Transverse-target-spin asymmetry in exclusive [omega]-meson electroproduction: The HERMES Collaboration

Abstract

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

Hard exclusive electroproduction of ...... mesons is studied with the HERMES spectrometer at the DESY laboratory by scattering 27.6 GeV positron and electron beams off a transversely polarized hydrogen target. The amplitudes of five azimuthal modulations of the single-spin asymmetry of the cross section with respect to the transverse proton polarization are measured. They are determined in the entire kinematic region as well as for two bins in photon virtuality and momentum transfer to the nucleon. Also, a separation of asymmetry amplitudes into longitudinal and transverse components is done. These results are compared to a phenomenological model that includes the pion pole contribution. Within this model, the data favor a positive ...... transition form factor.

Eur. Phys. J. C (2015) 75:600

DOI 10.1140/epjc/s10052-015-3825-7

Regular Article - Experimental Physics

HERMES Collaboration

A. Airapetian15,18, N. Akopov29, Z. Akopov7, E. C. Aschenauer8, W. Augustyniak28, A. Avetissian29, S. Belostotski21,H. P. Blok20,27, A. Borissov7, V. Bryzgalov22, G. P. Capitani13, G. Ciullo11,12, M. Contalbrigo11, P. F. Dalpiaz11,12,W. Deconinck7, R. De Leo2, E. De Sanctis13, M. Diefenthaler10,17, P. Di Nezza13, M. Dren15, G. Elbakian29,F. Ellinghaus6, L. Felawka25, S. Frullani23,24, D. Gabbert8, G. Gapienko22, V. Gapienko22, V. Gharibyan29,F. Giordano11,12,17, S. Gliske18, D. Hasch13, M. Hoek16, Y. Holler7, A. Ivanilov22, H. E. Jackson1, S. Joosten14,R. Kaiser16, G. Karyan29, T. Keri15, E. Kinney6, A. Kisselev21, V. Korotkov22, V. Kozlov19, V. G. Krivokhijine9,L. Lagamba2, L. Lapiks20, I. Lehmann16, P. Lenisa11,12, W. Lorenzon18, B.-Q. Ma3, S. I. Manaenkov21, Y. Mao3,B. Marianski28, H. Marukyan29, Y. Miyachi26, A. Movsisyan11,29, V. Muccifora13, Y. Naryshkin21, A. Nass10,M. Negodaev8, W.-D. Nowak8, L. L. Pappalardo11,12, R. Perez-Benito15, A. Petrosyan29, P. E. Reimer1,A. R. Reolon13, C. Riedl8,17, K. Rith10, G. Rosner16, A. Rostomyan7, J. Rubin17,18, D. Ryckbosch14, Y. Salomatin22,G. Schnell4,5,14,a, B. Seitz16, T.-A. Shibata26, M. Statera11,12, E. Steffens10, J. J. M. Steijger20, F. Stinzing10,S. Taroian29, A. Terkulov19, R. Truty17, A. Trzcinski28, M. Tytgat14, Y. Van Haarlem14, C. Van Hulse4,14,V. Vikhrov21, I. Vilardi2, C. Vogel10, S. Wang3, S. Yaschenko7,10, S. Yen25, B. Zihlmann7, P. Zupranski28

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 Department of Theoretical Physics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain

5 IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain

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

7 DESY, 22603 Hamburg, Germany

8 DESY, 15738 Zeuthen, Germany

9 Joint Institute for Nuclear Research, 141980 Dubna, Russia

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

11 Sezione di Ferrara, Istituto Nazionale di Fisica Nucleare, 44122 Ferrara, Italy

12 Dipartimento di Fisica e Scienze della Terra, Universit di Ferrara, 44122 Ferrara, Italy

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

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

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

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

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

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

19 Lebedev Physical Institute, 117924 Moscow, Russia

20 National Institute for Subatomic Physics (Nikhef), 1009 DB Amsterdam, The Netherlands

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

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

23 Gruppo Collegato Sanit, Sezione di Roma, Istituto Nazionale di Fisica Nucleare, 00161 Rome, Italy

24 Istituto Superiore di Sanit, 00161 Rome, Italy

25 TRIUMF, Vancouver, BC V6T 2A3, Canada

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

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

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

29 Yerevan Physics Institute, 375036 Yerevan, Armenia

Received: 4 September 2015 / Accepted: 1 December 2015 / Published online: 17 December 2015 The Author(s) 2015. This article is published with open access at Springerlink.com

F. Stinzing: Deceased.

a e-mail: mailto:[email protected]

Web End [email protected]

Abstract Hard exclusive electroproduction of mesons is studied with the HERMES spectrometer at the DESY laboratory by scattering 27.6 GeV positron and electron beams

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Web End = Transverse-target-spin asymmetry in exclusive -meson electroproduction

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off a transversely polarized hydrogen target. The amplitudes of ve azimuthal modulations of the single-spin asymmetry of the cross section with respect to the transverse proton polarization are measured. They are determined in the entire kinematic region as well as for two bins in photon virtuality and momentum transfer to the nucleon. Also, a separation of asymmetry amplitudes into longitudinal and transverse components is done. These results are compared to a phenomenological model that includes the pion pole contribution. Within this model, the data favor a positive transition form factor.

1 Introduction

In the framework of quantum chromodynamics (QCD), hard exclusive meson leptoproduction on a longitudinally or transversely polarized proton target provides important information about the spin structure of the nucleon. The process amplitude is a convolution of the leptonquark hard-scattering subprocess amplitude with soft hadronic matrix elements describing the structure of the nucleon and that of the meson. Here, factorization is proven rigorously only if the leptonquark interaction is mediated by a longitudinally polarized virtual photon [1,2]. The soft hadronic matrix elements describing the nucleon contain generalized parton distributions (GPDs) to parametrize its partonic structure. Hard exclusive production of vector mesons is described by GPDs H f and E f , where f denotes a quark avor or a gluon.These unpolarized, i.e., partonhelicitynonip distributions describe the photonparton interaction with conservation and ip of nucleon helicity, respectively. Both are of special interest, as they are related to the total angular momentum of partons, J f [3]. The GPDs H f are well constrained by existing experimental data. The GPDs Eu and Ed for up and down quarks, respectively, are partially constrained by nucleon form-factor data [4], while experimental information on sea-quark GPD Esea and gluon GPD Eg is scarce.For a recent review on the status of GPD determinations, see Ref. [5]. In contrast to leptoproduction of vector mesons with an unpolarized target, which is mainly sensitive to GPDs H f ,

vector-meson leptoproduction off a transversely polarized nucleon is sensitive to the interference between two amplitudes containing H f and E f , respectively, and thus opens access to E f .

For a transversely polarized virtual photon mediating the leptonquark interaction, there exists no rigorous proof of collinear factorization. In the QCD-inspired phenomenological GK model [68] however, factorization is also assumed for the transverse amplitudes. In this so-called modied perturbative approach [9], infrared singularities occur-ring in these amplitudes are regularized by quark transverse momenta in the subprocess, while the partons are still emitted and reabsorbed collinearly by the nucleon. By using the

quark transverse momenta in the subprocess, the transverse size of the meson is effectively taken into account. Using this approach, the GK model describes cross sections, spin density matrix elements (SDMEs), and spin asymmetries in exclusive vector-meson production for values of Bjorken-x below 0.2 [68]. The GPDs parametrized in the GK model were used in calculations of deeply virtual Compton scattering (DVCS) amplitudes, which led to good agreement with most DVCS measurements over a wide kinematic range [10]. In the most recent version of the model, the vertex function in the one-pion-exchange contribution is identied with the transition form factor [11]. Its magnitude is determined in a model-dependent way, while its unknown sign may be determined from comparisons with experimental data on spin asymmetries in hard exclusive leptoproduction.

Measurements of hard-exclusive production of various types of mesons are complementary to DVCS, as they allow access to various avor combinations of GPDs. Previous HERMES publications on measurements of azimuthal transverse-target-spin asymmetries include results on exclusive production of 0 [12] and + mesons [13] as well as on

DVCS [14].

In the present paper, the azimuthal modulations of the transverse-target-spin asymmetry in the cross section of exclusive electroproduction of mesons are studied. The available data allow for an estimation of the kinematic dependence of the measured asymmetry amplitudes on photon virtuality and four-momentum transfer to the nucleon. The measured asymmetry amplitudes are compared to the most recent calculations of the GK model using either possible sign of the transition form factor.

2 Data collection and process identication

The data were accumulated with the HERMES forward spectrometer [15] during the running period 20022005. The27.6 GeV positron (electron) beam was scattered off a transversely polarized hydrogen target, with the average magnitude PT of the proton-polarization component PT perpendicular to the beam direction being equal to 0.72. The lepton beam was longitudinally polarized, and in the analysis the data set is beam-helicity balanced. The meson is produced in the reaction

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

The same requirements to select exclusively produced mesons as in Ref. [16] are applied. The candidate events for exclusive -meson production are required to have exactly three charged tracks, i.e., the scattered lepton and

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50

Events/(4 MeV)

Events/(0.2 GeV)

entire kinematic region

40

40

30

20

20

10

0 5 10 15 20

E [GeV]

110 120 130 140 150 160 170

M (

) [MeV]

Fig. 1 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 vertical dashed line indicates the PDG value of the 0 mass [17]

two oppositely charged pions, and at least two clusters in the calorimeter not associated with a charged track. The 0 meson is reconstructed from two photon clusters with an invariant mass M( ) in the interval 0.11 GeV < M( ) < 0.16 GeV. Its distribution is shown in Fig. 1, where the t with a BreitWigner function yields 136.1

0.8 MeV (19 2 MeV) for the mass (width). The charged

hadrons and leptons are identied through the combined responses of four particle-identication detectors [15]. The three-pion invariant mass is calculated as M(+0) =

(p+ + p + p0 )2, where p are the four-momenta

of the charged and neutral pions. Events containing mesons are selected through the requirement 0.71 GeV < M(+0) < 0.87 GeV.

Further event-selection requirements are the following:

(i) 1.0 GeV2 < Q2 < 10.0 GeV2, where Q2 represents the negative square of the virtual-photon four-momentum. The lower value is applied in order to facilitate the application of perturbative QCD, while the upper value delimits the measured phase space;

(ii) t < 0.2 GeV2 in order to improve exclusivity, where

t = t tmin, t is the squared four-momentum transfer

to the nucleon and tmin represents the smallest kine

matically allowed value of t at xed virtual-photon

energy and Q2;

(iii) W > 3 GeV in order to be outside of the resonance region and W < 6.3 GeV in order to clearly delimit the

0

Fig. 2 Missing-energy distribution for exclusive -meson production. The unshaded histogram shows experimental data, while the shaded area shows the distribution obtained from a PYTHIA simulation of the SIDIS background. The vertical dashed line denotes the upper limit of the exclusive region

kinematic phase space, where W is the invariant mass of the photon-nucleon system;(iv) the scattered-lepton energy lies above 3.5 GeV in order to avoid a bias originating from the trigger.

In order to isolate exclusive production, the energy not accounted for by the leptons and the three pions must be zero within the experimental resolution. We require the missing energy to be in the interval 1.0 GeV < E < 0.8 GeV,

which is referred to as exclusive region in the following.

Here, the missing energy is calculated as E =

M2X M2p

2Mp ,

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

p+ p p

0 )2 the missing-mass squared, where p and q are the four-momenta of target nucleon and virtual photon, respectively. The distribution of the missing energy E is shown in Fig. 2. It exhibits a clearly visible exclusive peak centered about E = 0 . The shaded area

represents semi-inclusive deep-inelastic scattering (SIDIS) background events obtained from a PYTHIA [18] Monte-Carlo simulation that is normalized to the data in the region2 GeV < E < 20 GeV. The simulation is used to determine the fraction of background under the exclusive peak. This fraction is calculated as the ratio of the number of background events to the total number of events and amounts to about 21 %.

After application of all these constraints, the sample contains 279 exclusively produced mesons. This data sample is referred to in the following as data in the entire kinematic

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Events/(13 MeV)

60

50

40

30

20

10

720 740 760 780 800 820 840 860

M (

-

+ 0 ) [MeV]

Fig. 3 The +0 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 vertical dashed line indicates the PDG value of the mass [17]

region. The +0 invariant-mass distribution for this data sample is shown in Fig. 3. A BreitWigner t yields 785 2 MeV (52 5 MeV) for the mass (width).

3 Extraction of the asymmetry amplitudes

The cross section for hard exclusive leptoproduction of a vector meson on a transversely polarized proton target, written in terms of polarized photo-absorption cross sections and interference terms, is given by Eq. (34) in Ref. [19]. In this equation, the transverse-target-spin asymmetry AU T is decomposed into a Fourier series of terms involving sin(m S),

with m = 0, . . . , 3. The angles and S are the azimuthal

angles of the -production plane and of the component S

of the transverse nucleon polarization vector that is orthogonal to the virtual-photon direction. They are measured around the virtual-photon direction and with respect to the lepton-scattering plane (see Fig. 4). These denitions are in accordance with the Trento Conventions [20]. For the HERMES kinematics and acceptance in exclusive production, sin < 0.1 and cos > 0.99, which can be approximated by sin 0 and cos 1. Here, is the angle

between the lepton-beam and virtual-photon directions.

In this approximation, the angular-dependent part of Eq. (34) in Ref. [19] for an unpolarized beam reads:

W(, S) = 1 + Acos()UU cos() + Acos(2)UU cos(2)

+S[Asin(+S)U T sin( + S)

Fig. 4 Lepton-scattering and -production planes together with the azimuthal angles and S

+Asin(S)U T sin( S)

+Asin(S)U T sin(S)

+Asin(2S)U T sin(2 S)

+Asin(3S)U T sin(3 S)], (3)

where S = |S|. Here, AUU and AU T denote the ampli

tudes of the corresponding cosine and sine modulations as given in their superscripts. The rst letter in the subscript denotes unpolarized beam and the second letter U (T ) denotes unpolarized (transversely polarized) target. The above approximation in conjunction with the additional factor /2 0.4, where is the ratio of uxes of longitudinal and

transverse virtual photons, allows one to neglect the contribution of the sin(2 +S) modulation, appearing in Eq. (34) of

Ref. [19]. This approximation also makes the angular dependence of S disappear (see Eq. (8) of Ref. [19]), and S

PT = 0.72 is used in the following. Note that the modulation

sin( S) is the only one that appears at leading twist.

For exclusive production of mesons decaying into three pions, the angular distribution of the latter can be decomposed into parts corresponding to longitudinally (L) and transversely (T) polarized mesons:

W(, S, ) =

32 r0400 cos2() wL(, S)

+

34 (1 r0400) sin2() wT (, S). (4) Here, is the polar angle of the unit vector normal to the decay plane in the -meson rest frame, with the z-axis aligned opposite to the outgoing nucleon momentum [16]. The pre-factors r0400 and (1 r0400) represent the fractional

contribution to the full cross section by longitudinally and transversely polarized mesons, respectively [16]. The rst (second) term on the right-hand side of Eq. (4) represents the angular distribution of the longitudinally (transversely) polarized mesons, with

wL(, S) = 1 + AUU,L() + S AU T,L(, S),

wT (, S) = 1 + AUU,T () + S AU T,T (, S). (5)

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The functions AUU,K () and AU T,K (, S), with K = L

and K = T denoting longitudinal-separated and transverse-

separated contributions, respectively, are decomposed into a Fourier series in complete analogy to Eq. (3).

The function W(, S) is tted to the experimental angu

lar distribution using an unbinned maximum likelihood method. Here and in the following, the angle has to be added to the argument list of the function W, when appli

cable. The function to be minimized is the negative of the logarithm of the likelihood function:

ln L(R) =

N

[summationdisplay]

i=1

4 Results

The results for the ve AU T and two AUU amplitudes, as determined in the entire kinematic region, are shown in Table 1. These results are presented in Table 3 in two intervals of Q2 and t , with the denition of intervals together

with the average values of the respective kinematic variables given in Table 2. The results for the ve AU T amplitudes are also shown in Fig. 5, in two rows of ve panels each, where the upper and lower rows show the Q2 and t dependences,

respectively. Each panel shows as two lled circles the results in two kinematic bins, and as one open square the result in the entire kinematic region. The results are compared to calculations of the GK model [11,21], for both signs of the form factor. For completeness, also the model prediction without the pionpole contribution is included.

The model predictions differ substantially upon sign change of the form factor for the two amplitudes

Asin(S)U T and Asin(S)U T, in particular when considering the

t dependence. The data seem to favor a positive transition form factor.

Asymmetry amplitudes can be written in terms of SDMEs, as shown in the Appendix. By using Eqs. (9) and (10) and the earlier HERMES results on SDMEs [16],

Acos()UU = 0.13 0.04 0.08 Acos(2)UU = 0.03 0.04 0.01

are obtained, which are consistent within uncertainties with the results shown in Table 1.

The cross section for exclusive production of transversely polarized mesons dominates that for longitudinally polarized ones [16]. This is the reason why the 14-parameter t used here leads to still acceptable uncertainties for the results in the entire kinematic region on the transverse-separated asymmetry amplitudes, while those for the longitudinal-separated ones are so large that any inter-

Table 1 The amplitudes of the ve sine and two cosine modulations as determined in the entire kinematic region. The rst uncertainty is statistical, the second systematic. The results receive an additional 8.2 % scale uncertainty corresponding to the target polarization uncertainty

Amplitude

Asin(+S)U T 0.06 0.20 0.02 Asin(S)U T 0.12 0.19 0.03

Asin(S)U T 0.26 0.27 0.05 Asin(2S)U T 0.03 0.16 0.01

Asin(3S)U T 0.13 0.15 0.03 Acos()UU 0.01 0.11 0.10

Acos(2)UU 0.17 0.11 0.05

ln W(R; i, iS)[tildewide]

N(R)

. (6)

Here, R denotes the set of 7 asymmetry amplitudes of

the unseparated t or 14 asymmetry amplitudes of the longitudinal-to-transverse separated t and the sum runs over the N experimental-data events. The normalization factor

[tildewide]

N(R) =

NMC

j=1

[summationdisplay] W(R; j, jS) (7)

is determined using NMC events from a PYTHIA Monte-Carlo simulation, which are generated according to an isotropic angular distribution and processed in the same way as experimental data. The number of Monte-Carlo events in the exclusive region amounts to about 40,000.

Each asymmetry amplitude is corrected for the background asymmetry according to

Acorr =

Ameas fbg Abg 1 fbg

, (8)

where Acorr is the corrected asymmetry amplitude, Ameas is the measured asymmetry amplitude, fbg is the fraction of the SIDIS background and Abg is its asymmetry amplitude. While Ameas is evaluated in the exclusive region, Abg is obtained by extracting the asymmetry from the experimental SIDIS background in the region 2 GeV < E < 20 GeV.

The systematic uncertainty is obtained by adding in quadrature two components. The rst one, Acorr = Acorr

Ameas, is due to the correction by background amplitudes. In the most conservative approach adopted here, it is estimated as the difference between the asymmetry amplitudes Acorr and Ameas. This approach also covers the small uncertainty on fbg. The second component accounts for effects from detector acceptance, efciency, smearing, and misalignment. It is determined as described in Ref. [16]. An additional scale uncertainty arises because of the systematic uncertainty on the target polarization, which amounts to 8.2 %.

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Table 2 The denition of intervals and the mean values of the kinematic variables

Bin Q2 [GeV2] t [GeV2] W [GeV] xB

Entire kinematic bin 2.24 0.079 4.80 0.092

1.00 GeV2 < Q2 < 1.85 GeV2 1.39 0.084 4.69 0.064

1.85 GeV2 < Q2 < 10.00 GeV2 3.07 0.075 4.91 0.012

0.00 GeV2 < t < 0.07 GeV2 2.36 0.035 4.79 0.095 0.07 GeV2 < t < 0.20 GeV2 2.11 0.128 4.81 0.088

Table 3 Results on the kinematic dependences of the ve asymmetry amplitudes AU T and two amplitudes AUU . The rst two columns correspond to the t intervals 0.00 0.07 0.20 GeV2 and the last

two columns to the Q2 intervals 1.00 1.85 10.00 GeV2. The rst

uncertainty is statistical, the second systematic. The results receive an additional 8.2 % scale uncertainty corresponding to the target polarization uncertainty

Amplitude t = 0.035 GeV2 t = 0.128 GeV2 Q2 = 1.39 GeV2 Q2 = 3.07 GeV2

Asin(+S)U T 0.06 0.28 0.04 0.30 0.32 0.10 0.21 0.31 0.05 0.10 0.28 0.03 Asin(S)U T 0.02 0.28 0.03 0.22 0.27 0.06 0.02 0.30 0.03 0.18 0.26 0.03

Asin(S)U T 0.13 0.37 0.03 0.25 0.42 0.05 0.12 0.42 0.02 0.45 0.37 0.12 Asin(2S)U T 0.24 0.22 0.03 0.28 0.26 0.07 0.07 0.24 0.02 0.01 0.23 0.01

Asin(3S)U T 0.14 0.21 0.01 0.07 0.22 0.03 0.12 0.20 0.04 0.16 0.21 0.02 Acos()UU 0.05 0.15 0.06 0.09 0.17 0.16 0.04 0.15 0.10 0.04 0.16 0.11

Acos(2)UU 0.19 0.15 0.07 0.14 0.17 0.07 0.04 0.15 0.03 0.35 0.17 0.11

Asin (+ s)

UT

Asin (- s)

UT

0.4

0.3

0.4

0.2

Asin ( s)

UT

Asin (2- s)

UT

Asin (3- s)

UT

0.6

0.2

0.2

0.3

0.4

-0

0.1

0.2

0

0.2

-0

0.1

-0.2

-0.2

-0

-0.1

-0

-0.4

-0.4

-0.2

-0.2

-0.1

-0.4

-0.3

-0.2

0 2 4 0 2 4 0 2 4 0 2 4 0 2 4

Q2 [GeV2]

Q2 [GeV2]

Q2 [GeV2]

Q2 [GeV2]

Q2 [GeV2]

0.4

0.4

0.4

Asin (+ s)

UT

Asin (- s)

UT

Asin ( s)

UT

Asin (2- s)

UT

Asin (3- s)

UT

0.4

0.4

0.2

0.2

0.3

0.2

0.2

0

0

0.2

0

0.1

-0.2

-0.2

0

-0

-0.2

-0.4

-0.4

-0.2

-0.1

-0.4

-0.2

0 0.2 0 0.2 0 0.2 0 0.2 0 0.2

-t' [GeV2]

-t' [GeV2]

-t' [GeV2]

-t' [GeV2]

-t' [GeV2]

Fig. 5 The ve amplitudes describing the strength of the sine modulations of the cross section for hard exclusive -meson production. The full circles show the data in two bins of Q2 or t . The open squares

represent the results obtained for the entire kinematic region. The inner error bars represent the statistical uncertainties, while the outer ones indicate the statistical and systematic uncertainties added in quadrature.

The results receive an additional 8.2 % scale uncertainty corresponding to the target polarization uncertainty. The solid (dash-dotted) lines show the calculation of the GK model [11,21] for a positive (negative) transition form factor, and the dashed lines are the model results without the pion pole

pretation is precluded. Also, kinematic dependences can no longer be studied due to the large uncertainties. Therefore, for the transverse-separated asymmetry amplitudes only the results in the entire kinematic region are shown in Fig. 6

and Table 4 together with the corresponding predictions of the GK model [11,21]. Here, the large uncertainties prevent any conclusion on the sign of the transition form factor.

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0.2

0.4

0.3

0.3

Asin (+ s)

UT,T

Asin (- s)

UT,T

Asin (2- s)

UT,T

Asin (3- s)

UT,T

0.4

Asin ( s)

UT,T

0.2

0.2

-0

0.2

0.2

0.1

0.1

-0.2

0

0

-0

-0

-0.2

-0.4

-0.2

-0.1

-0.1

-0.4

-0.4

-0.2

-0.2

0 2 4 0 2 4 0 2 4 0 2 4 0 2 4

UT,T

Q2 [GeV2]

Q2 [GeV2]

Q2 [GeV2]

Q2 [GeV2]

Q2 [GeV2]

Asin (+ s)

Asin (- s)

Asin ( s)

UT,T

Asin (2- s)

0.2

0.4

0.3

0.3

0.4

0.2

0.2

UT,T

-0

UT,T

0.2

0.2

UT,T

Asin (3- s)

-0

0.1

0.1

-0.2

0

-0.2

-0

-0

-0.4

-0.4

-0.2

-0.1

-0.1

-0.6

-0.4

-0.2

-0.2

0 0.2 0 0.2 0 0.2 0 0.2 0 0.2

-t' [GeV2]

-t' [GeV2]

-t' [GeV2]

-t' [GeV2]

-t' [GeV2]

Fig. 6 As Fig. 5, but only for transversely polarized mesons

Table 4 Results on the ve asymmetry amplitudes AU T and two amplitudes AUU in the entire kinematic region, but separated into longitudinal and transverse parts. The rst column (K = L) gives the results for the

longitudinal components, while the second column, (K = T ), shows

the results for the transverse components. The rst uncertainty is statistical, the second systematic. The results receive an additional 8.2 % scale uncertainty corresponding to the target polarization uncertainty

Amplitude Longitudinal (K = L) Transverse (K = T )

Asin(+S)U T,K 0.16 0.92 0.02 0.14 0.29 0.05 Asin(S)U T,K 0.60 0.81 0.16 0.07 0.27 0.04

Asin(S)U T,K 0.08 1.06 0.03 0.21 0.38 0.01 Asin(2S)U T,K 0.38 0.71 0.11 0.10 0.21 0.02

Asin(3S)U T,K 0.21 0.56 0.10 0.07 0.20 0.01 Acos()UU,K 0.53 0.40 0.08 0.16 0.15 0.12

Acos(2)UU,K 0.60 0.39 0.17 0.37 0.15 0.10

5 Summary

In this Paper, results are reported on exclusive electroproduction off transversely polarized protons in the kinematic region 1 GeV2 < Q2 < 10 GeV2 and 0.0 GeV2 < t < 0.2

GeV2. The amplitudes of seven azimuthal modulations of the cross section for unpolarized beam are determined, i.e., of two cosine modulations for unpolarized target and ve sine modulations for transversely polarized target. Results are presented for the entire kinematic region as well as alternatively in two bins of t or Q2. Additionally, a separation

into asymmetry amplitudes for the production of longitudinally and transversely polarized mesons is done. A comparison of extracted asymmetry amplitudes to recent calcu-

lations of the phenomenological model of Goloskokov and Kroll favors a positive sign of the form factor.

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 4.0 International License (http://creativecommons.org/licenses/by/4.0/

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Web End =ons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Appendix: Relations between azimuthal asymmetry amplitudes and spin-density matrix elements

The full information on vector-meson leptoproduction is contained in the differential cross section d3

d Q2dtdx and the SDMEs in the Diehl representation [22]. Therefore, the azimuthal asymmetry amplitudes can be expressed in terms

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600 Page 8 of 8 Eur. Phys. J. C (2015) 75 :600

of the SDMEs. For scattering off an unpolarized target, the asymmetry amplitudes can be written in terms of the Diehl SDMEs u1212 or alternatively in terms of the SchillingWolf

SDMEs rni j [23] as

AcosUU = 2[radicalbig] (1 + ) Re[u0+]

= [radicalbig]2 (1 + ) [2r511 + r500], (9)

Acos2UU = Re[u+]

= [2r111 + r100]. (10)

Here, the abbreviated notation

u12 = u++12 + u12 + u0012 (11) is used, where 1, 2 denote the virtual-photon helicities and 1, 2 the vector-meson helicities. The symbol describes

the virtual-photon or vector-meson helicities 1, while the

symbol 0 describes longitudinal polarization. Equations (9) and (10) show that the asymmetry amplitudes can be calculated from the SchillingWolf SDMEs obtained in Ref. [16].

For scattering off a transversely polarized target, the asymmetry amplitudes can be expressed in terms of the Diehl SDMEs n1212 and s1212 as

Asin(+S)U T = ( /2) Im[n+ s+], (12)

Asin(S)U T = Im[n++ + n00], (13)

Asin(S)U T = [radicalbig] (1 + ) Im[n0+ s0+], (14)

Asin(2S)U T = [radicalbig] (1 + ) Im[n0+ + s0+], (15)

Asin(3S)U T = ( /2) Im[n+ + s+]. (16)

The abbreviated notations

n12 = n++12 + n12 + n0012, (17) s12 = s++12 + s12 + s0012 (18)

are analogous to those in Eq. (11). In this case, Schilling

Wolf SDMEs rni j [23] are not dened.

In order to get from Eqs. (9), (10) and (12)(16) expressions for the asymmetry amplitudes for the production of longitudinally polarized vector mesons, the terms with 1 =

2 = 0 have to be retained in Eqs. (9)(18), and the result has

to be divided by the SchillingWolf SDME r0400. For instance, Asin(2S)U T becomes

Asin(2S)U T,L =

(1 + )

r0400

Im[n000+ + s000+]

=

(1 + )

u00++ + u0000

Im[n000+ + s000+]. (19)

Correspondingly, for the production of transversely polarized vector mesons, the terms with 1 = 2 = 1 have to be

retained in Eqs. (9)(18), and the result has to be divided by (1 r0400). For instance, Asin(2S)U T becomes

Asin(2S)U T,T

=

(1 + )

1 r0400

Im[n++0+ + s++0+ + n0+ + s0+]

=

(1 + )

1 u00++ u0000

Im[n++0+ + s++0+ + n0+ + s0+].

(20)

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