We investigate the longitudinal and transverse polarized cross-sections of the leptoproduction of the ρ meson in the high energy limit. Our model is based on the computation of the impact factor γ^{*}(λ_{γ})→ ρ (λ_{ρ}) using the twist expansion in the forward limit which is expressed in the impact parameter space. This treatment involves in the final stage the twist 2 and twist 3 distribution amplitudes (DAs) of the ρ meson and the dipole scattering amplitude. Taking models that exist for the DAs and for the dipole cross-section. We get a phenomenological model for the helicity amplitudes. We compare our predictions with HERA data and get a fairly good description for large enough virtualities of the photon.
PACS number(s): 13.60.Le, 12.39.St, 12.38.Bx.
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Manuscript submitted on 25-11-2013 |
Original Manuscript | High Energy Rho Meson Leptoproduction § |
We study the high energy diffractive leptoproduction of ρ meson
where N is the nucleon target, λ_{ρ} and λ_{γ} are respectively the polarizations of the ρ meson and of the virtual photon. The longitudinal and transverse polarized cross-sections σ_{L} and σ_{T} of the process (1) can be expressed in terms of the helicity amplitudes, which are denoted as T_{λρλγ}. In the limit of high energy in the center of mass of the system, the helicity amplitudes can be factorized, using the factorization scheme, into the convolution of the impact factor
and the unintegrated gluon density^{1}
The t-channel gluon momenta, illustrated in Fig. (1), read
Fig (1) Impact factor representation of the helicity amplitudes. |
Assuming the virtuality of the photon Q (Q^{2} = -q^{2}) is large compared to the QCD scale ΛQCD, the impact factors
[http://dx.doi.org/10.1016/0550-3213(87)90057-5] ], using the collinear factorization on the light-cone. In this approach, the impact factors are parameterized by the leading twist DA of the ρ meson. This computation was extended in refs. [2Anikin IV, Ivanov DY, Pire B, Szymanowski L, Wallon S. On the description of exclusive processes beyond the leading twist approximation Phys Lett B 2010; 682: 413-8.
[http://dx.doi.org/10.1016/j.physletb.2009.11.040] , 3Anikin IV, Ivanov DY, Pire B, Szymanowski L, Wallon S. QCD factorization of exclusive processes beyond leading twist gamma*T → rho T impact factor with twist three accuracy Nucl Phys B 2010; 828: 1-68.
[http://dx.doi.org/10.1016/j.nuclphysb.2009.10.022] ] to obtain the
[http://dx.doi.org/10.1103/PhysRevD.84.054004] ] was based on the results of refs. [1Ginzburg IF, Panfil SL, Serbo VG. The possibility of the experimental investigation of the QCD pomeron Nucl Phys B 1987; 284: 685-705.
[http://dx.doi.org/10.1016/0550-3213(87)90057-5] , 3Anikin IV, Ivanov DY, Pire B, Szymanowski L, Wallon S. QCD factorization of exclusive processes beyond leading twist gamma*T → rho T impact factor with twist three accuracy Nucl Phys B 2010; 828: 1-68.
[http://dx.doi.org/10.1016/j.nuclphysb.2009.10.022] ] and used a model for the proton impact factor inspired from ref. [4Gunion JF, Soper DE. Quark counting and hadron size effects for total cross-sections Phys Rev D 1977; 15: 2617-1.
[http://dx.doi.org/10.1103/PhysRevD.15.2617] ]. The results of this study have led to the conclusion that the soft t-channel gluons have a sizable contribution, which calls for the implementation of the saturation effects in this perturbative approach.
For this aim, in ref. [6Besse A, Szymanowski L, Wallon S. The dipole representation of vector meson electroproduction beyond leading twist Nucl Phys B 2013; 867: 19-60.
[http://dx.doi.org/10.1016/j.nuclphysb.2012.09.011] ], we have performed calculations of the twist 2 and twist 3 impact factors in the impact parameter space. We have shown also the equivalence of obtained results with the ones in momentum space of ref. [3Anikin IV, Ivanov DY, Pire B, Szymanowski L, Wallon S. QCD factorization of exclusive processes beyond leading twist gamma*T → rho T impact factor with twist three accuracy Nucl Phys B 2010; 828: 1-68.
[http://dx.doi.org/10.1016/j.nuclphysb.2009.10.022] ]. The results in the impact parameter representation can be put in the form
where the functions
[http://dx.doi.org/10.1016/S0550-3213(98)00356-3] ] to get explicit expressions for the DAs. This model relies on the conformal expansion of the DAs to separate the longitudinal momentum dependence from the scale dependence in µ. It is customary to call “asymptotic” (AS) the results in the limit
Inserting the expressions (5, 6) for the impact factor in eq. (4) leads to
where
In ref. [8Besse A, Szymanowski L, Wallon S. Saturation effects in exclusive rhoT, rhoL meson electroproduction J High Energy Phys 2013; 1311: 062-108.], we have compared our predictions for the transverse and longitudinal polarized cross-sections, shown in Fig. (2), with the data from H1 [9Aaron FD [H1 Collaboration] et al. Measurement of diffractive scattering of photons with large momentum transfer at HERA J High Energy Phys 2010; 1005: 032-50.]. These predictions are obtained using the dipole scattering amplitude of ref. [10Albacete JL, Armesto N, Milhano JG, Quiroga-Arias P, Salgado CA. AAMQS A non-linear QCD analysis of new HERA data atsmall-x includingheavy quarks Eur Phys J C 2011; 71: 1705-5.
[http://dx.doi.org/10.1140/epjc/s10052-011-1705-3] ], which is based on numerical solutions of the running coupling Balitsky-Kovchegov (rcBK) equation [11Balitsky I. Quark contribution to the small-x evolution of color dipole Phys Rev D 2007; 75: 014001-21. [hep-ph/0609105]
[http://dx.doi.org/10.1103/PhysRevD.75.014001] ]. This model of dipole scattering amplitude allows to account for the saturation effects in our description of the ρ meson leptoproduction. Note that as we use a model of dipole cross-section already fitted on inclusive structure functions then we do not need to adjust value of any parameter. The results are in good agreement with the data for
Fig (2) Left: Total, WW and AS contributions to σ_{T} vs Q^{2} , compared to H1 [9Aaron FD [H1 Collaboration] et al. Measurement of diffractive scattering of photons with large momentum transfer at HERA J High Energy Phys 2010; 1005: 032-50.] data. Right: Total and AS twist 2 contributions to σ_{L} vs Q^{2} compared to H1 data. |
In Fig. (3), we show our predictions for the total cross-section σ of the diffractive leptoproduction of ρ meson and compared then with the data of H1 [9Aaron FD [H1 Collaboration] et al. Measurement of diffractive scattering of photons with large momentum transfer at HERA J High Energy Phys 2010; 1005: 032-50.] and ZEUS [12Chekanov S [ZEUS Collaboration], et al. Exclusive ρ° production in deep inelastic scattering at HERA PMC Phys A 2007; 1: 6-52.], as a function W. The W-dependence of our predictions is given by the dipole cross-section model [10Albacete JL, Armesto N, Milhano JG, Quiroga-Arias P, Salgado CA. AAMQS A non-linear QCD analysis of new HERA data atsmall-x includingheavy quarks Eur Phys J C 2011; 71: 1705-5.
[http://dx.doi.org/10.1140/epjc/s10052-011-1705-3] ]. In this way we obtain a good agreement between the predictions and the data for the W-dependence.
Fig (3) Predictions for the total cross-section σ vs W compared to H1 [9Aaron FD [H1 Collaboration] et al. Measurement of diffractive scattering of photons with large momentum transfer at HERA J High Energy Phys 2010; 1005: 032-50.] (left) and ZEUS [12Chekanov S [ZEUS Collaboration], et al. Exclusive ρ° production in deep inelastic scattering at HERA PMC Phys A 2007; 1: 6-52.] (right) data. |
The success of the model we have presented to describe the W- and the Q^{2}-dependencies with the proper normalizations for large enough Q^{2}, relies on the computations from first principles of the impact factors
The authors confirm that this article content has no conflicts of interest.
^{1 }We denote by
We thank B. Ducloué, K. Golec-Biernat, C. Marquet, S. Munier and B. Pire for interesting discussions and comments on this work. We thank the organizers, the Tel Aviv French ambassy and the French CEA (IPhT and DSM) for support. This work is supported by the P2IO consortium, the Polish Grant NCN No DEC-2011/01/B/ST2/03915, the Joint Research Activity Study of Strongly Interacting Matter (acronym HadronPhysics3, Grant 283286) under the Seventh Framework Programme of the European Community and the French grant ANR PARTONS (ANR-12-MONU-0008-01).
[1] | Ginzburg IF, Panfil SL, Serbo VG. The possibility of the experimental investigation of the QCD pomeron Nucl Phys B 1987; 284: 685-705. [http://dx.doi.org/10.1016/0550-3213(87)90057-5] |
[2] | Anikin IV, Ivanov DY, Pire B, Szymanowski L, Wallon S. On the description of exclusive processes beyond the leading twist approximation Phys Lett B 2010; 682: 413-8. [http://dx.doi.org/10.1016/j.physletb.2009.11.040] |
[3] | Anikin IV, Ivanov DY, Pire B, Szymanowski L, Wallon S. QCD factorization of exclusive processes beyond leading twist gamma*T → rho T impact factor with twist three accuracy Nucl Phys B 2010; 828: 1-68. [http://dx.doi.org/10.1016/j.nuclphysb.2009.10.022] |
[4] | Gunion JF, Soper DE. Quark counting and hadron size effects for total cross-sections Phys Rev D 1977; 15: 2617-1. [http://dx.doi.org/10.1103/PhysRevD.15.2617] |
[5] | Anikin I, Besse A, Yu D, et al. A phenomenological study of helicity amplitudes of high energy exclusive leptoproduction of the rho meson Phys Rev D 2011; 84: 054004-23. [http://dx.doi.org/10.1103/PhysRevD.84.054004] |
[6] | Besse A, Szymanowski L, Wallon S. The dipole representation of vector meson electroproduction beyond leading twist Nucl Phys B 2013; 867: 19-60. [http://dx.doi.org/10.1016/j.nuclphysb.2012.09.011] |
[7] | Ball P, Braun VM, Koike Y, Tanaka K. Higher twist distribution amplitudes of vector mesons in qcd formalism and twist three distributions Nucl Phys B 1998; 529: 323-82. [http://dx.doi.org/10.1016/S0550-3213(98)00356-3] |
[8] | Besse A, Szymanowski L, Wallon S. Saturation effects in exclusive rhoT, rhoL meson electroproduction J High Energy Phys 2013; 1311: 062-108. |
[9] | Aaron FD [H1 Collaboration] et al. Measurement of diffractive scattering of photons with large momentum transfer at HERA J High Energy Phys 2010; 1005: 032-50. |
[10] | Albacete JL, Armesto N, Milhano JG, Quiroga-Arias P, Salgado CA. AAMQS A non-linear QCD analysis of new HERA data atsmall-x includingheavy quarks Eur Phys J C 2011; 71: 1705-5. [http://dx.doi.org/10.1140/epjc/s10052-011-1705-3] |
[11] | Balitsky I. Quark contribution to the small-x evolution of color dipole Phys Rev D 2007; 75: 014001-21. [hep-ph/0609105] [http://dx.doi.org/10.1103/PhysRevD.75.014001] |
[12] | Chekanov S [ZEUS Collaboration], et al. Exclusive ρ° production in deep inelastic scattering at HERA PMC Phys A 2007; 1: 6-52. |