We present studies of double longitudinal spin asymmetries in semi-inclusive
deep inelastic scattering using a new dedicated Monte Carlo generator, which
includes quark intrinsic transverse momentum within the generalized parton
model based on the fully differential cross section for the process.
Additionally, we apply Bessel-weighting to the simulated events to extract
transverse momentum dependent parton distribution functions and also discuss
possible uncertainties due to kinematic correlation effects.Comment: 3rd Workshop on the QCD Structure of the Nucleon (QCD-N'12) October
  22-26, 2012, Bilbao, Spai

Aghasyan, M.

Avakian, H.

English

arXiv

We present studies of double longitudinal spin asymmetries in semi-inclusive deep inelastic scattering using a new dedicated Monte Carlo generator, which includes quark intrinsic transverse momentum within the generalized parton model based on the fully differential cross section for the process. Additionally, we apply Bessel-weighting to the simulated events to extract transverse momentum dependent parton distribution functions and also discuss possible uncertainties due to kinematic correlation effects

Aghasyan, Mher M.

Avakian, Harut A.

UNT Digital Library

IL NUOVO CIMENTO Vol. ?, N. ? ?Fully Differential Monte-Carlo Generator Dedicated to TMDs andBessel-Weighted AsymmetriesM. Aghasyan(1), H. Avakian(2)(1) INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy(2) JLab, 12000 Jefferson Ave, Newport News, VA 23606, USASummary. — We present studies of double longitudinal spin asymmetries in semi-inclusive deep inelastic scattering using a new dedicated Monte Carlo generator,which includes quark intrinsic transverse momentum within the generalized partonmodel based on the fully differential cross section for the process. Additionally,we apply Bessel-weighting to the simulated events to extract transverse momentumdependent parton distribution functions and also discuss possible uncertainties dueto kinematic correlation effects.PACS 13.60.-r, 13.87.Fh, 13.88.+e –PACS . – 14.20.Dh, 24.85.+p.1. – Fully differential SIDIS cross sectionThe transverse momentum dependent (TMD) partonic distribution and fragmentationfunctions play a crucial role in measuring and interpreting information towards a true 3-dimensional imaging of the nucleons. TMD PDFs can be accessed in several experiments,but the main source of information is semi-inclusive deep inelastic scattering (SIDIS) ofpolarized leptons off polarized nucleon. For SIDIS, the theoretical formalism is describedin a series of papers [1, 2] using a tree level factorization [3] where the standard momentumconvolution integral [4] relates the quark intrinsic transverse momentum in a nucleon tothe transverse momentum of the produced hadron PhT .In this work we present a model independent extraction of the ratio of polarized, g1L,and unpolarized, f1, TMD distributions using a Monte Carlo (MC) generator based onthe fully differential cross section, in which we reconstruct the transverse momentumof the final hadron after MC integration over the quark intrinsic transverse momen-tum. In the MC generator we used the SIDIS cross section described in Ref. [1]. TheBessel-weighted asymmetry, providing access to the ratio of Fourier transforms of g1Land f1 [5], has been extracted. The uncertainty of the extracted TMDs was estimatedusing unintegrated, transverse momentum non-factorized distribution and fragmentationfunctions.c© Società Italiana di Fisica 1arXiv:1303.3722v1  [hep-ph]  15 Mar 20132 M. AGHASYAN ETC.The MC generator software employs the general-purpose, self-adapting MC eventgenerator Foam [6] for drawing random points according to an arbitrary, user-defineddistribution in the n-dimensional space.We consider the following SIDIS process(1) `(l) +N(P )→ `(l′) + h(Ph) +X,were ` -is the scattered lepton, N is the proton target and h-is the observed hadron(four-momenta notations given in parentheses). The virtual photon momentum is definedq = l − l′ and its virtuality Q2 = −q2. Following the Trento conventions [7], we use thevirtual photon-nucleon center of mass system, where the virtual photon momentum q isalong the z axis and the proton momentum P is in the opposite direction. The detectedhadron h has momentum Ph. In the parton model the virtual photon scatters off anon-shell quark. The initial quark momentum k and scattered quark momentum k′ havethe same intrinsic transverse momentum component k⊥ with respect to the z axis. Theinitial quark has a fraction x of the proton’s light-cone momentum, while the producedhadron h has a light-cone momentum fraction z and transverse momenta component p⊥with respect to the scattered quark’s momentum k′ (see Ref. [1]). The fully differentialcross section used in the MC generator is then given by [1]:dσdxdydzd2p⊥d2k⊥= K(x, y)J(x,Q2, k⊥)××[∑qf1,q(x, k⊥)D1,q(z, p⊥) + λ√1− ε2g1L,q(x, k⊥)D1,q(z, p⊥)](2)where the summation runs over the quark flavors in the target. The kinematic factorsK(x, y) and ε, and the Jacobian J(x,Q2, k⊥) are defined in [1]. λ represents the productof lepton and nucleon helicities (λ = ±1). f1,q(x, k⊥) and g1L,q(x, k⊥) are the quark TMDdistributions andD1,q(z, p⊥) is the TMD fragmentation function for an unpolarized quarkq.In many phenomenological studies of the semi-inclusive deep inelastic scattering, thetransverse momentum dependence of distribution and fragmentation functions is fac-torized from the light-cone momentum dependence x and z, respectively. In our MCgenerator we use the Bessel-weighting method [5] to alleviate this approximations inthe extraction of TMD distributions from SIDIS measurements. We discuss the follow-ing modified Gaussian (MG) expressions for the TMD distributions and fragmentationfunction, inspired by AdS/QCD [8, 9], in which x and k⊥ (z and p⊥) are not factorized:(3) f1(x, k⊥) = f1(x)e− k2⊥x(1−x)〈k2⊥〉f1x(1− x)〈k2⊥〉f1, g1L(x, k⊥) = g1L(x)e− k2⊥x(1−x)〈k2⊥〉g1x(1− x)〈k2⊥〉g1(4) D1(z, p⊥) = D1(z)e− p2⊥z(1−z)〈p2⊥〉z(1− z)〈p2⊥〉TMD distributions non-factorized in x and k⊥ are also suggested by the diquark spec-tator model [10] and the NJL model [11]. For a test extraction, the x and z dependenceFULLY DIFFERENTIAL MONTE-CARLO GENERATOR DEDICATED TO TMDS AND BESSEL-WEIGHTED ASYMMETRIES3x0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45>2<k0.020.040.060.080.10.120.140.160.180.20.222/c2>=0.54 GeV2>, input <k2MG Eff <k2/c2>=0.75 GeV2>, input <k2MG Eff <kFig. 1. – (Color online) The effective 〈k2⊥(x)〉 in the modified Gaussian (MG) ansatz for the TMD distributions(see Eq. 3) as function of x for two different input parameters < k2⊥〉 obtained from MC events simulated withinthe range 0.50 < z < 0.52 and using 6 GeV (solid symbols) and 11 GeV (empty symbols) lepton beam energies.in Eqs. (3-4) we use the parametrizations f1(x) = (1 − x)3 x−1.313, g1L(x) = f1(x)x0.7and D1(z) = 0.8 (1 − z)2, with widths: 〈k2⊥〉f1 = 0.54 GeV2 [8], 〈k2⊥〉g1 = 0.8〈k2⊥〉f1and 〈p2⊥〉 = 0.14 GeV2. Fig. 1 illustrates the average effective transverse momentum〈k2⊥〉 = 〈k2⊥〉f1(1 − 〈x〉)〈x〉 as function of x from MC events for 6 GeV (solid symbols)and 11 GeV (empty symbols) incoming electron beam energies and produced hadronswithin 0.5 < z < 0.52. Note that the value for 〈k2⊥〉f1 obtained from the MC events isalways smaller than the implemented value due to energy and momentum conservation[12]. A similar non-flat dependence of the average quark transverse momenta versus xand z is also observed in NJL-jet model [11].2. – Bessel-weighted extraction of the double spin asymmetry ALLWe present the extraction of the double spin asymmetry ALL, defined as the ratioof the difference and the sum of electroproduction cross sections for antiparallel, σ+,and parallel, σ−, configurations of lepton and nucleon spins, using the Bessel-weightingprocedure described in [5] and applied in [13]. Within this approach, one can extract theFourier transform of the double spin asymmetry, AJ0(bTPhT )LL (bT ), defined as(5)AJ0(bTPhT )LL (bT ) =σ̃+(bT )− σ̃−(bT )σ̃+(bT ) + σ̃−(bT )=σ̃LL(bT )σ̃UU (bT )=√1− ε2∑q g̃q1(x, z2b2T )D̃q1(z, b2T )∑q f̃q1 (x, z2b2T )D̃q1(z, b2T ),4 M. AGHASYAN ETC.using measured double spin asymmetries as functions of PhT [14], for fixed x, y, and zbins. Here bT is the Fourier conjugate of the PhT . The Fourier transforms of the helicitydependent cross sections, σ±(bT ), can be extracted by integration (analytic models) orsummation (for data and MC) over the hadronic transverse momentum, weighted by aBessel function J0,(6) σ̃±(bT ) ' S± =N±∑i=1J0(bTPhT,i) .)-1(GeVTb0 1 2 3 4 5 6 7 8 9 100.340.360.380.40.420.44 z)T(b1 f~z)/T(b1 g~)-+S+)/(S--S+(S corr×)  -+S+)/(S--S+(Smaxline integrated up to k =  0.224, z= 0.510B xFig. 2. – (Color online) Bessel-weighted asymmetry vs bT with and without the correction, together withanalytical and numerical comparison from the MC with PDFs and FFs from Eq. 3. See the text for moredetails.The Bessel-weighted asymmetry obtained from the simulated events is shown in Fig. 2as function of bT with filled (red) circles, while the analytic expressiong̃1(x,zbT )f̃1(x,zbT )using〈k2⊥〉g1 and 〈k2⊥〉f1 from the fits to k2⊥ distributions from the same MC sample is depictedby the (blue) full line. For values bT < 6 GeV−1, which corresponds to about 1 fm,the Bessel-weighted asymmetries could be extracted with an accuracy of 2.5%, althoughwith a systematic shift. This clear systematic shift between the extracted and calculatedasymmetries is due to the kinematic restrictions introduced by energy and momentumconservation(1) as well as binning effects, which deform the Gaussian shapes of the k⊥(1) In the light-cone coordinate system the transverse component is less or equal to the mo-mentum component along the light-cone vector.FULLY DIFFERENTIAL MONTE-CARLO GENERATOR DEDICATED TO TMDS AND BESSEL-WEIGHTED ASYMMETRIES5and p⊥ distributions. In experiments, there is always a cutoff at high PhT due to accep-tance and the small cross section, as well as a cutoff at small PhT where the azimuthalangles are not well defined due to the experimental resolution. These restrictions in PhTdirectly affect the extracted k⊥ and p⊥ distributions and yield to the mentioned distor-tion of Gaussian shapes which result in the systematic shift between the extracted andthe calculated asymmetries. Obviously, this shift depends on experimentally introducedrestrictions for the accessible PhT range.We discuss two approaches which take these conditions into account, one correctsthe data (simulated MC events) the other applies limits on the integration range for theintrinsic transverse parton momenta when calculating the asymmetry. In order to correctthe data, the contributions from the missing PhT ranges outside the accessible values isestimated by choosing a certain model for the parton transverse momentum dependence,e.g. a Gaussian distribution. This model dependent correction of data is shown in Fig. 2by the (blue) filled squares. The extracted asymmetry does now match the theoreticalcurve for values bT < 6 GeV−1. Alternatively, limited numerical integration over intrinsictransverse momenta in the calculation of the asymmetry, where the integration limitscorrespond to the accessible experimental PhT range, yields a calculated asymmetry thatdescribes correctly the experimental situation without introducing a model dependence.This is shown in Fig. 2 by the (black) open squares.In summary, we used a fully differential Monte Carlo event generator to study howthe implemented quark transverse momentum distributions k⊥ and p⊥ are being changeddue to the energy and momentum conservation low and kinematic correlations withinthe generalized parton model. We then use the obtained k⊥ and p⊥ distributions toapply the Bessel-weighting strategy for the extraction of TMD distributions. As examplewe calculate the Bessel-weighted double spin asymmetry AJ0(bTPhT )LL (bT ). Within thebT range of bT < 6 GeV−1, which corresponds to about 1 fm, the Bessel-weightedasymmetries could be extracted within 2.5% accuracy.∗ ∗ ∗We thank M. Anselmino, U. D’Alesio, S. Melis, L. Gamberg, A. Prokudin, A. Kotzinian,S. Brodsky, B. Musch, D. Boer and H. Matevosyan for useful and stimulating discussions.This work was supported by the U.S. Department of Energy and the National ScienceFoundation, the Italian Istituto Nazionale di Fisica Nucleare.REFERENCES[1] Anselmino M., Boglione M., D’Alesio U., Kotzinian A., Murgia F. and ProkudinA., Phys. Rev. D, 71 (2005) 074006.URL http://link.aps.org/doi/10.1103/PhysRevD.71.074006[2] Boglione M., Melis S. and Prokudin A., Phys. Rev. D, 84 (2011) 034033.URL http://link.aps.org/doi/10.1103/PhysRevD.84.034033[3] Mulders P. J. and Tangerman R. D., Nucl. Phys.B, 461 (1996) 197.[4] Bacchetta A., Diehl M., Goeke K., Metz A., Mulders P. J. et al., JHEP, 0702(2007) 093.[5] Boer D., Gamberg L., Musch B. and Prokudin A., JHEP, 1110 (2011) 021.[6] Jadach S., Computer Physics Communications, 152 (2003) 55.[7] Bacchetta A., D’Alesio U., Diehl M. and Miller C. A., Phys. Rev.D, 70 (2004)117504.[8] Brodsky S. J., U of Warsaw, July 3-6, (2012) .6 M. AGHASYAN ETC.[9] de Téramond G. F. and Brodsky S. J., Phys. Rev. Lett., 102 (2009) 081601.URL http://link.aps.org/doi/10.1103/PhysRevLett.102.081601[10] Gamberg L. P., Goldstein G. R. and Schlegel M., Phys. Rev. D, 77 (2008) 094016.URL http://link.aps.org/doi/10.1103/PhysRevD.77.094016[11] Matevosyan H. H., Bentz W., Cloet I. C. and Thomas A. W., Phys.Rev.D, 85 (2012)014021.[12] Aghasyan M., publication in preparation, (2013) .[13] Lu Z. and Ma B.-Q., arXiv:1212.6864, (2012) .[14] Avakian H. et al., Phys. Rev. Lett., 105 (2010) 262002.

Fully Differential Monte-Carlo Generator Dedicated to TMDs and Bessel-Weighted Asymmetries

We present studies of double longitudinal spin asymmetries in semiinclusive deep inelastic scattering using a new dedicated Monte Carlo generator, which includes quark intrinsic transverse momentum within the generalized parton

model based on the fully differential cross section for the process. Additionally, we apply Bessel-weighting to the simulated events to extract transverse momentum dependent parton distribution functions and also discuss possible uncertainties due to kinematic correlation effects

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2013-11559-5Colloquia: QCDN12IL NUOVO CIMENTO Vol. 36 C, N. 5 Settembre-Ottobre 2013Fully differential Monte Carlo generator dedicated to TMDsand Bessel-weighted asymmetriesM. Aghasyan(1) and H. Avakian(2)(1) INFN, Laboratori Nazionali di Frascati - 00044 Frascati (RM), Italy(2) JLab - 12000 Jefferson Ave, Newport News, VA 23606, USAricevuto il 18 Aprile 2013Summary. — We present studies of double longitudinal spin asymmetries in semi-inclusive deep inelastic scattering using a new dedicated Monte Carlo generator,which includes quark intrinsic transverse momentum within the generalized partonmodel based on the fully differential cross section for the process. Additionally,we apply Bessel-weighting to the simulated events to extract transverse momentumdependent parton distribution functions and also discuss possible uncertainties dueto kinematic correlation effects.PACS 13.60.-r – Photon and charged-lepton interactions with hadrons.PACS 13.87.Fh – Fragmentation into hadrons.PACS 13.88.+e – Polarization in interactions and scattering.PACS 14.20.Dh – Protons and neutrons.1. – Fully differential SIDIS cross sectionThe transverse momentum dependent (TMD) partonic distribution and fragmentationfunctions play a crucial role in measuring and interpreting information towards a true 3-dimensional imaging of the nucleons. TMD PDFs can be accessed in several experiments,but the main source of information is semi-inclusive deep inelastic scattering (SIDIS) ofpolarized leptons off polarized nucleon. For SIDIS, the theoretical formalism is describedin a series of papers [1,2] using a tree level factorization [3] where the standard momentumconvolution integral [4] relates the quark intrinsic transverse momentum in a nucleon tothe transverse momentum of the produced hadron PhT .In this work we present a model independent extraction of the ratio of polarized, g1L,and unpolarized, f1, TMD distributions using a Monte Carlo (MC) generator based onthe fully differential cross section, in which we reconstruct the transverse momentumof the final hadron after MC integration over the quark intrinsic transverse momen-tum. In the MC generator we used the SIDIS cross section described in ref. [1]. TheBessel-weighted asymmetry, providing access to the ratio of Fourier transforms of g1Lc© Societa` Italiana di Fisica 195196 M. AGHASYAN and H. AVAKIANand f1 [5], has been extracted. The uncertainty of the extracted TMDs was estimatedusing unintegrated, transverse momentum non-factorized distribution and fragmentationfunctions.The MC generator software employs the general-purpose, self-adapting MC eventgenerator Foam [6] for drawing random points according to an arbitrary, user-defineddistribution in the n-dimensional space.We consider the following SIDIS process(l) + N(P ) → (l′) + h(Ph) + X,(1)where  is the scattered lepton, N is the proton target and h is the observed hadron(four-momenta notations given in parentheses). The virtual photon momentum is definedq = l − l′ and its virtuality Q2 = −q2. Following the Trento conventions [7], we use thevirtual photon-nucleon center of mass system, where the virtual photon momentum q isalong the z axis and the proton momentum P is in the opposite direction. The detectedhadron h has momentum Ph. In the parton model the virtual photon scatters off anon-shell quark. The initial quark momentum k and scattered quark momentum k′ havethe same intrinsic transverse momentum component k⊥ with respect to the z axis. Theinitial quark has a fraction x of the proton’s light-cone momentum, while the producedhadron h has a light-cone momentum fraction z and transverse momenta component p⊥with respect to the scattered quark’s momentum k′ (see ref. [1]). The fully differentialcross section used in the MC generator is then given by [1]:dσdxdydzd2p⊥d2k⊥= K(x, y)J(x,Q2, k⊥)[∑qf1,q(x, k⊥)D1,q(z, p⊥)(2)+λ√1− ε2g1L,q(x, k⊥)D1,q(z, p⊥)],where the summation runs over the quark flavors in the target. The kinematic factorsK(x, y) and ε, and the Jacobian J(x,Q2, k⊥) are defined in [1]. λ represents the productof lepton and nucleon helicities (λ = ±1). f1,q(x, k⊥) and g1L,q(x, k⊥) are the quarkTMD distributions and D1,q(z, p⊥) is the TMD fragmentation function for an unpolarizedquark q.In many phenomenological studies of the semi-inclusive deep inelastic scattering, thetransverse momentum dependence of distribution and fragmentation functions is fac-torized from the light-cone momentum dependence x and z, respectively. In our MCgenerator we use the Bessel-weighting method [5] to alleviate this approximations inthe extraction of TMD distributions from SIDIS measurements. We discuss the follow-ing modified Gaussian (MG) expressions for the TMD distributions and fragmentationfunction, inspired by AdS/QCD [8,9], in which x and k⊥ (z and p⊥) are not factorized:f1(x, k⊥) = f1(x)e− k2⊥x(1−x)〈k2⊥〉f1x(1− x)〈k2⊥〉f1, g1L(x, k⊥) = g1L(x)e− k2⊥x(1−x)〈k2⊥〉g1x(1− x)〈k2⊥〉g1(3)D1(z, p⊥) = D1(z)e− p2⊥z(1−z)〈p2⊥〉z(1− z)〈p2⊥〉.(4)FULLY DIFFERENTIAL MONTE CARLO GENERATOR DEDICATED TO TMDs ETC. 197x0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45>2<k0.020.040.060.080.10.120.140.160.180.20.222/c2>=0.54 GeV2>, input <k2MG Eff <k2/c2>=0.75 GeV2>, input <k2MG Eff <kFig. 1. – (Color online) The effective 〈k2⊥(x)〉 in the modified Gaussian (MG) ansatz for the TMDdistributions (see eq. (3)) as function of x for two different input parameters 〈k2⊥〉 obtained fromMC events simulated within the range 0.50 < z < 0.52 and using 6GeV (solid symbols) and11GeV (empty symbols) lepton beam energies.TMD distributions non-factorized in x and k⊥ are also suggested by the diquark spec-tator model [10] and the NJL model [11]. For a test extraction, the x and z dependencein eqs. (3), (4) we use the parametrizations f1(x) = (1− x)3 x−1.313, g1L(x) = f1(x)x0.7and D1(z) = 0.8 (1 − z)2, with widths: 〈k2⊥〉f1 = 0.54GeV2 [8], 〈k2⊥〉g1 = 0.8〈k2⊥〉f1and 〈p2⊥〉 = 0.14GeV2. Figure 1 illustrates the average effective transverse momentum〈k2⊥〉 = 〈k2⊥〉f1(1 − 〈x〉)〈x〉 as a function of x from MC events for 6GeV (solid symbols)and 11GeV (empty symbols) incoming electron beam energies and produced hadronswithin 0.5 < z < 0.52. Note that the value for 〈k2⊥〉f1 obtained from the MC eventsis always smaller than the implemented value due to energy and momentum conserva-tion [12]. A similar non-flat dependence of the average quark transverse momenta versusx and z is also observed in NJL-jet model [11].2. – Bessel-weighted extraction of the double spin asymmetry ALLWe present the extraction of the double spin asymmetry ALL, defined as the ratioof the difference and the sum of electroproduction cross sections for antiparallel, σ+,and parallel, σ−, configurations of lepton and nucleon spins, using the Bessel-weightingprocedure described in [5] and applied in [13]. Within this approach, one can extract theFourier transform of the double spin asymmetry, AJ0(bT PhT )LL (bT ), defined asAJ0(bT PhT )LL (bT )=σ˜+(bT )−σ˜−(bT )σ˜+(bT )+σ˜−(bT )=σ˜LL(bT )σ˜UU (bT )=√1−ε2∑q g˜q1(x, z2b2T )D˜q1(z, b2T )∑q f˜q1 (x, z2b2T )D˜q1(z, b2T ),(5)using measured double spin asymmetries as functions of PhT [14], for fixed x, y, and zbins. Here bT is the Fourier conjugate of the PhT . The Fourier transforms of the helicitydependent cross sections, σ±(bT ), can be extracted by integration (analytic models) or198 M. AGHASYAN and H. AVAKIAN)-1(GeVTb0 1 2 3 4 5 6 7 8 9 100.340.360.380.40.420.44 z)T(b1 f~z)/T(b1 g~)-+S+)/(S--S+(S corr×)-+S+)/(S--S+(Smaxline integrated up to k =  0.224, z= 0.510B xFig. 2. – (Color online) Bessel-weighted asymmetry vs bT with and without the correction,together with analytical and numerical comparison from the MC with PDFs and FFs fromeq. (3). See the text for more details.summation (for data and MC) over the hadronic transverse momentum, weighted by aBessel function J0,σ˜±(bT )  S± =N±∑i=1J0(bTPhT,i).(6)The Bessel-weighted asymmetry obtained from the simulated events is shown in fig. 2as a function of bT with filled (red) circles, while the analytic expressiong˜1(x,zbT )f˜1(x,zbT )using〈k2⊥〉g1 and 〈k2⊥〉f1 from the fits to k2⊥ distributions from the same MC sample is depictedby the (blue) full line. For values bT < 6GeV−1, which corresponds to about 1 fm,the Bessel-weighted asymmetries could be extracted with an accuracy of 2.5%, althoughwith a systematic shift. This clear systematic shift between the extracted and calculatedasymmetries is due to the kinematic restrictions introduced by energy and momentumconservation(1) as well as binning effects, which deform the Gaussian shapes of the k⊥and p⊥ distributions. In experiments, there is always a cutoff at high PhT due to accep-tance and the small cross section, as well as a cutoff at small PhT where the azimuthalangles are not well defined due to the experimental resolution. These restrictions in PhTdirectly affect the extracted k⊥ and p⊥ distributions and yield to the mentioned distor-tion of Gaussian shapes which result in the systematic shift between the extracted andthe calculated asymmetries. Obviously, this shift depends on experimentally introducedrestrictions for the accessible PhT range.(1) In the light-cone coordinate system the transverse component is less than or equal to themomentum component along the light-cone vector.FULLY DIFFERENTIAL MONTE CARLO GENERATOR DEDICATED TO TMDs ETC. 199We discuss two approaches which take these conditions into account, one correctsthe data (simulated MC events) the other applies limits on the integration range for theintrinsic transverse parton momenta when calculating the asymmetry. In order to correctthe data, the contributions from the missing PhT ranges outside the accessible values isestimated by choosing a certain model for the parton transverse momentum dependence,e.g. a Gaussian distribution. This model dependent correction of data is shown in fig. 2by the (blue) filled squares. The extracted asymmetry does now match the theoreticalcurve for values bT < 6GeV−1. Alternatively, limited numerical integration over intrinsictransverse momenta in the calculation of the asymmetry, where the integration limitscorrespond to the accessible experimental PhT range, yields a calculated asymmetry thatdescribes correctly the experimental situation without introducing a model dependence.This is shown in fig. 2 by the (black) open squares.In summary, we used a fully differential Monte Carlo event generator to study howthe implemented quark transverse momentum distributions k⊥ and p⊥ are being changeddue to the energy and momentum conservation low and kinematic correlations within thegeneralized parton model. We then use the obtained k⊥ and p⊥ distributions to apply theBessel-weighting strategy for the extraction of TMD distributions. As example we calcu-late the Bessel-weighted double spin asymmetry AJ0(bT PhT )LL (bT ). Within the bT range ofbT < 6GeV−1, which corresponds to about 1 fm, the Bessel-weighted asymmetries couldbe extracted within 2.5% accuracy.∗ ∗ ∗We thank M. Anselmino, U. D’Alesio, S. Melis, L. Gamberg, A. Prokudin, A.Kotzinian, S. Brodsky, B. Musch, D. Boer and H. Matevosyan for useful and stimu-lating discussions. This work was supported by the U.S. Department of Energy and theNational Science Foundation, the Italian Istituto Nazionale di Fisica Nucleare.REFERENCES[1] Anselmino M., Boglione M., D’Alesio U., Kotzinian A., Murgia F. andProkudin A., Phys. Rev. D, 71 (2005) 074006; http://link.aps.org/doi/10.1103/PhysRevD.71.074006.[2] Boglione M., Melis S. and Prokudin A., Phys. Rev. D, 84 (2011) 034033;http://link.aps.org/doi/10.1103/PhysRevD.84.034033.[3] Mulders P. J. and Tangerman R. D., Nucl. Phys. B, 461 (1996) 197.[4] Bacchetta A., Diehl M., Goeke K., Metz A., Mulders P. J. et al., JHEP, 02 (2007)093.[5] Boer D., Gamberg L., Musch B. and Prokudin A., JHEP, 10 (2011) 021.[6] Jadach S., Comput. Phys. Commun., 152 (2003) 55.[7] Bacchetta A., D’Alesio U., Diehl M. and Miller C. A., Phys. Rev. D, 70 (2004)117504.[8] Brodsky S. J., University of Warsaw, July 3-6 (2012).[9] de Te´ramond G. F. and Brodsky S. J., Phys. Rev. Lett., 102 (2009) 081601;http://link.aps.org/doi/10.1103/PhysRevLett.102.081601.[10] Gamberg L. P., Goldstein G. R. and Schlegel M., Phys. Rev. D, 77 (2008) 094016:http://link.aps.org/doi/10.1103/PhysRevD.77.094016.[11] Matevosyan H. H., Bentz W., Cloet I. C. and Thomas A. W., Phys. Rev. D, 85(2012) 014021.[12] Aghasyan M., in preparation (2013).[13] Lu Z. and Ma B.-Q., Phys. Rev. D, 87 (2013) 034037.[14] Avakian H. et al., Phys. Rev. Lett., 105 (2010) 262002.

Fully Differential Monte-Carlo Generator Dedicated to TMDs and Bessel-Weighted Asymmetries

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