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generalized parton distributions ( gpds ) have been introduced independently by mueller _ et al . _
@xcite , radyushkin @xcite and ji @xcite .
they provide unique information about the 3d structure and the spin structure of the nucleon .
they have been continuously at the heart of an intense theoretical and experimental activity as can be testified by the different reviews of this field @xcite .    deeply virtual compton scattering ( dvcs ) immediately appeared as the most promising channel to access gpds @xcite .
its crossed process , timelike compton scattering ( tcs ) , attracted attention later @xcite and , even if the experimental situation is not as mature as in the dvcs case , much progress @xcite is expected in forthcoming years .    with the jefferson lab upgrade at 12  @xmath1 and the beginning of the compass - ii experiment , the field of gpds will enter an era of unprecedented precision . in this work
we explore the consequences of the inclusion of next - to - leading order ( nlo ) gluon coefficient functions and nlo corrections to the quark coefficient functions entering both dvcs and tcs amplitudes .
firstly , we compute spacelike and timelike compton form factors ( cffs ) with two models of gpds .
secondly , we evaluate specific observables in kinematic conditions soon accessible in lepton nucleon collisions .
leading order ( lo ) and some nlo contributions to the dvcs and tcs amplitudes are shown on fig .  [
fig : feynamn - diagrams - compton - scattering - lo - and - nlo ] .
we denote the virtuality of the absorbed ( dvcs ) or emitted ( tcs ) virtual photon by @xmath2 , the skewness variables by @xmath3 ( dvcs ) and @xmath4 ( tcs ) , the momentum transfer on the nucleon by @xmath5 and the factorization scale by @xmath6 .
see ref .
@xcite for detailed notations . at lo , only quark gpds contribute to these processes , but nlo contributions include contributions due to quark and gluon gpds as well .
nlo coefficient functions have been computed for the first time in ref .
@xcite for the dvcs case , and in ref .
@xcite for the tcs case .
a first - principle relation between the dvcs and tcs coefficient functions at nlo has been established in ref .
@xcite .
the compton scattering amplitudes can be expressed in terms of cffs . generally speaking the quark cff @xmath7
is related to the unpolarized quark and gluon gpds @xmath8 and @xmath9 by the following relation : @xmath10 where we consider the singlet combination @xmath11 .
( [ eq : def - cff - h - all - orders ] ) has the following simple form at lo expressed in terms of the born coefficient function @xmath12 : @xmath13 and the more complex form at nlo : @xmath14 \nonumber \\ & & \quad +   \int_{-1}^{+1 } dx \ , h_g ( x,\xi , \mu_f ) \left [ c_1^g\big (   x , \xi , \alpha_s ( \mu_f ) \big ) + \frac{1}{2 } \ln \frac{| q^2 |}{\mu_f^2 } c_{\textrm{coll}}^g\big (   x , \xi , \alpha_s ( \mu_f ) \big ) \right ]    \label{eq : def - cff - h - nlo}\end{aligned}\ ] ] see ref .
@xcite for explicit expressions of the coefficient functions @xmath15 , @xmath16 and @xmath17 .    the differences between eqs .
( [ eq : def - cff - h - lo ] ) and ( [ eq : def - cff - h - nlo ] ) have some consequences regarding gpd extractions . indeed the expression of the imaginary part of the cff @xmath7 changes from @xmath18 to : @xmath19 where @xmath20 is a function of @xmath3 . the integral probes the gpd @xmath8 at values of @xmath21 and the whole expression involve gluon contributions .
there is no more direct link between the imaginary part of the cff @xmath7 and the value of the gpd @xmath8 on the cross - over line , even in the valence region where @xmath22 is expected to be small . while it is still possible and valuable to extract cffs in an almost model - independent way along the lines of ref .
@xcite , the interpretation of the extracted cff does not seem transparent anymore .
in this section we evaluate dvcs and tcs cffs with two gpd models based on double distributions ( dds ) @xcite .
dds naturally achieve one of the strongest constraints on gpds : the polynomiality of the mellin moments of gpds .
they also automatically restore usual pdfs in the forward limit .
the goloskokov  -  kroll ( gk ) model was developed in a series of papers @xcite to study deeply virtual meson electroproduction ( dvmp ) .
it has been recently tested againts dvcs data in a systematic way @xcite .
the gk model contains the following ingredients :    * the radyushkin dd factorized ansatz ( rdda ) . for @xmath23
= @xmath24 , sea or val : @xmath25^{n_i } } { ( 1 - | \beta | ) ^{2 n_i + 1}}.\end{aligned}\ ] ] * the expressions for @xmath26 and @xmath27 are the following : @xmath28    this model is built from the cteq6 m parton distribution function ( pdf ) set @xcite .      to avoid drawing conclusions
relying on a single gpd model , we built another gpd model in the rdda framework :    * we use mstw08 parton distribution functions @xcite .
* we assume factorized @xmath29-dependence : @xmath30 for @xmath31 and @xmath32 quarks : @xmath33 with @xmath34 and @xmath35 the @xmath31 and @xmath32 quark contributions to the proton form factor @xmath36 . for @xmath37 quark and gluons
a dipole ansatz was used .
* we add a d - term coming from chiral quark soliton model ( see refs  @xcite for details ) .      before using the gk model in order to compute cffs on a wide kinematic range at lo and nlo ,
let us remind briefly the phenomenological successes and limitations of the model .
[ fig : dvcs - kinematics ] pictures dvcs kinematics .
the angle @xmath38 between the leptonic and hadronic planes obey the trento convention @xcite .
following ref .
@xcite we consider a lepton beam with helicity @xmath39 and charge @xmath40 ( in units of @xmath41 ) and define combined beam - spin and charge asymmetries : @xmath42.\ ] ] single beam - spin asymmetry can be defined as well : @xmath43 hermes dvcs data are usually described through the fourier coefficients of the various measured asymmetries .
compass - ii will measure combined beam - spin and charge cross sections : @xmath44    these observables offer the interesting feature of different sensitivity to various combinations of cffs , as summarized in tab .
[ tab : normalized - cff - dependence ] . in spite of the fact that @xmath45 is not so small for typical kinematics in tab .
[ tab : typical - kinematics - dvcs - measurements ] , we do not take into account higher - twist contributions in our study , and restrict ourselves to the study of leading - order and next - to - leading order terms in the @xmath46 expansion .
ccc    ' '' ''    * experiment & * observable & * normalized cff dependence + [ 3pt ]   & * * *    ' '' ''    @xmath47 & @xmath48 + hermes &    ' '' ''    @xmath49 & @xmath50 +   &    ' '' ''    @xmath51 & @xmath52 +   &    ' '' ''    @xmath53 & @xmath54 + clas &    ' '' ''    @xmath55 & @xmath56 +   &    ' '' ''    @xmath57 & @xmath58 +    ccccc   & + [ 0pt]*experiment * &    ' '' ''    @xmath59 & @xmath2 [ gev@xmath60 & @xmath29 [ gev@xmath60 & @xmath61 + [ 3pt ] compass & 0.05 & 2.00 & -0.20 & 0.10 + hermes & 0.09 & 2.50 & -0.12 & 0.05 + clas & 0.19 & 1.25 & -0.19 & 0.15 +    figs .
[ fig : gk - hermes - bca ] and [ fig : gk - clas - bsa ] show the comparison of the predictions of the gk model ( no parameter was tuned ) to a selection of dvcs measurements .
the model is in good agreement with the data , the agreement being better at small @xmath3 ( which is the kinematic domain where the model was optimized for dvmp ) .
we computed the real and imaginary parts of the cff @xmath62 ( where @xmath63 is the electric charge of the quark @xmath64 in units of @xmath41 ) at lo and nlo on a wide kinematic range ( skewness varying between @xmath65 and 1 ) .
special attention was paid to the validation of the numerics : two independent codes were systematically compared and an accuracy of 0.1  % has been achieved in the range under scrutiny .    in all cases we have plot the lo cff , and the nlo result with quark contributions only , or both quark and gluon contributions .
the results are shown from figs .
[ fig : dvcs_re ] to [ fig : tcs_im ] .
we observe large nlo corrections , mostly due to gluon gpds .
surprisingly these corrections are maximal in the kinematic region of hermes and compass . in all our calculations ,
we choose @xmath66 .
the size of the corrections may depend on that choice , and this will be the subject of further studies . also the choice of a particular scale needs some theoretical justification .                      the effects of nlo corrections shown in fig .
[ fig : clas12-dvcs - cross - section ] are quite large in both considered gpd models although the value of @xmath3 is rather large .
in particular we see that the gluon contributions is by no means negligible .
the @xmath38-dependence of the interference term allows for an access to the real part of the cff @xmath67 . from fig .
[ fig : tcs_re ] we know that this quantity is subject to large nlo corrections . on fig .
[ fig : clas12-tcs - cross - section ] we indeed compare the pure bethe  -  heitler term , and the pure bethe  -  heitler term augmented by its interference with tcs at lo and nlo .          from fig .
[ fig : compass ] we observe that nlo effects should be highly visible at compass which probes a kinematic region populated by sea quarks and gluons .
deeply virtual compton scattering , both in its spacelike and timelike realizations , is the golden channel to extract gpds from measurements . using model - dependent evaluations we have demonstrated here , in the case of medium energy kinematics which will be explored in the near future at jlab and compass , that the inclusion of nlo corrections to the coefficient function could be an important issue .
the difference of these corrections between the spacelike and timelike regimes is so sizable that they can be promoted to the status of direct tests of the qcd understanding of the reactions . considering cff fitting , global fits of dvcs and tcs data will be needed to separate quark and gluon contributions and allow an accurate interpretation of extracted data .    finally , let us emphasize that we do not consider as a weakness of the current physics program the apparent importance of nlo contributions to compton form factors and dvcs or tcs observables . although it certainly opens the way to a challenging verification that nnlo corrections are either under control or subject to a legitimate resummation procedure @xcite , it points to the very positive fact that the compass - ii and jlab12 experiments may constrain gluon gpds . the 3d tomography of the gluonic structure of the nucleon may thus be scrutinized through the @xmath29-dependence of compton form factors extracted from near future experimental data .
deeply virtual compton scattering in the near forward kinematic region is the golden access to generalized parton distributions . 
 we studied the @xmath0 corrections to the scattering amplitude for both spacelike and timelike kinematics relevant respectively to the leptoproduction of a real photon and to the photoproduction of a lepton pair . 
 it turns out that these corrections are phenomenologically important and that the gluonic contributions are by no means negligible , even in the moderate energy range of jlab12 and of the compass - ii experiment at cern .