Harut Avakian Jefferson Lab, Jefferson Avenue, Newport News, Virginia 23606, USA

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1 Harut Avakian Jefferson Lab, 2000 Jefferson Avenue, Newport News, Virginia 23606, USA Leonard Gamberg Division of Science, Penn State Berks, Reading, PA 960, USA Patrizia Rossi INFN, Laboratori Nazionali di Frascati, Frascati, Italy Jefferson Lab, 2000 Jefferson Avenue, Newport News, Virginia 23606, USA Division of Science, Penn State Berks, Reading, PA 960, USA We review te concept of Bessel weigted asymmetries for semi-inclusive deep inelastic scattering and focus on te cross section in Fourier space, conjugate to te outgoing adron s transverse momentum, were convolutions of transverse momentum dependent parton distribution functions and fragmentation functions become simple products. Individual asymmetric terms in te cross section can be projected out by means of a generalized set of weigts involving Bessel functions. Te procedure is applied to studies of te double longitudinal spin asymmetry in semi-inclusive deep inelastic scattering using a new dedicated Monte Carlo generator wic includes quark intrinsic transverse momentum witin te generalized parton model. We observe a few percent systematic offset of te Bessel-weigted asymmetry obtained from Monte Carlo extraction compared to input model calculations, wic is due to te limitations imposed by te energy and momentum conservation at te given energy and ard scale Q 2. We find tat te Bessel weigting tecnique provides a powerful and reliable tool to study te Fourier transform of TMDs wit controlled systematics due to experimental acceptances and resolutions wit different TMD model inputs. QCD Evolution 205 -QCDEV May 205 Jefferson Lab (JLAB, Newport News Virginia, USA Speaker. c Copyrigt owned by te autor(s under te terms of te Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0. ttp://pos.sissa.it/

2 . Introduction One of te main goals of adron pysics is to provide a partonic description of te treedimensional (3D momentum structure of protons and neutrons. Te nucleon structure is being studied wit many dedicated experiments, recent (HERMES at DESY, Halls A,B and C experiments at Jefferson Lab, running (COMPASS at CERN, STAR and PHENIX at RHIC, approved (JLab 2 GeV upgrade [], COMPASS-II [2] or planned (Electron Ion Collider [3, 4, 5]. Te 3D momentum picture of te nucleon is described by te Transverse Momentum Dependent parton distribution functions and fragmentation functions. Tese Transverse Momentum Dependent distribution and fragmentation functions (collectively ere called TMDs can be accessed in several types of processes, one of te most important is single particle adron production in Semi-Inclusive Deep Inelastic Scattering (SIDIS of leptons on nucleons. A significant amount of data on spin-azimutal distributions of adrons in SIDIS, providing access to TMDs as been accumulated in recent years by several collaborations, including HERMES, COMPASS and Halls A,B and C at Jefferson Lab [6, 7, 8, 9, 0,, 2, 3, 4, 5]. At least an order of magnitude more data is expected in coming years of running of JLab 2 []. QCD factorization teorems generically allow one to separate te sort and long distance pysics, and tus encode information about internal nucleon structure in individual TMDs [6, 7, 8, 9]. In te kinematical domain were transverse momentum of te produced adron is muc smaller tan te ard scale Q, and on te order of Λ QCD, i.e. Λ 2 QCD P2 Q2, te SIDIS cross section can be expressed in terms of structure functions wic are given by convolutions of TMDs were transverse momentum (k dependence of TMDs is integrated over and related to te measured value of P. A reliable metod to directly access te k dependence of TMDs is very desirable. In a paper by Boer, Gamberg, Musc, and Prokudin [20], a new tecnique as been proposed called Bessel weigting, wic relies on a model-independent deconvolution of structure functions in terms of Fourier transforms of TMDs from observed azimutal moments in SIDIS wit polarized and unpolarized targets. A first application of te Bessel weigting Ref. [20] was carried out by Agasyan, Avakian, De Sanctis, Gamberg, Mirazita, Musc, Prokudin, Rossi in Ref. [2]. In tese proceedings we will recapitulate findings of Refs. [20, 2]. 2. Bessel Weigting Te SIDIS cross section can be expressed in a model independent way wit leptonic L µν and adronic tensors W µν [22, 23, 24, 25, 26, 27], dσ dxdydψ dzdφ d P 2 = α2 y 2 xyq 2 ( ε ( + γ2 L µν W µν (2. 2x 2

3 or in terms of a set of 8 structure functions dσ dxdydψ dzdφ d P 2 = α2 y 2 xyq 2 2( ε + 2ε( + ε cosφ F cosφ UU ( { + γ2 F UU,T + ε F UU,L 2x + ε cos(2φ F cos2φ UU + λ e 2ε( ε sinφ F sinφ LU ] + S [ 2ε( + ε sinφ F sinφ UL + ε sin(2φ F sin2φ UL ] + S λ e [ ε 2 F LL + 2ε( ε cosφ F cosφ LL + S [ ( sin(φ φ S F sin(φ φ S UT,T + ε F sin(φ φ S UT,L + ε sin(φ + φ S F sin(φ +φ S UT + ε sin(3φ φ S F sin(3φ φ S UT + 2ε( + ε sinφ S F sinφ S UT + 2ε( + ε sin(2φ φ S F sin(2φ φ S UT [ + S λ e ε 2 cos(φ φ S F cos(φ φ S LT + 2ε( ε cosφ S F cosφ S LT ]} + 2ε( ε cos(2φ φ S F cos(2φ φ S LT, (2.2 were te first two subscripts of te structure functions F XY indicate te polarization of te beam and target, and in certain cases, a tird sub-script in F XY,Z indicates te polarization of te virtual poton. Te structure functions depend on te te scaling variables x, z, te four momentum Q 2 = q 2, were q = l l is te momentum of te virtual poton, and l and l are te 4-momenta of te incoming and outgoing leptons, respectively. P is te transverse momentum component of te produced adron wit respect to te virtual poton direction. Te scaling variables ave te standard definitions, x = Q 2 /2(P q, y = (P q/(p l, and z = (P P /(P q. Furter, in Eq. (2.2 α is te fine structure constant; te angle ψ is te azimutal angle of l around te lepton beam axis wit respect to an arbitrary fixed direction [26], and φ is te azimutal angle between te scattering plane formed by te initial and final momenta of te electron and te production plane formed by te transverse momentum of te observed adron and te virtual poton, wereas φ S is te azimutal angle of te transverse spin in te scattering plane [28]. Finally, ε is te ratio of longitudinal and transverse poton fluxes [27]. At tree-level (parton-model of te ard poton-quark scattering process, and to leading order in te /Q expansion, te adronic tensor can be written in factorized form as [24, 29, 27] 2MW µν = e 2 a a { d 2 p d 2 K T δ (2 (zp + K T P Tr Φ(x, p γ µ (z, K T γ ν}. (2.3 ] 3

4 Te quark-quark correlator [30, 7] in te above equation is defined as Φ i j (p,p,s d 4 b (2π 4 eip b P,S ψ j (0U [C b ]ψ i (b P,S. (2.4 First we use te representation of te δ-function along wit te following definitions, W µν (P Φ i j (x,zb T i j (z, b T δ (2 (zp + K T P = d 2 b T (2π 2 eib T (zp +K T P, (2.5 d 2 b T (2π 2 e ib T P W µν (b T, (2.6 d 2 p e izb T p db b Φ i j (x, p = (2π eixp+ P,S ψ j (0U [C b ]ψ i (b P,S, b + =0 (2.7 d 2 K T e ib T K T i j (z, K T, (2.8 to re-write te leading term in te adronic tensor in Fourier space, 2M W µν = e 2 a Tr ( Φ(x,zb T γ µ (z, b T γ ν. (2.9 a Te advantage of te b T space representation is clear: te adronic tensor is no longer a convolution of p and K T dependent functions but a simple product of b T -dependent functions. Tis motivates us to re-write te entire cross section in terms of te Fourier transform dσ dxdydψ dz dφ P d P = d 2 b T (2π 2 e ib T P { α 2 xyq 2 y 2 ( ε } ( + L γ2 µν W µν. (2.0 2x Te advantage of tis procedure is a possibility to access b T -dependent TMDs directly experimentally by projecting wit appropriate weigts. Indeed, we find tat generically te SIDIS 4

5 cross-section takes te following form: dσ dxdydφ S dz dφ P d P = α2 y 2 xyq 2 ( ε { ( + γ2 d bt 2x (2π b T + J 0 ( b T P F UU,T + ε J 0 ( b T P F UU,L + 2ε( + ε cosφ J ( b T P F cosφ UU + ε cos(2φ J 2 ( b T P F cos(2φ UU + λ e 2ε( ε sinφ J ( b T P F sinφ LU ] + S [ 2ε( + ε sinφ J ( b T P F sinφ UL + ε sin(2φ J 2 ( b T P F sin2φ UL ] + S λ e [ ε 2 J 0 ( b T P F LL + 2ε( ε cosφ J ( b T P F cosφ LL + S [ ( sin(φ φ S J ( b T P F sin(φ φ S UT,T + ε F sin(φ φ S UT,L + ε sin(φ + φ S J ( b T P F sin(φ +φ S UT + ε sin(3φ φ S J 3 ( b T P F sin(3φ φ S UT + 2ε( + ε sinφ S J ( b T P F sinφ S UT + 2ε( + ε sin(2φ φ S J 2 ( b T P F sin(2φ φ S UT + S λ e [ ε 2 cos(φ φ S J ( b T P F cos(φ φ S LT + 2ε( ε cosφ S J 0 ( b T P F cosφ S LT + 2ε( ε cos(2φ φ S J 2 ( b T P F cos(2φ φ S LT ] ]}. (2. Te structure of te cross section is wat one gets from a multipole expansion in b T -space followed by a Fourier transform. Eac of te structure functions F XY,Z in b T -space corresponds to te Hankel (or Fourier-Bessel transform of te corresponding structure function F XY,Z in te usual momentum space representation of te cross section. Te combinations sin(nφ +...J n ( b T P and cos(nφ +...J n ( b T P act as basis functions of te combined transform to ( P,φ - space. Due to te fact tat te multipole expansion of te pysical cross section terminates, only a finite number of terms appear in te cross section, wit J 3 being te Bessel function of igest order. Te structures F XY,Z are functions of b T, x and z, but no longer depend on te angular variables. Introducing a sort-and notation for products P[ f (n D (m ] x B e 2 a (zm b T n (zm b T m f a(n (x,z 2 b 2 T D a(m (z, b 2 T, (2.2 a te leading twist tree level analysis reveals tat te Fourier transformed structures in te cross 5

6 section are simple products of TMD PDFs and TMD FFs F sin(φ φ S UT,T D (0 F UU,T = P[ f (0 = P[ f ( T F LL = P[ g (0 L F cos(φ φ s LT = P[ g ( T F sin(φ +φ S UT = P[ (0 F cos(2φ UU F sin(2φ UL F sin(3φ φ S UT = P[ ( = P[ ( L ], (2.3 D (0 ], (2.4 D (0 ], (2.5 D (0 = (2 P[ T 4 ], (2.6 H ( ], (2.7 H ( ], (2.8 H ( ], (2.9 H ( ]. (2.20 Transverse momentum weigted SSA [24, 3, 25] provide a means to disentangle in a model independent way te cross sections and asymmetries in terms of te transverse (momentum moments of TMD PDFs. Generally tey are given by A W XY = 2 d P P dφ dφ S W ( P,φ,φ S ( dσ (φ,φ S dσ (φ,φ S d P P dφ dφ S (dσ (φ,φ S + dσ (φ,φ S, (2.2 were te labels X,Y represent te polarization, un (U, longitudinally (L and transversely (T of te beam and target, respectively. Te angles φ S and φ specify te directions of te adron spin polarization and te transverse adron momentum respectively, relative to te lepton scattering plane. Te cross sections dσ and dσ correspond to te cases wit opposite transverse spin polarization of te target adron. We ave introduced te sort-and notation W wic is a function containing various powers and P as well as angular dependences of te form sin(mφ ± nφ S or cos(mφ ± nφ S. For te conventional weigted Sivers asymmetry, W w sin(φ φ S, were w = P /zm. Based on te expansion of te SIDIS cross section in terms of Bessel functions J n of transverse momentum and impact parameter in Eq. (2., we exploit te ortogonality to generalize te weigting procedure. Now te weigting is of te form A W d P P dφ dφ S W ( P,φ,φ S,B T ( dσ dσ XY (B T = 2 d P P dφ dφ S J 0 ( P B T (dσ + dσ, (2.22 were te weigt function W corresponds to tat of conventional weigted asymmetries, except tat we replace ( 2 n P n J n ( P B T n!. (2.23 B T Taking te asymptotic form of te Bessel function te conventional weigts [3, 25] wic are P n appear as te leading term of te Taylor expansion of te rigt and side of Eq. (2.23. Furtermore we note tat te parameter B T > 0 regularizes UV divergences in moments of TMD PDFs and FFs. More importantly, we will sow tat te parameter B T > 0 allows us to scan TMD PDFs and TMD FFs in Fourier space. In fact, te form of Eq. (2.22 already indicates tat 6

7 te weigting implements a Fourier-decomposition of te cross section in transverse momentum space. We will illustrate tis metod for te Sivers Bessel weigted asymmetry. One can see from Eq. (2. tat te appropriate weigt for te Sivers asymmetry is W = 2J ( P B T zmb T sin(φ φ S, i.e., w = 2J ( P B T zmb T, (2.24 corresponding to P /zm in te limit P /B T. Ten te Bessel-weigted Sivers asymmetry is 2J ( P B T 2J zmb sin(φ φ S d P P A T dφ dφ ( P B T S zmb UT (B T = 2 T sin(φ φ S ( dσ dσ d P P dφ dφ S J 0 ( P B T (dσ + dσ, (2.25 were te axially symmetric denominator is given by d P P dφ dφ S J 0 ( P B T and from Eq. (2. te numerator is d bt (2π b T J 0 ( b T P F UU,T, (2.26 2J ( P B T d P P dφ dφ S sin 2 d bt (φ φ s zmb T (2π b T J ( b T P F sin(φ φ s UT. (2.27 Finally, making use of te closure relation of te Bessel function we obtain f te Bessel weigted Sivers asymmetry, 2J ( P B T zmb sin(φ φ s A T UT,T (B T = 2 a e 2 a H sin(φ φs UT,T (Q 2, µ 2 f (a T (x,z 2 BT 2 ; µ2,ζ D (0a (z,bt 2 ; µ2, ˆζ a e 2 a H UU,T (Q 2, µ 2 f (0a (x,z 2 BT 2 ; µ2,ζ D (0a (z,bt 2 ; µ2, ˆζ. (2.28 One can see from Eq tat te weiged asymmetry is related to a product of Fourier transformed Sivers function and unpolarised TMD fragmentation function. Details of te metod can be found in Ref [20]. 3. Feasibility studies We studied experimental feasibility of te metod in Ref. [2]. Consider te unpolarized and double longitudinal structure functions, F UU,T = x a e 2 f a a (x,z 2 b 2 T D a (z, b 2 T, F LL = x e 2 a g a L(x,z 2 b 2 T D a (z, b 2 T, (3. a were te Fourier transform of te TMDs are defined as f (x, b 2 T = d 2 k e ib T k f (x, k 2 = 2π dk k J 0 ( b T k f (x, k 2, (3.2 7

8 D(z, b 2 T = d 2 p e ib T p D(z, p 2 = 2π Now we form te double longitudinal spin asymmetry d p p J 0 ( b T p D(x, p 2. (3.3 A J 0(b T P LL (b T = σ + (b T σ (b T σ + (b T + σ (b T σ LL(b T σ UU (b T = ε 2 a e 2 a g a L (x,z2 b 2 T D a (z,b2 T a e 2 a f a (x,z2 b 2 T D a (z,b2 T, (3.4 Note tat in our definition b T is Fourier conjugate variable to P [20]. A Monte Carlo generator is a crucial component in testing experimental procedures suc as tose described in Eq. (3.4. In order to ceck te Bessel weigting tecnique we need a Monte Carlo tat generates events in pase space wit different TMD model input, wic includes explicit dependence on intrinsic parton transverse momentum k and p. In te Monte Carlo generator software, we used te general-purpose, self-adapting event generator, Foam [32], for drawing random points according to an arbitrary, user-defined distribution in n-dimensional space. Implementing te Monte Carlo we generate kinematical distributions in x, z, k, and p of SIDIS events for several model inputs of TMDs. Tese distributions are ten used to ceck te consistency of dependence of extracted quantities under different model assumptions, including, for example Gaussian and non-gaussian distributions in transverse momentum. In case te dependence is assumed to be a Gaussian, x and z dependent widts are assumed, so tat TMDs take te following form, ( f (x, k 2 = f (x k 2 (x exp k2 f k 2 (x, (3.5 f ( g L (x, k 2 = g L (x k 2 (x exp k2 g k 2 (x, (3.6 g ( D (z, p 2 = D (z exp p2 (z p 2 (z, (3.7 p 2 were f (x and D(z are corresponding collinear parton distribution and fragmentation functions and te widts are x and z dependent functions. In our studies we adopt te modified Gaussian distribution functions and fragmentation functions from Eq. (3.5-(3.7, in wic x and k dependencies are inspired by AdS/QCD results [33, 34], wit k 2 (x = Cx( x and p2 (z = Dz( z, were te constants C and D may be different for different flavors and polarization states (see for example [35]. Similarly suc non-factorized x,k distribution functions are also suggested by te diquark spectator model [36] and te NJL-jet model [37]. For te x and z dependence in Eqs. (3.5 and (3.7 we use te parametrizations, f (x = ( x 3 x.33, g L (x = f (xx 0.7, and D (z = 0.8( z 2, using input values C = 0.54 GeV 2 and D = 0.5 GeV 2. We also assume tat k 2 g L = 0.8 k 2 f ; tis assumption is consistent wit lattice studies [38] and experimental measurements [4]. 8

9 As an example of a non-gaussian k distribution we implement te following one inspired by te sape of te resulting distribution in te ligt-cone quark model [39, 40] f (x, k 2 = f (x/ ( k k k6. (3.8 were te coefficients for g L (x, k 2 are cosen in suc a way tat effectively k 2 g L / k 2 f = 0.8. We ten generate events using te cross section for bot Gaussian and non-gaussian initial distributions respectively. Note tat te generator we construct is implemented wit on mass-sell partons and wit four momentum conservation imposed. Wile tis coice is not compulsory we adopt it as it allows us to fully reconstruct kinematics for a given event. At te same time limitations due to available pase space integration will modify te reconstructed distributions wit respect to te input distributions. We analyze te effect of te available pase space in te Monte Carlo on te average k 2 for finite beam energies as a function of x by calculating te effective k2 from te following formula, k 2 (x = d 2 k k 2 dσ MC d 2 k dσ MC = N j= k2 j N, (3.9 were te index j runs over te N Monte Carlo generated events. Note, dσ MC is te cross section of te Monte Carlo simulation modified by imposing te four momenta conservation and on-sell condition for initial quark. Te Monte Carlo generated events are used like experimental events to extract bot te Bessel weigted asymmetry, A J 0(b T P LL, and te ratio of te Fourier transform of g L and f using te Bessel weigting metod described in [20]. Te results are ten compared to te Monte Carlo input. Te Bessel moments are extracted from te Monte Carlo wit 6 GeV beam energy using bot te modified Gaussian type of functions (see Eqs. (3.5-(3.7 and power law-tail like function (see Eq. (3.8. Te numerical results of our studies are summarized and displayed in Figs. and 2 for te modified Gaussian distribution function and for te power law-tail like distribution function inputs respectively. In te left panel of Fig. we sow te Bessel-weigted asymmetry versus b T. Te blue curve labeled BW Input, is te asymmetry calculated analytically using te rigt and side of Eq. (3.4 and te Fourier transformed input distribution functions. We now compare various distributions generated from te Monte Carlo. We plot te generated distribution (full red points labeled BW(P Generated, and te black triangles labeled BW(P Sm + Acc, wic represents te same extraction after experimental smearing and acceptance (using te CLAS detector [4], wic is a quasi-4π detector wit less tan % momentum resolution in te presented bin x = 0.22, and z = 0.5. Analyzing our MC results wit four momenta conservation and target mass correction, we are able to distinguis two effects in te left panels of Figs. and 2:. Solid (blue curve versus triangular (green data points: Te distributions realized in te MC simulation differ from te input distributions. In te MC, te four-momentum conservation does not allow te variables k and p to be sampled independently over te wole integration range, as it would ave to be done to reproduce te unmodified generalized parton model. Te actual k and p distributions realized by te MC differ from te analytic 9

10 input distributions Eqns. (3.5-(3.8 noticeably, especially in teir widts. Te solid (blue curve in te left panel of Figs. and 2 is calculated from te input distributions according to te generalized parton model; te FFs on te rigt and side of Eq. (3.4 cancel exactly in te single flavor scenario. Tus te solid curve can be compared to te triangle saped (green data points, wic ave also been calculated from a ratio of TMD PDFs, Eq. (??, albeit wit te actual distributions realized in te MC. 2. Triangular (green data points vs. circular (red data points: inadequacy of te generalized parton model to describe te data. In a single flavor scenario, te distribution functions D a cancel exactly on te rigt and side of Eq. (3.4. Terefore, tere sould not be any difference between te full asymmetry A J 0(b T P LL (b T of Eq. (3.4 and te ratio of TMD PDFs. However, we do observe a difference between te circular (red data points and te triangular (green data points in te left panels of Figs. and 2. Again, te four-momentum conservation we ave implemented is te reason for te observed difference. Since k and p are no longer sampled and te rigt and side of Eq. (3.4 looses te prerequisites for its derivation and is violated to some degree. Terefore, we see only an incomplete cancellation of FFs for te Monte Carlo events. To an experimentalist wo is concerned about systematic errors attributed to te observables e or se extracts, te first of te two effects above is not an issue. Te purpose of te generalized parton model is to provide a parametrization of te data one observes. Any effect of te underlying BW Input BW (k BW (P BW (P Generated Sm + Acc ratio..05 BW (k BW (k BW (P / BW (P / BW (P Generated Sm+Acc Generated / BW input b T (GeV b T (GeV Figure : Left panel:te ratio of Fourier transforms g L / f and te Bessel weigted asymmetry A J 0(b T P LL plotted versus b T. Te solid curve (blue is te Fourier transform of te input to te Monte Carlo given by Eq. (3.4, red points are generated Monte Carlo events, and triangles down (black represent results of Monte Carlo events after experimental smearing and acceptance at x = 0.22, and z = 0.5. Te triangles up wit dased curve (green are results of te Monte Carlo witout inclusion of fragmentation functions. Rigt panel: Ratios tat represent accuracies of our results. 0

11 BW Input BW (k BW (P BW (P Generated Sm + Acc b T (GeV ratio BW (k / BW (P BW (k / BW (P BW (P Generated Sm+Acc Generated / BW input b T (GeV Figure 2: Te same as in Fig. ( but ere from te power-law tail distribution function based on te Monte Carlo. scattering mecanism tat can be absorbed into te distributions does not contradict te validity of te model. Te only concern one migt ave is tat te distributions become beam energy/q 2 dependent, an issue tat sould be addressed using TMD evolution equations. On te oter and, te second effect presented above can be taken as an indication for systematic uncertainties. If, indeed, te pysical reality does not generate events in accordance wit te functional sape of te generalized parton model, ten using te model for te extraction of distributions necessarily involves systematic errors. Again, we point out tat it is unclear weter te modifications we ave implemented in our MC bring us closer to te pysical reality. Noneteless, te modifications are reasonable and so we believe tey can give us a int about te order of magnitude of systematic errors from te corresponding approximations in te model. One can ten estimate tat for calculations suc as tose performed in Ref. [42], systematic errors in te comparison wit experimental data for b T < 6GeV are of te order of a few percent. For te data wit b T > 6GeV, te effects of four-momentum conservation (difference between red and green points becomes more pronounced, and a fit of data using te generalized parton model witout manifest four-momentum conservation terefore becomes less accurate. 4. Conclusions We find tat te Bessel weigting tecnique provides a powerful and reliable tool to study te Fourier transform of TMDs wit controlled systematics due to experimental acceptances and resolutions wit different TMD models inputs. We plan to expand our studies wit more advanced parton sower and fragmentation mecanisms, as well as to include nuclear modifications in our Monte Carlo and extraction procedure.

12 References [] J. Dudek et al., Eur.Pys.J. A48, 87 (202, arxiv: [2] COMPASS, F. Gauteron et al., (200. [3] H. Gao et al., Eur.Pys.J.Plus 26, 2 (20, arxiv: [4] D. Boer et al., (20, arxiv: [5] A. Accardi et al., (202, arxiv: [6] HERMES, A. Airapetian et al., Pys. Rev. Lett. 94, (2005, ep-ex/ [7] COMPASS, V. Y. Alexakin et al., Pys. Rev. Lett. 94, (2005, ep-ex/ [8] COMPASS, E. S. Ageev et al., Nucl. Pys. B765, 3 (2007, ep-ex/ [9] HERMES, A. Airapetian et al., Pys. Lett. B693, (200, arxiv: [0] Te COMPASS, M. G. Alekseev et al., Pys. Lett. B692, 240 (200, arxiv: [] M. G. Alekseev et al., Eur. Pys. J. C70, 39 (200, arxiv: [2] COMPASS Collaboration, C. Adolp et al., Pys.Lett. B73, 0 (202, arxiv: [3] H. Mkrtcyan et al., Pys. Lett. B665, 20 (2008, arxiv:ep-p/ [4] CLAS, H. Avakian et al., Pys. Rev. Lett. 05, (200, arxiv:ep-ex/ [5] Te Jefferson Lab Hall A Collaboration, X. Qian et al., Pys.Rev.Lett. 07, (20, arxiv: , 6 pages, 2 figures, 2 tables, publised in PRL. [6] J. C. Collins and D. E. Soper, Nucl. Pys. B93, 38 (98. [7] J. C. Collins and D. E. Soper, Nucl.Pys. B94, 445 (982. [8] J. C. Collins, D. E. Soper, and G. F. Sterman, Nucl.Pys. B250, 99 (985. [9] J. Collins, (20, Foundations of Perturbative QCD, Cambridge, UK: Univ.Pr. [20] D. Boer, L. Gamberg, B. Musc, and A. Prokudin, JHEP 0, 02 (20, arxiv: [2] M. Agasyan et al., JHEP 03, 039 (205, arxiv: [22] M. Gourdin, Nucl. Pys. B49, 50 (972. [23] A. Kotzinian, Nucl. Pys. B44, 234 (995, ep-p/ [24] P. J. Mulders and R. D. Tangerman, Nucl. Pys. B46, 97 (996, ep-p/ [25] D. Boer and P. J. Mulders, Pys. Rev. D57, 5780 (998, ep-p/ [26] M. Diel and S. Sapeta, Eur. Pys. J. C4, 55 (2005, arxiv:ep-p/ [27] A. Baccetta et al., JHEP 02, 093 (2007, arxiv:ep-p/ [28] A. Baccetta, U. D Alesio, M. Diel, and C. A. Miller, Pys. Rev. D70, 7504 (2004, arxiv:ep-p/ [29] D. Boer, P. J. Mulders, and F. Pijlman, Nucl. Pys. B667, 20 (2003, ep-p/ [30] D. E. Soper, Pys. Rev. Lett. 43, 847 (979. [3] A. Kotzinian and P. Mulders, Pys.Lett. B406, 373 (997, arxiv:ep-p/

13 [32] S. Jadac, Computer Pysics Communications 52, 55 (2003, arxiv:pysics/ [33] S. J. Brodsky, U of Warsaw, July 3-6 (202. [34] G. F. de Teramond and S. J. Brodsky, Pys.Rev.Lett. 02, 0860 (2009, arxiv: [35] A. Signori, A. Baccetta, M. Radici, and G. Scnell, JHEP 3, 94 (203, arxiv: [36] L. P. Gamberg, G. R. Goldstein, and M. Sclegel, Pys. Rev. D77, (2008, arxiv:ep-p/ [37] H. H. Matevosyan, W. Bentz, I. C. Cloet, and A. W. Tomas, Pys.Rev. D85, 0402 (202, arxiv:.740. [38] B. U. Musc, P. Hagler, J. W. Negele, and A. Scafer, Pys.Rev. D83, (20, arxiv:0.23. [39] B. Pasquini and S. Boffi, Pys.Rev. D76, 0740 (2007, arxiv: [40] B. Pasquini, S. Boffi, A. Efremov, and P. Scweitzer, (2009, arxiv: [4] CLAS Collaboration, B. Mecking et al., Nucl.Instrum.Met. A503, 53 (2003. [42] Z. Lu and B.-Q. Ma, Pys.Rev. D87, (203, arxiv: