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TRANSVERSE MOMENTUM DEPENDENT PARTON DISTRIBUTIONS AND AZIMUTHAL ASYMMETRIES IN LIGHT-CONE QUARK MODELS B. Pasquini 1 † ,S.Boffi, 1 ,A.V.Efremov 2 , and P. Schweitzer 3 (1) Dipartimento di Fisica Nucleare e Teorica, Università degliStudidiPavia,and Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, I-27100 Pavia, Italy (2) Joint Institute for Nuclear Research, Dubna,141980 Russia (3) Department of Physics, University of Connecticut, Storrs, CT 06269, USA † pasquini@pv.infn.it Abstract We review the information on the spin and orbital angular momentum structure of the nucleon encoded in the T-even transverse momentum dependent parton distributions within light-cone quark models. Model results for azimuthal spin asymmetries in semi-inclusive lepton-nucleon deep-inelastic scattering are discussed, showing a good agreement with available experimental data and providing predictions to be further tested by future CLAS, COMPASS and HERMES data. 1 TMDs and Light-Cone CQMs A convenient framework for the analysis of hadronic states is quantization on the lightcone. The proton state, for example, can be represented as a superposition of light-cone wave functions (LCWFs), one for each of the Fock components (qqq), (qqq¯q), ... of the nucleon state. This light-cone representation has a number of simplifying properties [1]. In particular it allows one to describe the hadronic matrix elements which parametrize the soft-contribution in inclusive and exclusive reactions in terms of overlap of LCWFs with different parton configurations. In principle, there is an infinite number of LCWFs in such an expansion. However, there are many situations where one can confine the analysis to the contribution of the Fock components with a few partons. For example, light-cone models limited to the minimal Fock-space configuration of valence quarks are able to reproduce the main features of the hadron electromagnetic form factors [2] as well to account for the behaviour of the hadron structure functions in deeply inelastic processes at large values of the Bjorken variable x [3, 4]. Here the LCWFs of constituent quark models (CQMs) will be used to describe transverse momentum dependent parton distributions (TMDs) which are a natural extension of standard parton distributions from one to three-dimensions in momentum space. In particular, to disentangle the spin-spin and spin-orbit quark correlations encoded in the different TMDs, we expand the three-quark LCWF in a basis of eigenstates of orbital angular momentum. In the light-cone gauge A + = 0, such an expansion involves six independent amplitudes corresponding to the different combinations of quark helicity and orbital angular momentum. Explicit expressions for the light-cone amplitudes have been obtained in Ref. [5] representing the light-cone spinors of the quarks through the unitary 102
Melosh rotations which boost the rest-frame spin into the light-cone. Furthermore, assuming SU(6) symmetry, the light-cone amplitudes have a particularly simple structure, with the spin and isospin dependence factorized from a momentum-dependent function which is spherically symmetric. Under this assumption the orbital angular momentum content of the wave function is fully generated by the Melosh rotations and therefore matches the analytical structure expected from model-independent arguments [6]. The model dependence enters the choice of the momentum-dependent part of the LCWF, as obtained, for example, from the eigenvalue equation of the Hamiltonian with a specific potential model for the bound state of the three quarks. Using a more phenomenological description, we choose this part by assuming a specific functional form with parameters fitted to hadronic structure constants. This is the strategy adopted also in Ref. [7] through a fit of the LCWF to the anomalous magnetic moments of the nucleon. The same wave function was also used to predict many other hadronic properties [8], providing a good description of available experimental data, and being able to capture the main features of hadronic structure functions, like parton distributions [4], generalized parton distributions (GPDs) [3] and TMDs [5]. The eight leading twist TMDs, f1, f ⊥ 1T ,g1, g1T ,g ⊥ 1L ,h1, h ⊥ 1T ,h⊥ 1L ,andh⊥ 1 , are defined in terms of the same quark correlation functions entering the definition of ordinary parton distributions, but without integration over the transverse momentum. Among them, the Boer-Mulders h⊥ 1 [9] and the Sivers f ⊥ 1T [10] functions are T-odd, i.e. they change sign under “naive time reversal”, which is defined as usual time reversal, but without interchange of initial and final states. Since non-vanishing T-odd TMDs require gauge boson degrees of freedom which are not taken into account in our light-cone quark model, our model results will be discussed only for the T-even TMDs. Projecting the correlator for quarks of definite longitudinal (sL) or transverse (sT ) polarizations, one obtains in nucleon states described by the polarization vector S =(SL, ST ) the following spin densities in the momentum space ˜ρ(x, k 2 T ,sL, S) = 1 � f1 + S 2 i T ɛijk j 1 T m f ⊥ 1T + sLSL g1L + sL S i T ki 1 T m g � 1T , (1) ˜ρ(x, k 2 T , sT , ST ) = 1 � f1 + S 2 i T ɛ ij k j 1 T m f ⊥ 1T + s i T ɛ ij k j 1 T m h⊥ 1 + s i T S i T h1 � , (2) + s i T (2k i T k j T − k2T δ ij )S j T 1 2m 2 h⊥ 1T + SL s i T k i T 1 m h⊥ 1L where the distribution functions depend on x and k 2 T . The unpolarized TMD f1, the helicity TMD g1L, and the transversity TMD h1 in Eqs. (1) and (2) correspond to monopole distributions in the momentum space for unpolarized, longitudinally and transversely polarized nucleon, respectively. They can be obtained from the overlap of LCWFs which are diagonal in the orbital angular momentum, but probe different transverse momentum and helicity correlations of the quarks inside the nucleon. All the other TMDs require a transfer of orbital angular momentum between the initial and final state. In particular, g1T and h⊥ 1L correspond to quark densities with specular configurations for the quark and nucleon spin: g1T describes longitudinally polarized quarks in a transversely polarized nucleon, while h⊥ 1L gives the distribution of transversely polarized quarks in longitudinally polarized nucleon. Therefore, g1T requires helicity flip of the nucleon which is not 103
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XIII Advanced Research Workshop on
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ã�� [539.12.01 + 539.12 ... 14
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Analytic perturbation theory and pr
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Measurements of GEp/GMp to high Q 2
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WELCOME ADDRESS Alexei Sissakian Di
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he joined the group of prof. Przemy
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proton. Dividing these 90 ◦ cm p-
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p-p scattering angle fixed at exact
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tuning for each ring. The Workshop
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was only about 10 5 per second, but
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[10] E.F. Parker et al., Phys. Rev.
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MICROSCOPIC STERN-GERLACH EFFECT AN
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DeGrand-Miettinen model predicted P
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TOWARDS STUDY OF LIGHT SCALAR MESON
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104xdBR(φ--> π η 0 -1 γ )/dm, G
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RECURSIVE FRAGMENTATION MODEL WITH
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Figure 2: String decaying into pseu
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pT -distributions in the quark frag
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4 Inclusion of spin-1 mesons For a
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CONSTITUENT QUARK REST ENERGY AND W
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〈r 2 ch 〉≈1fm2 . Now with Eqs
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TOWARDS A MODEL INDEPENDENT DETERMI
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Common for the difference cross sec
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TRANSVERSITY GPDs FROM γN → πρ
Page 53 and 54: The scattering amplitude of the pro
Page 55 and 56: INFRARED PROPERTIES OF THE SPIN STR
Page 57 and 58: 5 Acknowledgement B.I. Ermolaev is
Page 59 and 60: We shall call this model the “sec
Page 61 and 62: y f ν V = f l V = f u V = f d gz V
Page 63 and 64: 2. We remind, that the celebrated G
Page 65 and 66: of the resonance A, θV � 39 o .
Page 67 and 68: • Right-handed EW currents. The c
Page 69 and 70: pesum- (p Entries 0 + pe)-(p-pe) Me
Page 71 and 72: CROSS SECTIONS AND SPIN ASYMMETRIES
Page 73 and 74: R(ρ) 5 4 3 2 1 0 2 3 4 5 6 7 8 9 1
Page 75 and 76: TWO-PHOTON EXCHANGE IN ELASTIC ELEC
Page 77 and 78: and depend on three parameters : fN
Page 79 and 80: O(αs) SPIN EFFECTS IN e + e −
Page 81 and 82: 3 Polarization results for mq → 0
Page 83 and 84: Since one has (↑↓) pc Born =(
Page 85 and 86: Figure 1: A typical lowest order Fe
Page 87 and 88: sin φ S (π + ) A UT 1.0 0.8 0.6 0
Page 89 and 90: form the agreement with these data
Page 91 and 92: in settling this dynamical issue. G
Page 93 and 94: SPIN CORRELATIONS OF THE ELECTRON A
Page 95 and 96: the two-photon system equaling 1).
Page 97 and 98: ANOTHER NEW TRACE FORMULA FOR THE C
Page 99 and 100: Compare the gain of time and effort
Page 101 and 102: where the triplet and octet axial c
Page 103: [2] Y. Prok et al. [CLAS Collaborat
Page 107 and 108: ky �GeV� 0.4 0.2 0.0 �0.2 �
Page 109 and 110: [2] B. Pasquini, and S. Boffi, Phys
Page 111 and 112: The appearance of both |+〉 and |
Page 113 and 114: 2.2 Intrinsic Transverse Motion Qua
Page 115 and 116: are still complex: for a given corr
Page 117 and 118: This last is required to complete t
Page 119 and 120: [16] P.G. Ratcliffe and O.V. Teryae
Page 121 and 122: Figure 1: a)[left] μpG p E /Gp M (
Page 123 and 124: 4 Conclusion We introduced a simple
Page 125 and 126: √ δΔq8 x for Δq8 and γΔGx fo
Page 127 and 128: understanding of polarized light qu
Page 129 and 130: They are inverse powers of Q 2 kine
Page 131 and 132: accounts approximately for the TM a
Page 133 and 134: POTENTIAL FOR A NEW MUON g-2 EXPERI
Page 135 and 136: When the momentum increases (p >p0)
Page 137 and 138: EVOLUTION EQUATIONS FOR TRUNCATED M
Page 139 and 140: 4 Perspectives Evolution equations
Page 141 and 142: NEW DEVELOPMENTS IN THE QUANTUM STA
Page 143 and 144: statistical approach compared to th
Page 145 and 146: POLARIZED HADRON STRUCTURE IN THE V
Page 147 and 148: g 3 A =Δuv(Q 2 ) − Δdv(Q 2 ), g
Page 149 and 150: AXIAL ANOMALIES, NUCLEON SPIN STRUC
Page 151 and 152: 3. Vector current of strange quarks
Page 153 and 154: ORBITAL MOMENTUM EFFECTS DUE TO A L
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The elastic scattering S-matrix in
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IDENTIFICATION OF EXTRA NEUTRAL GAU
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for final l + l − events (l = μ,
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QUARK INTRINSIC MOTION AND THE LINK
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where wP = − m 2M 2 −p1 cos ω
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where g q 2x − ξ 1(x, pT )= πM2
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EXPERIMENTAL RESULTS
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01 00 11 01 01 01 00 11 01 01 01 00
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where C1, C2 and A1 - signals from
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Figure 1: Layout of experiment targ
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According to requirements of the de
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Electron energy is measured at ATLA
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Fig. 2 shows the results of the lon
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e γ* e’ x+ ξ x− ξ A A’ e e
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cos 0φ C A φ cos C A cos 2φ C A
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5 Summary HERMES has measured signi
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(1.205 ± 0.0012) GeV/c, 10 10 p/s,
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was done in the interval ”−t”
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If we admit a non-zero quark transv
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which is trivial in collinear LO ap
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In figure 3 the expected errors on
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[12] A. V. Efremov and O. V. Teryae
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Figure 1: The COMPASS spectrometer.
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Comparing this formula to the inclu
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Figure 5: Comparison of COMPASS inc
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[9] B. Adeva et al. (SMC Coll.), Ph
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q T sin( φ-φ ) M AUT 0.14 0.12 0.
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proton beam is planned. References
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Run Year √ s (GeV) Polarization R
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ackground fraction, and asymmetry i
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At √ s = 200 GeV, ALL(pp → π
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[6] D. de Florian, W. Vogelsang, an
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higher energies, where the cross se
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Figure 3: The experimental results
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kinematic range to low values of Bj
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3 Semi-Inclusive DIS Collins and Si
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Unpolarized target asymmetries. The
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4 Summary Since the end of data-tak
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polarization as well as a construct
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Finally the COMPASS reconstruction
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tained in the NLO QCD approximation
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with only modern NN potential are r
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(a) (b) Figure 3: (a) T20 data take
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of electroproduction from polarized
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As is seen in Fig. 1 values of the
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SEMI-INCLUSIVE DIS AND TRANSVERSE M
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The yellow band in Fig. 1 is the re
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At leading twist, a third asymmetry
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THE COMPLETION OF SINGLE-SPIN ASYMM
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A N , % 30 20 10 0 0 0.2 0.4 0.6 0.
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Unexpectedly large single spin asym
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THE GENERALIZED PARTON DISTRIBUTION
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, E) H ~ (H, I Im C 5 4 3 2 1 Q = 1
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Reducing to a common denominator (
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LAMBDA PHYSICS AT HERMES S. Belosto
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analysis was carried out reconstruc
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SPIN PHYSICS AT NICA A. Nagaytsev J
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original software packages (MC simu
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MEASUREMENTS OF TRANSVERSE SPIN EFF
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to “near-side” and “away-side
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THE FIRST STAGE OF POLARIZATION PRO
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In paper [12] the study was made of
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emphasis to increase the statistics
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Table 2: The parameters of the main
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POLARIZATION MEASUREMENTS IN PHOTOP
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With a polarized electron beam inci
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Figure 3: Preliminary helicity asym
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INVESTIGATION OF DP-ELASTIC SCATTER
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framework with the use of deuteron
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SPIN PHYSICS WITH CLAS Y. Prok 12,
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1.3 Flavor Decomposition of the Hel
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Γ p 1 0.15 0.125 0.1 0.075 0.05 0.
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The upcoming energy upgrade of Jeff
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y the nearly 1000 combined citation
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� R = −Pt/Pl τ(1 + ε)/2ε; he
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4 Results of GEp-III Experiment In
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6 Theoretical Developments The earl
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lz = 0 (along the direction of the
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models have recently been challenge
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[32] Chung P. L. and F. Coester, Ph
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48 ± 3% for two beams. The proton
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is constant with cos θ∗ , as exp
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In summary, we made measurements on
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angle between the direction of the
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The sensitivity of the data to the
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COMPASS could be converted into a f
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3.1.2 Beam charge and spin asymmetr
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contributions enter only beyond lea
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References [1] D. Mueller et al, Fo
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l’ l p q p h H (z) 1 h (x) 1 l l
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3. Data selection. The data selecti
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φ sin3 a 0.06 0.04 0.02 0 -0.02 -0
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TRANSVERSE SPIN AND MOMENTUM EFFECT
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model. Both the cos φh and the cos
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D cos φ A 0 -0.1 -0.2 -0.3 + h -2
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4.3 The Sivers asymmetry According
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[3] A. Airapetian et al. [HERMES co
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2. Delta-Sigma Experimental Set-up.
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6. Estimate of the count rate in tr
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2 Analysis Formalism Spin-dependent
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The MC production comprises three s
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6 High pT hadron pair analysis for
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[2] V. Y. Alexakhin et al. [COMPASS
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2 Towards polarized Antiprotons For
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4 Spin-Filtering Experiments at COS
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to test the theoretical models of t
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C1 C2 π CH1,2 CH3,4 CH5,6 111 000
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not even resemble the data behavior
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to the slope of the Rosenbluth plot
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The differential cross section of t
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with zero for all mass ranges, with
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more precise and dedicated measurem
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oth types of experiment results are
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is recorded, a good particle identi
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error of the extracted asymmetries
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2.6 Summary and Outlook The first d
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386
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PROTON BEAM POLARIZATION MEASUREMEN
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N A 0.06 0.05 0.04 0.03 0.02 0.01 0
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Intensity profile (arb. units) 1 0.
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[4] H. Okada et al., Proc. of the 1
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The amplitudes of the signals and t
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The following results has been obta
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interaction vanishes and a single Z
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an amorphous NH3 for low and high,
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and depends on a choice of � ℓ1
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3 The conditions for transparent sp
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where angles α and β are defined
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forward angles, where vector Ay and
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espectively. The open symbols repre
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RF dipole (transverse B): εBdl = 1
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i PV / PV i PV / PV 1 0.5 0 -0.5 -1
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The energies of the states Ψ1 −
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field P ≈ +1. If we add the trans
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econstruction provide more accurate
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y detector saturation from pC colli
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2 �p+ 8 He analyzing power measur
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3.3 Effect of valence neutrons on s
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i.e. n0(θ +2π) =n0(θ) . There ar
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w and ˙ɛ. For example, we use par
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436
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REGULARIZATION OF SOURCE OF THE KER
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The corresponding phase transition
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ABOUT SPIN PARTICLE SOLUTION IN BOR
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where the functions fi(s, η) must
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SPINDYNAMICS I.B. Pestov JINR, 1419
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State of hadron matter is defined t
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REMARK ON SPIN PRECESSION FORMULAE
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It is easy to see that for a (massi
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DYNAMICS OF SPIN IN NONSTATIC SPACE
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Schwinger gauge. Probably for the f
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DSPIN-09 WORKSHOP SUMMARY Jacques S
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to some preliminary results reporte
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A new measurement of the polarizati
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new NLO QCD parametrization of the
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Name (Institution, Town, Country) E
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