12 U11

2 0 12

U11U10. (37)

The expressions for the phase difference U between the UPE amplitudes U11 and U10 follow immediately from Eq. (37):

cos U = u2/ (u2)2 + (u3)2, (38) sin U = u3/ (u2)2 + (u3)2, (39)

tan U = u3/u2 =

r811 + r811 r511 + r511

. (40)

The phase differences obtained for the entire kinematic region are U = (126 12 2) degrees for proton and

U = (100 61 3) degrees for deuteron data.

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Im r3

r04

1-1 0.2

r1 11

0.1

proton

1-1 0.2

0.15

0.1

0.05

-0.2

-0.1

-0.4

-0.05

-0.2

1 2 3 4 1 2 3 4 1 2 3 4

Q2 [GeV2]

r04

r1 11

0.1

1-1 0.2

Im r3

1-1 0.4

0.2

0.1

-0.1

-0.2

-0.4

-0.1

0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3

-t' [GeV2]

Fig. 11 Q2 and t dependences of class-E SDMEs. Otherwise as for Fig. 7

Fig. 12 Comparison of SDMEs in exclusive and 0 [20]

electroproduction at HERMES for the entire kinematic region.The average values of the kinematic variables in exclusive 0 production are

Q2 = 1.95 GeV2, W = 4.8 GeV, and t = 0.13 GeV2

Im r3 1-1

r0400

A * VM

r1 1-1

Im r2 1-1

Re r5 10

Im r6 10

Im r7 10

Re r8 10

Re r0410

Re r1 10

Im r2 10

r5 00

r1 00

Im r3 10

r8 00

r5 11

r5 1-1

Im r6 1-1

Im r7 1-1

r8 11

r8 1-1

r041-1

r1 11

-0.5 -0.4 -0.3 -0.2 -0.1

0 0.1 0.2 0.3 0.4 0.5 0.6

SDME values

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3110 Page 14 of 25 Eur. Phys. J. C (2014) 74:3110

u 1

2.5

u 2

0.3

u 3

0.5

0.2

( ) proton (total)

0.25

1.5

0.1

-0.25

-0.5

0.5

-0.1

-0.75

-0.2

1 2 3 4 1 2 3 4 1 2 3 4

Q2 [GeV2]

u 1

2.5

u 2

0.5

0.1

u 3

0.25

1.5

-0.25

-0.1

-0.5

0.5

-0.75

-0.2

0 0.1 0.2 0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3

-t' [GeV2]

Fig. 13 The Q2 and t dependences of u1, u2, and u3. The open symbols represent the values over the entire kinematic region. Otherwise as for

Fig. 7

The phase difference N between the NPE amplitudes T11 and T00 can be calculated as follows [20]:

cos N =

2 (Re{r

5 10

} Im{r610}) r0400(1 r0400 + r111 Im{r211}). (41)

The phase differences obtained for the entire kinematic region are |N | = (51 5 14) degrees and |N | =

(50 7 16) degrees for proton and deuteron data, respec

tively. Using polarized SDMEs, in principle also the sign of N can be determined from the following equation:

sin N =

2 (Re{r

8 10

} + Im{r710}) r0400(1 r0400 + r111 Im{r211}), (42)

which is given in Ref. [20]. For the present data, the large experimental uncertainties of the polarized SDMEs make it impossible to determine the sign of N .

5.6 Longitudinal-to-transverse cross-section ratio

Usually, the longitudinal-to-transverse virtual-photon differential cross-section ratio

R =

dL( L V )

dT ( T V )

is experimentally determined from the measured SDME r0400 using the approximated equation [20]

r0400

1 r0400

. (43)

This relation is exact in the case of SCHC. The Q2 dependence of R for the meson is shown in the left panel of

Fig. 14, where also for comparison the same dependence for the 0 meson [20] is shown. For 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 0 production, this ratio is about four times smaller, and for the meson this ratio is almost independent of Q2. The t dependence

of R is shown in the right panel of Fig. 14. The comparison of the proton data to the GK model calculations with and without inclusion of the pion-pole contribution demonstrates the clear need to include the pion pole. The data are well described by the model and appear to follow the t depen

dence suggested by the model when the pion-pole contribution is included. This implies that transverse and longitudinal virtual-photon cross sections have different t dependences.

Hence the usual high-energy assumption that their ratio can be identied with the corresponding ratio of the integrated cross sections does not hold in exclusive 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.

5.7 The UPE-to-NPE asymmetry of the transverse cross section

The UPE-to-NPE asymmetry of the transverse differential cross section is dened as [29]

P =

d NT dUTd NT + dUT

d NT /dUT 1 d NT /dUT + 1

= (1 + R)(2r111 r100), (44)

where NT and UT denote the part of the cross section due to NPE and UPE, respectively. Substituting Eq. (43) in Eq. (44)

leads to the approximate relation

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Eur. Phys. J. C (2014) 74:3110 Page 15 of 25 3110

1.25

( ) 0, proton (total)( ) 0, deuteron (total)

( ) , proton (total)( ) , deuteron (total)

0.75

0.5

0.25

0 1 2 3 4 0 0.1 0.2 0.3

-t' [GeV2]

Q2 [GeV2]

Fig. 14 The Q2 (left) and t (right) dependences of the longitudinal

to-transverse virtual-photon differential cross-section ratio for exclusive and 0 electroproduction at HERMES, where the t bin covers the

interval [0.00.2] GeV2 for production and [0.00.4] GeV2 for 0

production [20]. The symbols that are parenthesized in the legend represent the value of R in the entire kinematic region. Otherwise as for Fig. 7

0.5

2r111 r100 1 r0400

. (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 Q2 and

t dependences of the UPE-to-NPE asymmetry of the trans-verse differential cross section for exclusive production are presented. Again, the GK model calculation appears to fully account for the unnatural-parity contribution and shows very good agreement with the data both in shape and magnitude.

5.8 Hierarchy of amplitudes

In order to develop a hierarchy of amplitudes, in the following a number of relations between individual helicity amplitudes is considered. The resulting hierarchy is given in Eqs. (62) and (64) below.

5.8.1 U10 versus U11

From Eqs. (35) and (37), the relation

2(u22 + u23)u1 |

U11U10|

|U11|2 + 2 |U10|2

= |

( ) proton (total)( ) deuteron (total)

-0.5

-1

3 3.5 4 4.5

Q2 [GeV2]

0.5

1 1.5 2 2.5

-0.5

-1

-0.05 0 0.05 0.1

0.15 0.2 0.25 0.3

-t' [GeV2]

Fig. 15 The Q2 and t dependences of the UPE-to-NPE asymmetry

P of the transverse differential cross section for exclusive electroproduction at HERMES. The open symbols represent the values over the entire kinematic region. Otherwise as for Fig. 7

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 |U10/U11| is due to the large mea

sured value of u3. However, as this value is compatible with zero within about one standard deviation of the total uncer-

U10/U11|1 + 2 |U10/U11|2

(46)

is obtained. Using the measured values of those SDMEs that determine u1, u2, and u3, the following amplitude ratio is estimated:

|U10|

|U11|

2(u22 + u23)u1 0.2. (47)

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3110 Page 16 of 25 Eur. Phys. J. C (2014) 74:3110

tainty, the contribution of u3 in Eq. (47) can be neglected, which leads to the value of 0.06 as lower bound on |U10/U11|.

5.8.2 T11 versus U11

With the above considerations, it follows from Eq. (35) that the contribution of |U10/U11|2 is only a few percent and

hence will be neglected everywhere. Then, in particular, the relation

u1 2|U11|2/N (48)

is valid with a precision of a few percent.

Equations (79) show that the nucleon-helicity-ip amplitudes T1

12 1 12 (T0

2 0 12 ) are suppressed by a factor of about

t /M compared to the amplitude T11 (T00) with diag

onal transitions ( N = N ). Therefore, the second-order

contributions of the amplitudes T

2 12 for any will be

neglected compared to any bilinear product of T00 and T11. In this approximation, the relation

2[Im{r211} r211]

u1 = 1

T11

U11

From Eqs. (79), it follows that the terms

T01T 10 and

U01U10 on the right-hand side of Eqs. (52, 53) are suppressed by a factor (t )/M2 compared to T11T 00 and will

be neglected. The simplest consequence of Eqs. (52, 53) is the relation

[Re{r510} Im{r610}]2 + [Im{r710} + Re{r810}]2

2N 2 |

T11|2|T00|2. (54)

Dividing this relation by u21/8 and using Eq. (48), one gets the formula of interest:

[Re{r510} Im{r610}]2 + [Im{r710} + Re{r810}]2

u21/8

T11|2|T00|2

|U11|4

. (55)

Using numerical SDME values from Table 1 and |T11/U11| =

0.6, the estimate |T00/U11| 0.5 is obtained, which is in

agreement with the previous estimate. However, as the polarized SDMEs Im{r710} and Re{r810} have very large uncertain

ties, the latter result is less reliable than the former. Omitting the contribution of the polarized SDMEs in Eq. (55) leads to the inequality

8[Re{r510} Im{r610}]2

u21

< |T11|2|T00|2

|U11|4

(49)

follows from Eqs. (31) and (48). Substituting numerical values for the SDMEs in Eq. (49) leads to the estimate

|T11/U11| 0.6.

5.8.3 T00 versus U11

Using Eq. (48) and the expression for r0400 from Refs. [3,20] yields

2r0400u1 =

[ |T00|2 + |T01|2 + |U01|2] |U11|2

, (56)

which provides the lower limit of 0.3 for the same ratio

|T00/U11|. This result combined with the former estimate

leads to the boundaries 0.3 < |T00/U11| < 0.6.

5.8.4 T00 versus T01

In order to estimate the value of |T01|, the quantity

(r500)2 + (r800)2 r0400 =

. (50)

Neglecting in the numerator of the right-hand side of Eq. (50) all positive terms except |T00|2, the inequality of interest is

obtained:

2r0400u1 >

|T00|2

|U11|2

. (51)

Using for the estimate = 0.8 and values of SDMEs from

Table 1 yields the result |T00/U11| < 0.6.

The same ratio can be estimated from other SDMEs. Using expressions for the SDMEs from [3,20], the following equations can be written:

Re{r510} Im{r610}

T01T 00|

[ |T00|2 + |T01|2 + |U01|2]

(57)

can be formed. Neglecting in the denominator of the right-hand side of Eq. (57) all the terms except |T00|2, the inequal

ity

(r500)2 + (r800)2 r0400

< 2|

T01T 00| |T00|2

(58)

is obtained. The sum in the numerator of the right-hand side of Eq. (58) is

T01T 00 = T012 1 12 T 01

2 0 12 + T0

2 1 12 T 0

1 Re[T11T 00 + T01T 10 U01U10], (52)

Im{r710} + Re{r810}

2 0 12 (59)

1 Im[T11T 00 + T10T 01 U10U01]. (53)

according to Eq. (11). If the rst product on the right-hand side of Eq. (59) dominates, then inequality (58) becomes

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Eur. Phys. J. C (2014) 74:3110 Page 17 of 25 3110

simpler:

(r500)2 + (r800)2 r0400

< 2

T01|

|T00|

is assumed to be dominant, Eq. (60) has to be replaced by

(r500)2 + (r800)2r0400

|T012 1 12 T 012 0 12 |

|T00|2

. (60)

Numerically, this yields the estimate |T01/T00| 0.3. The

dominant contribution to this number comes from the polarized SDME r800 that is compatible with zero within about one standard deviation of the total uncertainty. Retaining only the contribution of the unpolarized SDME r500 in Eq. (60)

gives the following result: |T01/T00| > 0.1. The experimen

tal accuracy of the presented data is not sufcient to provide a reliable estimate for the upper bound to the ratio |T01/T00|.

As shown in Appendix A, the upper limits for

(|T01|2 + |U01|2)

|T00|2

|T012 1 12 |

|T00|

|T012 0 12 |

|T00|

. (63)

The nucleon-helicity-ip amplitude T0

12 0 12 is smaller than the helicity-conserving amplitude T00 T0

12 0 12 by a factor of about t /M 0.3 (see Eq. 9). Substituting this factor

for |T0

2 0 12 /T00|, using = 0.8 and the measured SDME

values, the nal estimate |T0

2 1 12 | |T00| is obtained. This

result shows that the nucleon-helicity-ip amplitude T0

2 1 12

(61)

could be of the same order of magnitude as T00, while the values of T01 and U01 could be as given in the previous estimates.

As the SDME r100, which is proportional to

[|U01|2

|T01|2], was measured to be compatible with zero, the value

of |U0

2 1 12 | should be about the same as that of |T0

are 1.3 0.7 for the proton and 1.1 1.2 for the deuteron.

In the below consideration the estimate based on Eq. (60), namely |T01/T00| 0.3, is assumed to be realistic.

The numerator in the denition of r100 is

2 1 12 |.

Then, the values of |T0

2 1 12 |, |U0

2 1 12 |, and |T00| are of the

same order of magnitude, so that the hierarchy of amplitudes becomes

|U11|2 > |T00|2 |T11|2 |T012 1 12 |2 |U012 1 12 |2

|U10|2 |T01|2 |U01|2 |T10|2, |T11|2, |U11|2, (64)

where again negligibly small amplitudes are neglected. Note that the usually used Eq. (43) for R is not applicable in this case. The estimation performed in Appendix A shows that the accuracy of the presented data is not sufcient to decide between hierarchies (62) and (64). The best way to get information on the amplitudes T0

2 1 12 and U0

[|U01|2

|T01|2]. The values of r100 are compatible with zero within

two standard deviations of the total experimental uncertainty, hence |U01| cannot be much larger than |T01|.

Considering the SDME combinations (r511 r511) and

(Im{r811}r811), which are proportional to the real and imag

inary parts of

T10(T11 T11), respectively, it is possible

in principle to estimate the value of |T10|. Since these com

binations are compatible with zero within one standard deviation of the total uncertainty, it can be concluded that |T10|

is negligibly small compared to the large amplitude moduli

|U11|, |T11|, and |T00|.

5.8.5 Resulting hierarchy of amplitudes

As a result, the following hierarchy is obtained:

|U11|2 > |T00|2 |T11|2

|U10|2 |T01|2 |U01|2

|T10|2, |T11|2, |U11|2, (62)

where negligibly small amplitudes are neglected.

However, there exists a possible alternative for the hierarchy presented on the second line of Eq. (62), if the helicityip amplitudes T0

2 1 12 and U0

12 1 12 are of the same order of magnitude as the helicity-conserving amplitudes T00 and T11.

2 1 12 is to

study electroproduction of mesons on transversely polarized protons, where these amplitudes contribute linearly to the angular distribution.

6 Summary

Exclusive electroproduction is studied at HERMES using a longitudinally polarized lepton beam and unpolarized hydrogen and deuterium targets in the kinematic region Q2 >

1.0 GeV2, 3.0 GeV < W < 6.3 GeV, and t < 0.2 GeV2.

The average kinematic values are Q2 = 2.42 GeV2, W = 4.8 GeV, and t = 0.080 GeV2. Using an

unbinned maximum likelihood method, 15 unpolarized and, for the rst time, 8 polarized spin density matrix elements are extracted. The kinematic dependences of all 23 SDMEs are presented for proton and deuteron data. No signicant differences between proton and deuteron results are seen.

Indeed, the sum

T01T 00 in Eq. (58) is the sum of two products, T01

2 1 12 T 01

2 0 12 and T0

2 1 12 T 0

2 0 12 , according to Eq. (59).

In order to obtain Eq. (60) from Eq. (58), the dominance of the rst product was assumed. If instead the second product

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3110 Page 18 of 25 Eur. Phys. J. C (2014) 74:3110

The SDMEs are presented in ve classes corresponding to different helicity transitions between the virtual photon and the meson. While the values of class-A and B SDMEs agree with the hypothesis of s-channel helicity conservation, the class-C SDME r500 indicates a violation of this hypothesis.

The values of those class-D SDMEs that correspond to the transition L T also indicate a small violation of the

hypothesis of s-channel helicity conservation.Using the SDMEs r111 and Im{r211}, it is shown that

for exclusive -meson production the amplitude of the UPE transition T T is larger than the NPE amplitude for

the same transition, i.e., |U11|2 > |T11|2. The importance of

UPE transitions is also shown by a combination of SDMEs denoted u1. This suggests that at HERMES energies in exclusive electroproduction the quark-exchange mechanism, or 0, a1... exchanges in Regge phenomenology, plays a signicant role.

The phase shift between those UPE amplitudes that describe transverse production by transverse and longitudinal virtual photons, U11 for T T and U10 for L T ,

respectively, as well as the magnitude of the phase difference between the NPE amplitudes T11 and T00 is determined for the rst time.

The ratio R between the differential longitudinal and transverse virtual-photon cross-sections is determined to be R = 0.250.030.07 for the meson, which is about four

times smaller than in the case of the 0 meson. In contrast to the case of the 0 meson, R shows only a weak dependence on Q2 for the meson.

The UPE-to-NPE asymmetry 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 T T transition domi

nates over the two NPE amplitudes describing the L L

and T 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 production at HERMES energies the common high-energy assumption of identifying R with the ratio of the integrated longitudinal and transverse cross sections.

Acknowledgments We are grateful to Sergey Goloskokov and Peter Kroll for fruitful discussions on the comparison between our data and

their model calculations. We gratefully acknowledge the DESY management for its support and the staff at DESY and the collaborating institutions for their signicant effort. This work was supported by the Ministry of Education and Science of Armenia; the FWO-Flanders and IWT, Belgium; the Natural Sciences and Engineering Research Council of Canada; the National Natural Science Foundation of China; the Alexander von Humboldt Stiftung, the German Bundesministerium fr Bildung und Forschung (BMBF), and the Deutsche Forschungsgemeinschaft (DFG); the Italian Istituto Nazionale di Fisica Nucleare (INFN); the MEXT, JSPS, and G-COE 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 the UPV/EHU under program UFI 11/55; 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).

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Funded by SCOAP3 / License Version CC BY 4.0.

Appendix A: Estimate of T0

2 1 12 and U0

2 1 12 values

The normalization factor N is given by (see, e.g., [3,20])

N = NT + NL, (65)

with

NT=

(|T11|2 + |T01|2 + |T11|2 + |U11|2

+ |U01|2 + |U11|2), (66)

NL=

(|T00|2 + 2|T10|2 + 2|U10|2). (67)

Using Eqs. (6567) and the expression dening r0400 [3,20],

r0400 =

1 ( |T00|2 + |T01|2 + |U01|2), (68)

the exact relation

1 r0400 =

1 [|T11|2 + |U11|2 + |T11|2 + |U11|2

+ 2 (|T10|2 + |U10|2)] (69)

is obtained. Neglecting, as usual,

[|T11|2 + |U11|2 + |T10|2 + |U10|2] in this expression, we get the approximate

relation

1 r0400

1 (|T11|2 + |U11|2). (70) Neglecting also the small nucleon-helicity-ip amplitudes

2 1 12 and U1

2 1 12 in Eq. (30) and then subtracting it from

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Eq. (70), the relation

1 r0400 + r111 Im{r211}

2 T11|2 (71)

is obtained. After neglecting in Eq. (68) only the nucleonhelicity-ip amplitude T0

2 0 12 , it can be rewritten as

r0400

and deuteron data in exclusive 0-meson production [20], one obtains A 0.22 0.09 and A 0.28 0.09, respec

tively. This shows that in this case the probability for the amplitudes T0

2 1 12 and U0

12 1 12 to be of the same order of magnitude as T00 is small.

1 |T00|2 +

Appendix B: SDMEs for proton and deuteron

See Tables 1, 2, 3, 4, 5, 6, 7, 8 and 9.

(|T01|2 + |U01|2)]. (72)

Multiplying this equation by Eq. (71) and dividing it by Eq. (54) with a factor of four, the equation of interest reads

+ A

Table 1 The 23 unpolarized and polarized SDMEs from the proton and deuteron data. The rst uncertainty is statistical, the second systematic

Element Proton Deuteron

r0400 0.168 0.018 0.036 0.160 0.024 0.038 r111 0.175 0.029 0.039 0.215 0.036 0.047

Im r211 0.171 0.029 0.023 0.248 0.037 0.039

Re r510 0.037 0.009 0.012 0.045 0.010 0.014 Im r610 0.061 0.008 0.012 0.043 0.010 0.009

Im r710 0.109 0.075 0.021 0.021 0.087 0.004 Re r810 0.169 0.075 0.035 0.083 0.083 0.017

Re r0410 0.010 0.012 0.002 0.020 0.014 0.005 Re r110 0.014 0.019 0.005 0.016 0.022 0.009

Im r210 0.039 0.018 0.007 0.003 0.023 0.002 r500 0.042 0.015 0.012 0.036 0.019 0.014 r100 0.006 0.029 0.008 0.107 0.036 0.023

Im r310 0.059 0.047 0.012 0.038 0.056 0.008 r800 0.142 0.110 0.029 0.017 0.131 0.004 r511 0.059 0.012 0.022 0.025 0.015 0.015 r511 0.043 0.014 0.006 0.021 0.018 0.001

Im r611 0.036 0.014 0.008 0.056 0.019 0.013

Im r711 0.092 0.117 0.018 0.113 0.135 0.028

r811 0.079 0.089 0.017 0.097 0.103 0.020 Im r811 0.060 0.110 0.012 0.150 0.125 0.034

r0411 0.004 0.018 0.004 0.060 0.023 0.016

r111 0.014 0.024 0.004 0.037 0.030 0.007 r311 0.023 0.076 0.010 0.122 0.089 0.025

r0400(1 r0400 + r111 Im{r211})/4

[Re{r510} Im{r610}]2 + [Im{r710} + Re{r810}]2

, (73)

where the quantity A is dened in Eq. (61). The value of A is

close to zero, if |T0

2 1 12 |2 and |U0

12 1 12 |2 are much smaller than |T00|2, and it should be of the order of one if they are

comparable to |T00|2. Since the uncertainties of the polarized

SDMEs Im{r710} and Re{r810} are large, the use of Eq. (73)

for the present data is not very successful. Indeed, using for numerical calculations = 0.8 and the values for the SDMEs

in Eq. (73) from Table 1 we get A = 0.56 0.20 and A = 0.50 1.8 for the proton and deuteron data, respec

tively. In contrast, in 0-meson production, the corresponding values of A [20], 0.031 0.084 and 0.064 0.068,

exclude practically the possibility that the amplitudes T0

12 1 12

and U0

2 1 12 are comparable to the dominant amplitudes U11,

T00 and T11.

If the contribution of [Im{r710}+Re{r810}] in the denomina

tor of the right-hand side of Eq. (73) is neglected, the useful inequality

r0400(1 r0400 + r111 Im{r211})

4[Re{r510} Im{r610}]2

(74)

can be obtained. The numerical estimates A 1.3 0.7

and A 1.1 1.2 for the proton and deuteron data, respec

tively, show that the possibility for the values of |T0

2 1 12 |2

and |U0

2 1 12 |2 to be of the same order of magnitude as |T00|2

is not excluded by the presented results on SDMEs. For comparison, when applying Eq. (74) to the results on proton

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3110 Page 20 of 25 Eur. Phys. J. C (2014) 74:3110

Table 2 The 23 unpolarized and polarized SDMEs for the proton data in Q2 intervals: 1.001.572.5510.00 GeV2.

The rst uncertainty is statistical, the second systematic

Element Q2 = 1.28 GeV2 Q2 = 2.00 GeV2 Q2 = 4.00 GeV2

r0400 0.164 0.034 0.022 0.166 0.030 0.044 0.179 0.031 0.036 r111 0.032 0.050 0.032 0.175 0.049 0.037 0.314 0.053 0.090

Im r211 0.172 0.048 0.027 0.133 0.050 0.043 0.163 0.057 0.029 Re r510 0.038 0.016 0.018 0.022 0.015 0.010 0.053 0.015 0.022

Im r610 0.062 0.015 0.012 0.069 0.012 0.014 0.046 0.014 0.013 Im r710 0.163 0.139 0.030 0.006 0.125 0.009 0.170 0.128 0.042

Re r810 0.088 0.143 0.021 0.078 0.137 0.028 0.280 0.119 0.067 Re r0410 0.005 0.021 0.004 0.060 0.020 0.011 0.016 0.019 0.022

Re r110 0.005 0.032 0.013 0.090 0.031 0.012 0.073 0.034 0.016 Im r210 0.012 0.030 0.012 0.042 0.030 0.003 0.036 0.034 0.016 r500 0.031 0.029 0.001 0.029 0.025 0.012 0.068 0.027 0.016 r100 0.009 0.049 0.011 0.039 0.049 0.013 0.032 0.053 0.015

Im r310 0.044 0.096 0.008 0.047 0.076 0.009 0.073 0.076 0.018 r800 0.147 0.210 0.039 0.035 0.196 0.026 0.197 0.171 0.045 r511 0.074 0.020 0.021 0.050 0.020 0.012 0.070 0.021 0.029 r511 0.047 0.024 0.007 0.078 0.025 0.021 0.008 0.025 0.009

Im r611 0.070 0.025 0.013 0.015 0.024 0.017 0.043 0.026 0.026 Im r711 0.326 0.223 0.058 0.161 0.198 0.030 0.046 0.204 0.023

r811 0.276 0.171 0.049 0.120 0.155 0.021 0.312 0.144 0.080 Im r811 0.507 0.212 0.093 0.026 0.188 0.005 0.185 0.178 0.063

r0411 0.004 0.032 0.000 0.023 0.031 0.003 0.008 0.031 0.014

r111 0.063 0.040 0.015 0.037 0.041 0.012 0.003 0.044 0.012 r311 0.074 0.153 0.013 0.110 0.131 0.021 0.088 0.124 0.024

Table 3 The 23 unpolarized and polarized SDMEs for the proton data in t intervals:

0.000 0.044 0.105

0.200 GeV2. The rst uncertainty is statistical, the second systematic

Element t = 0.021 GeV2 t = 0.072 GeV2 t = 0.147 GeV2

r0400 0.136 0.027 0.036 0.197 0.032 0.027 0.212 0.036 0.032 r111 0.239 0.043 0.023 0.141 0.048 0.043 0.120 0.060 0.048

Im r211 0.220 0.045 0.033 0.138 0.050 0.015 0.111 0.057 0.012 Re r510 0.015 0.013 0.008 0.032 0.015 0.010 0.081 0.018 0.025

Im r610 0.051 0.012 0.012 0.077 0.013 0.013 0.058 0.015 0.018 Im r710 0.143 0.121 0.037 0.340 0.123 0.071 0.277 0.146 0.073

Re r810 0.151 0.125 0.039 0.232 0.127 0.044 0.151 0.136 0.039 Re r0410 0.022 0.018 0.004 0.010 0.020 0.006 0.006 0.023 0.002

Re r110 0.020 0.030 0.007 0.013 0.032 0.001 0.029 0.035 0.011 Im r210 0.017 0.029 0.008 0.003 0.029 0.005 0.125 0.033 0.023 r500 0.016 0.023 0.029 0.059 0.027 0.011 0.100 0.031 0.012 r100 0.032 0.047 0.033 0.067 0.050 0.024 0.106 0.053 0.067

Im r310 0.063 0.073 0.010 0.076 0.082 0.018 0.121 0.090 0.036 r800 0.155 0.179 0.033 0.138 0.197 0.026 0.442 0.191 0.115 r511 0.059 0.018 0.012 0.051 0.020 0.015 0.068 0.024 0.048 r511 0.034 0.022 0.002 0.060 0.024 0.007 0.052 0.030 0.011

Im r611 0.010 0.022 0.000 0.090 0.024 0.020 0.020 0.028 0.009

Im r711 0.027 0.176 0.004 0.244 0.197 0.046 0.601 0.233 0.165

r811 0.136 0.145 0.023 0.155 0.150 0.029 0.038 0.169 0.010 Im r811 0.182 0.181 0.046 0.085 0.180 0.017 0.055 0.210 0.025

r0411 0.006 0.029 0.003 0.007 0.030 0.006 0.023 0.036 0.008

r111 0.009 0.037 0.005 0.023 0.040 0.006 0.033 0.047 0.029 r311 0.016 0.111 0.006 0.160 0.134 0.036 0.154 0.156 0.054

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Table 4 The 23 unpolarized and polarized SDMEs for the deuteron data in Q2 intervals: 1.001.572.5510.00 GeV2.

The rst uncertainty is statistical, the second systematic

Element Q2 = 1.28 GeV2 Q2 = 2.00 GeV2 Q2 = 4.00 GeV2

r0400 0.148 0.043 0.025 0.132 0.041 0.053 0.186 0.040 0.034 r111 0.045 0.063 0.030 0.347 0.058 0.075 0.258 0.072 0.070

Im r211 0.232 0.063 0.045 0.216 0.065 0.063 0.313 0.073 0.056 Re r510 0.059 0.020 0.021 0.056 0.017 0.015 0.025 0.020 0.014

Im r610 0.034 0.018 0.006 0.039 0.016 0.009 0.055 0.021 0.015 Im r710 0.174 0.160 0.032 0.225 0.150 0.044 0.068 0.156 0.015

Re r810 0.026 0.154 0.005 0.197 0.148 0.039 0.020 0.140 0.004 Re r0410 0.004 0.027 0.007 0.020 0.024 0.011 0.040 0.025 0.012

Re r110 0.039 0.037 0.019 0.052 0.037 0.015 0.025 0.046 0.008 Im r210 0.014 0.037 0.013 0.003 0.036 0.012 0.028 0.049 0.004 r500 0.074 0.033 0.007 0.050 0.032 0.012 0.006 0.035 0.031 r100 0.079 0.061 0.028 0.077 0.059 0.012 0.143 0.073 0.048

Im r310 0.124 0.107 0.031 0.009 0.095 0.002 0.016 0.096 0.004 r800 0.186 0.248 0.041 0.024 0.242 0.005 0.088 0.211 0.019 r511 0.027 0.026 0.013 0.054 0.025 0.018 0.001 0.030 0.011 r511 0.040 0.031 0.005 0.049 0.031 0.010 0.021 0.036 0.009

Im r611 0.062 0.031 0.016 0.050 0.032 0.004 0.057 0.035 0.021 Im r711 0.399 0.250 0.079 0.053 0.236 0.011 0.003 0.234 0.001

r811 0.332 0.193 0.059 0.103 0.184 0.020 0.022 0.164 0.005 Im r811 0.260 0.234 0.075 0.051 0.216 0.033 0.129 0.200 0.029

r0411 0.043 0.040 0.013 0.005 0.039 0.008 0.150 0.040 0.040

r111 0.009 0.048 0.003 0.027 0.051 0.011 0.104 0.060 0.012 r311 0.006 0.174 0.001 0.337 0.157 0.071 0.021 0.141 0.005

Table 5 The 23 unpolarized and polarized SDMEs for the deuteron data in t intervals:

0.000 0.044 0.105

0.200 GeV2. The rst uncertainty is statistical, the second systematic

Element t = 0.021 GeV2 t = 0.071 GeV2 t = 0.147 GeV2

r0400 0.153 0.034 0.031 0.147 0.041 0.036 0.215 0.050 0.028 r111 0.167 0.054 0.029 0.298 0.063 0.074 0.238 0.074 0.083

Im r211 0.281 0.056 0.044 0.198 0.064 0.036 0.309 0.070 0.067 Re r510 0.030 0.015 0.010 0.043 0.018 0.012 0.070 0.024 0.024

Im r610 0.050 0.016 0.008 0.045 0.017 0.010 0.030 0.022 0.011 Im r710 0.067 0.130 0.010 0.041 0.150 0.008 0.201 0.179 0.055

Re r810 0.062 0.136 0.015 0.406 0.153 0.078 0.011 0.143 0.003 Re r0410 0.032 0.022 0.004 0.020 0.025 0.006 0.050 0.030 0.014

Re r110 0.028 0.035 0.002 0.007 0.038 0.009 0.001 0.045 0.001 Im r210 0.060 0.034 0.012 0.082 0.038 0.022 0.020 0.048 0.016 r500 0.007 0.027 0.021 0.036 0.033 0.018 0.089 0.043 0.012 r100 0.092 0.057 0.043 0.117 0.055 0.039 0.145 0.080 0.005

Im r310 0.009 0.081 0.001 0.160 0.099 0.033 0.059 0.119 0.016 r800 0.029 0.209 0.004 0.302 0.223 0.063 0.211 0.256 0.058 r511 0.030 0.022 0.008 0.032 0.027 0.011 0.022 0.032 0.038 r511 0.029 0.027 0.000 0.025 0.032 0.004 0.014 0.042 0.013

Im r611 0.077 0.028 0.022 0.063 0.033 0.014 0.008 0.035 0.009

Im r711 0.157 0.208 0.023 0.411 0.238 0.085 0.087 0.267 0.024

r811 0.005 0.163 0.001 0.018 0.182 0.007 0.325 0.186 0.089 Im r811 0.165 0.193 0.024 0.100 0.228 0.040 0.172 0.229 0.047

r0411 0.021 0.034 0.001 0.052 0.041 0.022 0.140 0.048 0.052

r111 0.009 0.045 0.005 0.013 0.053 0.005 0.145 0.059 0.038 r311 0.030 0.132 0.011 0.083 0.165 0.029 0.247 0.177 0.068

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Table 6 The 23 unpolarized and polarized SDMEs in the Diehl representation [4] for proton and deuteron data in the entire kinematic region. The rst uncertainty is statistical, the second systematic

Element Proton Deuteron

u00++ + u0000 0.168 0.018 0.036 0.160 0.024 0.038 Re u000+ 0.010 0.012 0.002 0.020 0.014 0.005

u00+ 0.004 0.018 0.004 0.060 0.023 0.016 Re (u0+0+ u00+) 0.014 0.024 0.004 0.037 0.030 0.007

Re (u0+++ u0++ + 2 u0+00) 0.006 0.029 0.008 0.107 0.036 0.023 Re u0++ 0.014 0.019 0.005 0.016 0.022 0.009

Re (u00+ u+00+) 0.175 0.029 0.039 0.215 0.036 0.047 Re u0++ 0.039 0.018 0.007 0.003 0.023 0.002 u++ 0.171 0.029 0.023 0.248 0.037 0.039

Re (u++0+ + u 0+) 0.059 0.012 0.022 0.025 0.015 0.015

Re u+0+ 0.042 0.015 0.012 0.036 0.019 0.014

Re (u+++ + u+00) 0.037 0.009 0.012 0.045 0.010 0.014 Re u+++ 0.043 0.014 0.006 0.021 0.018 0.001

Re u+0+ 0.061 0.008 0.012 0.043 0.010 0.009

u++ 0.036 0.014 0.008 0.056 0.019 0.013 Im u000+ 0.059 0.047 0.012 0.038 0.056 0.008

Im (u0+0+ u00+) 0.023 0.076 0.010 0.122 0.089 0.025 Im (u0+++ u0++) 0.109 0.075 0.021 0.021 0.087 0.004

Im (u00+ u+00+) 0.092 0.117 0.018 0.113 0.135 0.028

Im (u++0+ + u 0+) 0.079 0.089 0.017 0.097 0.103 0.020

Im u+0+ 0.142 0.110 0.029 0.017 0.131 0.004

Im u+++ 0.169 0.075 0.035 0.083 0.083 0.017 Im u+0+ 0.060 0.110 0.012 0.150 0.125 0.034

Table 7 The denition of intervals and the mean values for kinematic variables

Bin Q2 [GeV2] t [GeV2] W [GeV] xB Overall 2.42 0.080 4.80 0.097

1.00 GeV2 < Q2 < 1.57 GeV2 1.28 0.082 4.87 0.059

1.57 GeV2 < Q2 < 2.55 GeV2 2.00 0.079 4.78 0.085

Q2 > 2.55 GeV2 4.00 0.078 4.91 0.147

0.000 GeV2 < t < 0.044 GeV2 2.38 0.021 4.73 0.0970.044 GeV2 < t < 0.105 GeV2 2.49 0.072 4.78 0.0990.105 GeV2 < t < 0.200 GeV2 2.39 0.147 4.85 0.095

123

Eur. Phys. J. C (2014) 74:3110 Page 23 of 25 3110

Table8Thecorrelationmatrixforthe23SDMEsobtainedfromthehydrogentargetdata.ThecolumnheadingsdonotindicatetherealandimaginarypartsofanySDMEsinordertokeepthe

tablecompact

110.000.010.030.020.000.000.000.020.020.000.000.000.010.030.020.080.010.070.110.260.030.031.00

SDMEr04

10r

00r

11r

10r

100.000.010.020.000.010.000.010.010.010.030.010.010.030.030.030.340.081.00

Imr7

110.030.030.010.010.000.010.020.030.000.020.020.030.020.020.050.060.320.071.00

100.020.020.010.020.030.020.010.000.010.010.000.010.000.010.010.100.120.130.040.000.011.00

00 r8

11r

10r

11r

10r

0.180.030.080.160.110.070.000.070.061.00

11 r5

000.480.120.010.120.370.080.080.090.100.381.00

Rer5

110.040.070.180.010.030.070.150.130.030.230.030.061.00

Imr6

0.140.050.120.040.140.150.140.100.150.010.030.140.091.00

Imr6

100.000.010.040.020.010.000.020.050.000.040.000.010.030.010.061.00

Imr3

110.020.030.010.020.030.010.040.040.050.040.020.010.010.010.010.070.070.020.261.00

0.070.040.000.030.090.030.010.010.010.030.060.010.010.000.010.110.030.080.050.381.00

Rer8

11r

10r

110.020.120.030.150.000.130.010.070.100.320.050.020.070.011.00

Imr3

110.030.010.030.000.000.030.010.010.080.040.020.020.010.050.010.061.00

Imr7

10r

11r

10r

00r

11r

110.200.010.010.020.020.031.00

Imr2

110.230.060.030.000.010.010.130.051.00

10r

11r

10r

0.040.170.020.000.081.00

00 r1

100.090.200.080.010.000.040.041.00

Imr2

100.110.340.070.040.130.240.100.020.130.030.151.00

11r

00r

11r

110.030.061.00

0.070.080.060.381.00

Rer1

10r

110.020.010.371.00

11r

10r

00r

100.071.00

r04

00r

SDMEr04

001.00

Rer04

r04

123

3110 Page 24 of 25 Eur. Phys. J. C (2014) 74:3110

Table9Thecorrelationmatrixforthe23SDMEsobtainedfromthedeuteriumtargetdata.ThecolumnheadingsdonotindicatetherealandimaginarypartsofanySDMEsinordertokeepthe

tablecompact

110.000.020.000.010.010.000.050.020.030.010.020.030.000.020.030.040.030.050.110.210.030.001.00

SDMEr04

10r

00r

11r

10r

100.030.020.000.030.030.030.030.000.010.010.040.030.000.030.020.440.091.00

Imr7

110.020.010.020.020.010.010.010.010.000.030.020.030.010.010.020.080.430.051.00

100.020.030.010.010.030.010.000.020.010.020.010.040.040.030.020.080.080.170.000.010.061.00

00 r8

11r

10r

11r

10r

0.210.030.020.200.130.090.020.050.051.00

11 r5

000.510.130.030.150.400.090.080.100.070.401.00

Rer5

110.010.130.260.020.040.100.160.110.020.320.060.151.00

Imr6

0.090.020.090.070.060.120.090.240.140.010.000.160.081.00

Imr6

100.020.020.000.010.040.050.020.040.020.020.010.000.000.050.031.00

Imr3

110.020.030.030.030.000.030.020.010.010.040.020.000.000.010.030.050.140.050.241.00

0.030.020.010.010.000.020.010.050.020.010.000.020.000.020.020.060.020.030.020.401.00

Rer8

11r

10r

110.010.080.020.200.030.130.010.090.270.320.040.020.080.081.00

Imr3

110.010.000.010.010.020.030.010.030.040.030.000.020.000.020.050.101.00

Imr7

10r

11r

10r

00r

11r

110.210.090.030.020.010.071.00

Imr2

110.200.060.040.010.040.050.110.051.00

10r

11r

10r

0.020.220.090.000.121.00

100.010.230.060.040.010.040.021.00

Imr2

100.130.380.140.070.110.230.170.000.120.050.161.00

11r

00r

11r

110.030.091.00

10r

0.000.010.381.00

11 r1

000.070.060.070.401.00

Rer1

11r

10r

00r

100.111.00

r04

00r

SDMEr04

001.00

Rer04

r04

123

Eur. Phys. J. C (2014) 74:3110 Page 25 of 25 3110

References

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123

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Abstract

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(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Exclusive electroproduction of ...... mesons on unpolarized hydrogen and deuterium targets is studied in the kinematic region of ...... GeV......, 3.0 GeV ...... 6.3 GeV, and ...... GeV....... Results on the angular distribution of the ...... meson, including its decay products, are presented. The data were accumulated with the HERMES forward spectrometer during the 1996-2007 running period using the 27.6 GeV longitudinally polarized electron or positron beam of HERA. The determination of the virtual-photon longitudinal-to-transverse cross-section ratio reveals that a considerable part of the cross section arises from transversely polarized photons. Spin density matrix elements are presented in projections of ...... or ....... Violation of ......-channel helicity conservation is observed for some of these elements. A sizable contribution from unnatural-parity-exchange amplitudes is found and the phase shift between those amplitudes that describe transverse ...... production by longitudinal and transverse virtual photons, ...... and ......, is determined for the first time. A hierarchy of helicity amplitudes is established, which mainly means that the unnatural-parity-exchange amplitude describing the ...... transition dominates over the two natural-parity-exchange amplitudes describing the ...... and ...... transitions, with the latter two being of similar magnitude. Good agreement is found between the HERMES proton data and results of a pQCD-inspired phenomenological model that includes pion-pole contributions, which are of unnatural parity.

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Title

Spin density matrix elements in exclusive [omega] electroproduction on 1H and 2H targets at 27.5 GeV beam energy

Author

Airapetian, A; Akopov, N; Akopov, Z; Augustyniak, W; Avetissian, A; Blok, H P; Borissov, A; Bryzgalov, V; Capiluppi, M; Capitani, G P; Cisbani, E; Ciullo, G; Contalbrigo, M; Dalpiaz, P F; Deconinck, W; De Leo, R; De Sanctis, E; Diefenthaler, M; Di Nezza, P; Düren, M; Ehrenfried, M; Elbakian, G; Ellinghaus, F; Etzelmüller, E; Fabbri, R; Felawka, L; Frullani, S; Gabbert, D; Gapienko, G; Gapienko, V; Garibaldi, F; Gavrilov, G; Gharibyan, V; Hartig, M; Hasch, D; Holler, Y; Hristova, I; Ivanilov, A; Jackson, H E; Joosten, S; Kaiser, R; Karyan, G; Keri, T; Kinney, E; Kisselev, A; Korotkov, V; Kozlov, V; Kravchenko, P; Krivokhijine, V G; Lagamba, L; Lapikás, L; Lehmann, I; Lenisa, P; Lorenzon, W; Ma, B-q; Mahon, D; Manaenkov, S I; Mao, Y; Marianski, B; Marukyan, H; Movsisyan, A; Murray, M; Naryshkin, Y; Nass, A; Nowak, W-d; Pappalardo, L L; Perez-benito, R; Petrosyan, A; Reimer, P E; Reolon, A R; Riedl, C; Rith, K; Rostomyan, A; Ryckbosch, D; Schäfer, A; Schnell, G; Schüler, K P; Seitz, B; Shibata, T-a; Stahl, M; Stancari, M; Statera, M; Steffens, E; Steijger, J J; M; Taroian, S; Terkulov, A; Truty, R; Trzcinski, A; Tytgat, M; Van Haarlem, Y; Van Hulse, C; Vikhrov, V; Vilardi, I; Wang, S; Yaschenko, S; Yen, S; Zeiler, D; Zihlmann, B; Zupranski, P

Pages

1-25

Publication year

2014

Publication date

Nov 2014

Publisher

Springer Nature B.V.

ISSN

14346044

e-ISSN

14346052

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1140/epjc/s10052-014-3110-1

ProQuest document ID

1874841196

Spin density matrix elements in exclusive [omega] electroproduction on 1H and 2H targets at 27.5 GeV beam energy

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