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the high precision of current collider data requires comparable precision in the phenomenological predictions . the state of the art in high - energy calculations is at next - to - next - to - leading order ( nnlo ) in quantum chromodynamics ( qcd ) . consequently , major efforts have been undertaken to produce nnlo parton distribution functions ( pdfs ) from a global analysis of the available data . these include the ct14nnlo pdfs  @xcite as well as others  @xcite , all of which include lhc data in the determination of the pdfs . in this paper we describe the introduction of qed evolution at leading order ( lo ) with the next - to - leading order ( nlo ) qcd evolution in the same cteq global analysis package that was used to produce the ct14 pdfs  @xcite . past studies of qed effects in global analysis have been done by the mrst  @xcite and the nnpdf  @xcite groups . we have checked our code against other qed@xmath5qcd evolution codes  @xcite and find good agreement . the mrst and nnpdf analyses used different approaches for modeling the photon pdf . the mrst group used a parametrization for the photon pdf based on radiation off of `` primordial '' up and down quarks , with the photon radiation cut off at low scales by constituent or current quark masses  @xcite . the nnpdf group used a more general photon parametrization , which was then constrained by high - energy @xmath6 , @xmath7 and drell - yan data at the lhc  @xcite . they found constraints on the size of the photon pdf , which was still consistent with zero at the initial scale of @xmath8 gev . as discussed by martin and ryskin  @xcite , the photon pdf has a large _ elastic _ contribution in which the proton remains intact , in addition to the _ inelastic _ contribution in which the proton breaks into a multihadron final state . neither mrst nor nnpdf addresses these separate contributions to the photon pdf , although we can assume that the nnpdf photon is _ inclusive _ , containing both inelastic and elastic components , since it was constrained using inclusive drell - yan and vector boson data . given the limited amount of data to constrain the shape of the photon pdf , we will use a generalization of the mrst approach . we parametrize the inelastic contribution to the photon at the initial scale gev is the same as that used in the standard ct14 pdf sets , and was chosen to be just below the input charm pole mass of @xmath9 gev . ] @xmath1 gev by @xmath10 where @xmath11 is the convolution of the quark - to - photon splitting function @xmath12 with the `` primordial '' quark distribution @xmath13 , which we take to be the initial ct14 nlo up and down valence distributions . we then set @xmath14 to obtain a single parameter family of photon distributions , which we can label by their initial inelastic momentum fraction @xmath15 . for comparison , in analogy with the mrst approach , we will also show results for a `` current mass '' ( cm ) photon distribution , given by defining @xmath16 , and setting the @xmath17 to the quark current masses ; _ i.e. , _ @xmath18 mev and @xmath19 mev . we will constrain the inelastic photon pdf using data on deep inelastic scattering ( dis ) with isolated photons from the zeus collaboration  @xcite . the advantage of using this process is that the initial - state photon contributions are at leading order in the perturbation expansion . in contrast , the initial - state photon contribution to drell - yan or @xmath6 and @xmath7 production is suppressed by factors of @xmath20 relative to the leading quark - antiquark production . however , to use the dis - plus - photon data we will first need to address some technical issues relating to the combination of different subprocess contributions to the observed final state . the organization of this paper is as follows : in sec . [ sec : qedevol ] we describe the inclusion of qed evolution in the cteq global analysis code and give more details about our initial pdf parametrizations . in sec . [ sec : constraints ] we discuss constraints on the photon pdf coming from the ct14 global analysis data set and from the zeus dis with isolated photon data . in this section we present a new calculation for the dis - plus - isolated - photon process , which consistently combines the photon - initiated contribution with the quark - initiated contribution . we show that this data gives significant constraints on the initial photon pdf . in sec . [ sec : conclude ] we discuss our findings and give conclusions . we also include an appendix where we show comparisons between our qcd@xmath5qed evolution code and other publicly - available codes . in this section we discuss the implementation of the qed evolution and the initial photon pdf in the context of the cteq - tea global analysis program . the evolution of the pdfs , @xmath21 , including qed contributions at leading order ( lo ) and qcd contributions at higher orders , is described by the equations : @xmath22 where @xmath23 , the indices @xmath24 and @xmath25 run over active quark flavors , and the convolution is defined by @xmath26 the qcd splitting functions , given by @xmath27 are known up to @xmath28 , next - to - next - to - leading order ( nnlo )  @xcite . the lo qed splitting functions can be extracted from the lo qcd splitting functions , giving @xmath29\nonumber\\ \tilde{p}^{(0)}_{\gamma q}(z)&=&\frac{1+(1-z)^2}{z}\\ \tilde{p}^{(0)}_{\gamma \gamma}(z)&=&-\frac{2}{3}n_c\sum_i e_i^2\,\delta(1-z)\ , \end{aligned}\ ] ] where @xmath30 is the number of colors . we have modified the fortran nlo evolution code evolve , which was used for previous cteq - tea pdfs ( cteq6 - 6.6  @xcite-@xcite and ct09  @xcite ) , to include the lo qed contributions . this code solves the evolution equations directly in @xmath31-space , so that the only new technical issue introduced by the qed corrections is the separation of the quark singlet distributions into separate up and down contributions , based on the quark charges . we have checked our evolution code against the public qcd@xmath5qed codes , partonevolution  @xcite and apfel  @xcite , and we find good agreement . details of this comparison are given in the appendix . the initial photon pdf at the scale @xmath4 is a nonperturbative input that must be obtained by a fit to data . even a choice of zero initial photon pdf is ambiguous , since it depends on the arbitrary scale @xmath4 . so far there have been two different approaches to the initial photon pdf . in the mrst analysis  @xcite the initial photon pdf was given by an ansatz , obtained from radiation off primordial valence up and down quark distributions , cut off at low scales given by the current quark masses , @xmath32 mev and @xmath33 mev , or by constituent quark masses @xmath34 mev . alternatively , the nnpdf approach  @xcite was to use a general parametrization for the initial photon pdf to be constrained by high - energy @xmath6 , @xmath7 , and drell - yan production at the lhc .    in this work we will use a generalization of the mrst ansatz , but we must first address a subtlety of the photon pdf . unlike the case for colored partons , the photon pdf has a large elastic component , in which the proton remains intact  @xcite . this is in addition to the inelastic component , in which the proton dissociates . the elastic component can be parametrized by the equivalent photon approximation ( epa )  @xcite , which involves an integration over the proton electromagnetic form factors . for this work we focus on the inelastic component , which we parametrize by a radiative ansatz , but with free parameters to be fit by data . given the weak constraints from data on the photon pdf , we find it useful to limit the number of parameters to one or two for the time being . we shall see that the zeus dis - plus - isolated - photon data  @xcite constrains the inelastic photon pdf roughly in the range @xmath35 for @xmath36 gev@xmath37 . we parametrize the inelastic contribution to the initial photon pdfs in the proton and neutron by @xmath38 where @xmath39 and @xmath40 are `` primordial '' valence - type distributions in the proton , and the initial photon pdf in the neutron is obtained by an approximate isospin symmetry . defining @xmath16 , we can trade the parameters @xmath41 for mass scales @xmath17 , and we see that the nonperturbative inputs @xmath42 are modeled by the radiation of a single photon off the `` primordial '' quarks , with a collinear cutoff given by the scales @xmath17 . the mrst initial photon pdfs can be obtained from this parametrization by setting @xmath43 gev , using the functions for @xmath44 and @xmath45 given in ref . @xcite , and letting @xmath46 and @xmath47 be either the constituent or current quark masses . for our analysis , we use @xmath1 gev as in ct14 , and we set @xmath48 the initial up and down ct14 nlo valence distributions in the proton . the presence of the photon pdf violates isospin between the neutron and proton . continuing with the radiative ansatz and working to first order in @xmath49 , we can neglect the isospin violation in the gluon and sea - quark pdfs  @xcite and use @xmath50 for the valence quarks at first order in @xmath49 , the radiative ansatz plus approximate isospin symmetry implies @xmath51 this suggests a consistent set of pdfs for the valence quarks in the neutron given by @xmath52 note that eqs . ( [ eq : photonpdf ] ) and ( [ eq : valencen ] ) together ensure that if the number and momentum sum rules ( including the photon contribution ) are satisfied for the pdfs in the proton , they are automatically satisfied for the pdfs in the neutron , regardless of the choices for @xmath39 and @xmath40 . again , for our analysis , we choose @xmath44 and @xmath45 to equal the initial up and down ct14 nlo valence distributions in the proton . thus , from eqs . ( [ eq : photonpdf])-([eq : sea ] ) and ( [ eq : valencen ] ) we can obtain the quark , gluon , and photon pdfs in both the proton and neutron in terms of the parametrization of the quark and gluon pdfs in the proton , plus the two additional parameters @xmath53 and @xmath54 .    for this paper , we shall make the further simplification of @xmath14 , which corresponds to cutting off the radiation from both valence quarks at the same scale . with this choice , everything is specified by one additional parameter , which can be taken to be the cutoff scale @xmath55 , defined by @xmath56 . alternatively , we can trade this parameter for the initial inelastic photon momentum fraction in the proton : @xmath57 for the remainder of this paper , unless otherwise specified , the photon pdfs will be in this one - parameter radiation ansatz labeled by @xmath0 . for comparison purposes , we will make one exception to this by defining a `` current mass '' ( cm ) photon pdf , analogous to the mrst current mass pdf , and given by @xmath16 with @xmath18 mev and @xmath19 mev . for this choice the initial inelastic photon momentum fraction is determined to be @xmath58% . for all other partons in our analysis we use the ct14 nlo initial distributions , except that to maintain a total momentum fraction of 1 , we re - normalize the initial up , down , and strange sea - quark distributions , to account for the additional photon momentum fraction . given that the relevant photon momentum fractions are very small , we find that this reduces the sea - quark distributions by typically less than 1% , and it is inconsequential in our analysis . ( the sea - quark distributions are reduced by 0.9% for @xmath59% , and they are reduced by 1.6% for the cm photon pdf . ) in fig .  [ fig : xfq ] we plot the quantity @xmath60 for three representative photon pdfs , relative to the quark and gluon pdfs , at the scales @xmath61 gev and @xmath62 gev . we note that the effect of the initial photon pdf and the qed evolution on the quark and gluon pdfs is imperceptible in these plots . for the photon pdf , we plot for @xmath63% ( @xmath64 gev ) and for @xmath59% ( @xmath65 mev ) , and for the cm photon pdf .     for @xmath61 gev ( left ) and @xmath62 gev ( right ) . three representative photon pdfs are plotted : the `` current mass '' photon pdf ( @xmath66 , red dotted ) , and photon pdfs with initial inelastic photon momenta fractions of @xmath63 and 0.14% ( @xmath67 , blue dashed , and @xmath68 , green dot - dashed , respectively ) . the effects of the different initial photon pdfs on the quark and gluon pdfs are imperceptible in these plots . [ fig : xfq],title="fig:",scaledwidth=47.0% ]   for @xmath61 gev ( left ) and @xmath62 gev ( right ) . three representative photon pdfs are plotted : the `` current mass '' photon pdf ( @xmath66 , red dotted ) , and photon pdfs with initial inelastic photon momenta fractions of @xmath63 and 0.14% ( @xmath67 , blue dashed , and @xmath68 , green dot - dashed , respectively ) . the effects of the different initial photon pdfs on the quark and gluon pdfs are imperceptible in these plots . [ fig : xfq],title="fig:",scaledwidth=47.0% ] and @xmath69 for @xmath70 gev ( black dots ) , 3.2 gev ( red dashes ) , and 85 gev ( blue solid ) . the left plot is for zero initial photon momentum and the right plot is for the cm photon pdf . [ fig : isospin],title="fig:",scaledwidth=47.0% ]   and @xmath69 for @xmath70 gev ( black dots ) , 3.2 gev ( red dashes ) , and 85 gev ( blue solid ) . the left plot is for zero initial photon momentum and the right plot is for the cm photon pdf . [ fig : isospin],title="fig:",scaledwidth=47.0% ]    the constraints on the photon pdf from the dis and tevatron data , used in the ct14 analysis , are relatively weak . these come from two main sources : isospin violation effects in nuclear scattering and constraints from the momentum sum rule . in general , isospin violation will arise through qed evolution , as well as from the initial conditions given by eq . ( [ eq : valencen ] ) . this isospin violation can be seen in fig . [ fig : isospin ] , where we plot @xmath71 and @xmath69 for several values of @xmath72 for the case where the initial photon pdf is zero , and for the case where the initial photon is the cm choice . note that the isospin violation is small and most important at large @xmath31 . given that cuts of @xmath73 gev@xmath37 , applied to enforce perturbativity in the calculations , typically require @xmath74 , we expect constraints from isospin violation to be small in the present data , as observed in the mrst analysis  @xcite . constraints from the momentum sum rule arise because any momentum carried by the photon implies less momentum available for the quark and gluon pdfs . in this way , constraints from data on the colored parton pdfs indirectly impact the photon pdf . we have performed a preliminary analysis using the data sets included for ct10  @xcite . for a fixed initial photon momentum fraction , with the photon pdf parametrized as discussed in sec . ii , we minimized the global @xmath75 by varying the quark and gluon pdfs . using the usual cteq - tea choice of @xmath76 tolerance , we obtain a limit on the photon momentum fraction of @xmath77% at the 90% confidence level , which is similar in magnitude to the results found by the mrst and nnpdf analyses . the best fit for the initial photon momentum fraction from this global analysis is @xmath78% , but with only a small change of @xmath79 , relative to the fit with @xmath63% . for comparison , we find the elastic contribution to the initial photon momentum fraction , as calculated in the equivalent photon approximation , to be @xmath80% .    unfortunately , this limit on the initial photon momentum fraction is much larger than one would expect for a photon pdf . in the analysis of the nnpdf group , additional constraints were made on the initial photon pdf by including lhc data on high - energy @xmath6 , @xmath7 , and drell - yan production , and comparing with theoretical predictions that included photon initial - state contributions . although the photon - induced contribution to these processes is small compared to the dominant quark - antiquark annihilation subprocess , the precision of these measurements was enough to substantially increase the constraints on the photon pdf  @xcite . however , the small relative contribution of the photon - photon subprocess puts a stringent requirement on the precision needed for both experimental and theoretical analyses . any small misjudgment of systematic errors on the experimental side , or uncalculated higher - order corrections on the theoretical side could have a significant effect on the extraction of the photon pdf . in particular , given that the initial photon pdf is nominally of order @xmath49 , one might expect that the uncalculated @xmath81 quark - initiated contributions to drell - yan production would contribute at the same level as the photon - initiated contributions . for this reason , we consider a different experimental process , isolated photon production in dis , to constrain the photon pdf .      at the partonic level , the process of dis with isolated photon occurs at lo through compton scattering of a photon coming from the proton off the lepton , as shown in fig . [ fig : feyndiags](a ) . thus , this process probes the photon pdf at lo , having no large backgrounds with which to compete . however , the quark - initiated subprocess shown in figs . [ fig : feyndiags](b ) and ( c ) , while formally suppressed by @xmath82 , is just as large because of the small size of @xmath83 relative to @xmath84 . in fact , if we consider the photon pdf to be @xmath82 , then the photon - initiated subprocess and the quark - initiated subprocess are actually the same order in @xmath49 . thus , the correct way to calculate the cross section for dis with isolated photons is to include both subprocesses consistently without double - counting .    in the literature there have been two approaches to calculations of the process @xmath2 . the calculation of mrst  @xcite , which was preceded by studies of blmlein _ et al . _ @xcite , included just the photon - initiated contribution of fig . [ fig : feyndiags](a ) . the calculation of gehrmann - de ridder , gehrmann , and poulsen ( ggp )  @xcite included just the quark - initiated contributions of figs . [ fig : feyndiags](b , c ) . in the ggp analysis it was found convenient to make the lorentz - invariant separation of the cross section into three components , depending on the fermion line off which the final - state photon is emitted : @xmath85 for emission off the lepton line , given by the square of the partonic amplitude in fig . [ fig : feyndiags](b ) ; @xmath86 for emission off the quark line , given by the square of the partonic amplitude in fig . [ fig : feyndiags](c ) ; and @xmath87 for the interference between the two sets of diagrams . actually represents two feynman diagrams , where the final - state photon is emitted off the initial - state lepton or quark as well as off the final - state lepton or quark . ] in the ggp calculation a cut on the outgoing quark was necessary to remove the divergence in the amplitude as the photon off - shellness went to zero in the @xmath85 amplitude . a hybrid calculation was also considered by the zeus collaboration in their analysis of the dis - plus - isolated photon data  @xcite , where the @xmath85 component of the quark - initiated subprocess of ggp was replaced by the photon - initiated subprocess of mrst .    . for each diagram shown there is an additional diagram where the photon is emitted off the initial - state lepton or quark . [ fig : feyndiags],scaledwidth=80.0% ]    in this section we introduce a consistent and systematic method of combining the photon- and quark - initiated subprocesses , which also reduces the factorization scale dependence of either calculation . first , consider the calculation of the differential cross section as a power series in @xmath49 without consideration of the relative sizes of @xmath83 and @xmath84 . it can be written as a convolution over partonic cross sections @xmath88 where each of the partonic cross sections can be expanded in a power series in @xmath49 , @xmath89 the diagram in fig . [ fig : feyndiags](a ) corresponds to a lo contribution ( @xmath90 ) , while the diagrams in figs . [ fig : feyndiags](b , c ) correspond to nlo contributions ( @xmath91 ) . through nlo in @xmath49 the cross section can still be written as a sum of @xmath85 , @xmath86 , and @xmath87 components , @xmath92 where the @xmath85 component also includes the photon - initiated contribution , @xmath93\ , \ ] ] using the modified minimal subtraction ( @xmath94 ) scheme , we can factorize the initial - state singularity from the nlo quark - initiated subprocess into the definition of the photon pdf , leaving a nlo subprocess cross section , @xmath95 where the first term on the right - hand side , @xmath96 , is the hard partonic quark - induced subprocess , and the second term is the collinear subtraction counterterm . here , we have distinguished the renormalization scale @xmath97 from the factorization scale @xmath72 , and we explicitly note that the initial - state collinear singularity cancels within the @xmath85 component . calculating everything in dimensional regularization with @xmath98 dimensions , the combined @xmath85 component of the nlo quark - initiated subprocess cross section , @xmath99 is finite as @xmath100 .    in principle , there are additional virtual and real contributions at nlo in @xmath49 , besides the quark - initiated contributions . however , all other nlo terms are proportional to @xmath83 , which is in fact suppressed by an amount of order @xmath49 relative to @xmath84 , as seen in the previous section . thus , by keeping the photon - initiated contribution at lo and the quark - initiated contributions at nlo , and including the collinear - subtraction counterterm of eq . ( [ eq : ap ] ) we have a well - defined calculation in the @xmath94 scheme , while maintaining all contributions of the same size in @xmath49 . note that this is reminiscent of the acot scheme  @xcite for including charm and bottom quark pdfs , although for the photon , there is no equivalent of the fixed - flavor scheme , due to its zero mass and the consequently nonperturbative photon pdf . the calculations of the @xmath86 and @xmath87 components are identical to those in the ggp calculation . for the kinematics of interest to us , the @xmath87 component is negligible , but it is included for completeness . the @xmath86 contribution has a final - state singularity , when the photon and final - state quark become collinear , which can be handled by including a fragmentation contribution and an associated collinear subtraction counterterm in the @xmath94 scheme . thus , we obtain @xmath101 where @xmath102 is the hard partonic subprocess , @xmath103 is the lo subprocess cross section for @xmath104 , and @xmath105 is the fragmentation function for finding the photon ( with momentum @xmath106 ) in the quark @xmath107 ( with momentum @xmath107 ) , with momentum fraction @xmath108 at the fragmentation scale @xmath109 . the singularities in the first two terms on the right - hand side of this equation cancel as @xmath100 . we will discuss the choice for the fragmentation function in the next subsection . we have calculated the @xmath85 and the @xmath86 components of the cross section , using the subtraction method to handle the collinear divergences  @xcite in the hard cross - section term . following this method , we subtract a term with a two - particle final state mapped onto a 3-particle final - state , with the third particle phase space unintegrated , and then add back the exact same term with the third particle phase space integrated out . the subtraction term is designed to have the same collinear - singular limit as the hard term in the same region of phase space of the third particle , so that the hard cross section term minus the subtraction is integrable in @xmath110 dimensions , while the @xmath111 singularities in the remaining terms cancel . using this method we obtain for the @xmath85 quark - initiated contribution : @xmath112 where @xmath113\ , \,d\hat\sigma_\gamma^{(0)}(z\xi ) \ .\label{eq : ap3}\ ] ] the hard term minus the subtraction term can be written as an integral over the phase space of the additional quark : @xmath114 where @xmath115 with @xmath116 and @xmath117 the scattering angles of the final - state quark in the initial parton - parton center - of - momentum frame . note that the hard term on the right - hand side is treated with three - body final - state phase space , while the subtraction term is treated with two - body final - state phase space . similarly , we obtain for the @xmath86 contribution : @xmath118 where @xmath119\bigg\}\,d\hat\sigma_{eq\rightarrow e^\prime q^\prime}\ .\nonumber\end{aligned}\ ] ] the hard term minus the subtraction term can be written @xmath120 where in this case we have found it convenient to use a different parametrization of the final - state quark phase space . letting @xmath121 be the hard partonic subprocess , we use @xmath122 , where @xmath123 is the angle between @xmath124 and @xmath125 in the @xmath126 center - of - momentum frame and @xmath127 is the azimuthal angle between the @xmath128 plane and the @xmath129 plane . as before , the hard term on the right - hand side is treated with three - body final - state phase space , while in this case the subtraction term and the fragmentation term are treated in the limit where the final - state photon and quark are collinear , with momenta satisfying @xmath130 and @xmath131 . the zeus experiment  @xcite used proton and lepton beam energies of @xmath132 gev and @xmath133 gev , respectively , corresponding to a center - of - mass energy and rapidity of @xmath134 respectively . ( we neglect the proton mass , @xmath135 , in all calculations here . ) for the process @xmath2 , with momentum satisfying , @xmath136 , one can define the standard dis variables that describe the kinematics of the scattered lepton , @xmath137 , @xmath138 , and @xmath139 . the zeus collaboration measured distributions for two leptonic variables @xmath140 and @xmath31 , and for two photonic variables , @xmath141 and @xmath142 , the transverse energy and pseudorapidity of the photon , respectively . the collaboration combined data that were 59.1% @xmath143 and 40.9% @xmath144 scattering . note that the sign of the charged lepton has no effect on the @xmath85 or @xmath86 components of the calculation , but the combination of the two charged lepton contributions produces a significant cancellation of the already small @xmath87 component , so that it is negligible in the analysis . the kinematic region defined by the experiment was @xmath145 the cut on the final - state lepton angle , @xmath146 , can be written in terms of its rapidity as @xmath147    there are two additional cuts that require discussion . the experimentalists reported a cut of @xmath148 gev , where @xmath149 . naively , this cut looks problematic because it would remove the photon - initiated contribution , which occurs at exactly @xmath150 . however , upon closer investigation it appears that this cut was only applied to the theoretical and monte carlo calculations . to quote from ref . @xcite , `` the keen reader will note that no such cut was applied at the detector level . this proved impossible due to the poor @xmath151 resolution at detector level and poor description of the data by mc ... '' . the relevant detector - level cut was the requirement of at least one reconstructed track , well separated from the lepton , which was used to ensure some hadronic activity and to remove deeply virtual compton scattering events . after this cut , it was found that the number of events in the monte carlo calculations with @xmath152 gev was negligible . for our purposes , we interpret the forward track cut as a requirement to tag inelastic events , and we include no explicit @xmath151 cut . note that the forward track cut should equally remove elastic isolated photon events , and so remove the contribution from the elastic part of the photon pdf . in this way , the zeus data probe only the inelastic part of the photon pdf , and therefore , we only include this inelastic contribution in comparison with the experimental data . in doing so we have made the approximation @xmath153 ; _ i.e. , _ the elastic and inelastic components of the photon pdf evolve separately . this approximation is good because @xmath154 changes very little from @xmath4 to @xmath140 due to the rapid falloff of the proton electromagnetic form factor , while the inelastic contribution evolves additively , @xmath155 up to corrections suppressed by extra factors of @xmath49 . we have verified by explicit calculation that this additive approximation replicates the consistently evolved inclusive photon pdf , with errors that are far smaller than other theoretical uncertainties that we will discuss below . the second additional important cut is the isolation cut on the photon , enforcing that 90% of the energy in the jet containing the photon belongs to the photon , where jets are formed with the @xmath156 cluster algorithm with parameter @xmath157 . we will model this isolation cut in our calculation in two different ways . first , we can model the experimental cut at the parton level , requiring @xmath158 if the photon - quark separation satisfies @xmath159 . for later reference , we call this the `` sharp '' isolation cut . since this does not completely remove the quark - photon collinear singularity , the theoretical calculation of the @xmath86 component will depend on the choice of the quark - to - photon fragmentation function @xmath160 . for this , we use the lo fragmentation function determined by the aleph collaboration  @xcite , parametrized by @xmath161\ , \nonumber\end{aligned}\ ] ] where @xmath162 gev and @xmath163 . note that this parametrization of @xmath160 is an exact solution to the evolution equation at @xmath82 , so that the dependence on the fragmentation scale @xmath109 cancels exactly in our calculation . in the ggp analysis  @xcite , other parametrizations of @xmath160 with different assumptions were compared , with only a small effect on the calculated cross sections . one of the disadvantages of having a dependence on the quark - to - photon fragmentation function , in addition to the uncertainties due to the phenomenological fit to the function , is that it assumes that the cross section is inclusive in the fragmentation remnant . in our calculation , the combination of the experimental constraints on the photon and on the lepton indirectly imposes constraints on the remnant quark in the process . therefore , we also consider an alternative model of the experimental isolation cut , replacing it with a `` smooth '' isolation cut  @xcite , so as to avoid the necessity of the fragmentation contribution . the smooth isolation cut is given by requiring that the hadronic energy @xmath164 inside all cones of radius @xmath165 around the photon direction satisfy @xmath166 where we take @xmath167 and @xmath168 . these values of @xmath169 and @xmath170 are chosen to ensure that the photon will contain at least 90% of the energy inside a cone of @xmath171 centered on the photon , just as for the experimental isolation cut . however , the smooth cut does not translate exactly to the experimental isolation cut , because it requires the photon to carry a greater fraction of the energy as the cone size @xmath172 becomes smaller . in practice , because of this , the theoretical calculation with the smooth cut is better behaved than with the sharp cut . in addition , the smooth isolation cut is more restrictive than the sharp isolation ; for a strictly positive - definite differential cross section , the smooth isolation prescription must always give a smaller predicted cross section . in this way , a comparison of the two calculations can give some indication of the theoretical uncertainty due to the isolation cut . finally , we note that the jet - clustering algorithm also includes the electron , so that the isolation cut effectively imposes @xmath173 .     and correspond to the total prediction ( light gray solid ) , the @xmath86 component ( blue hashed ) , the @xmath85 component ( dark red solid ) , and the photon - initiated contribution only ( green hashed ) . also shown are the zeus data points with combined statistical and systematic errors . [ fig : scaledep],title="fig:",scaledwidth=47.0% ]   and correspond to the total prediction ( light gray solid ) , the @xmath86 component ( blue hashed ) , the @xmath85 component ( dark red solid ) , and the photon - initiated contribution only ( green hashed ) . also shown are the zeus data points with combined statistical and systematic errors . [ fig : scaledep],title="fig:",scaledwidth=47.0% ]      before discussing the comparison of the theory with the data , it is useful to understand the theoretical uncertainties of the calculation by studying the factorization scale dependence and the dependence on the isolation prescription . in fig . [ fig : scaledep ] we plot the differential cross sections for @xmath174 and @xmath175 as functions of @xmath141 and @xmath142 , respectively , while varying the factorization scale within @xmath176 . here we have used the sharp isolation cut and calculated with zero initial inelastic photon pdf at @xmath1 gev . the four bands on each of the two plots correspond to the photon - initiated contribution only ( green hashed band ) , @xmath85 component ( dark red solid band ) , @xmath86 component ( blue hashed band ) , and the total calculation ( light gray solid band ) . the @xmath87 contribution is imperceptible on the scale of these plots . from these plots we learn several important facts . first , the scale dependence of the @xmath85 component is reduced dramatically compared to the photon - initiated contribution alone . this large scale dependence of the photon - initiated contribution cancels greatly with that of the collinear subtraction counterterm in the combined @xmath85 component . second , the @xmath85 and @xmath86 components dominate in different regions of phase space . for instance , the cross sections are most sensitive to the @xmath85 component , and consequently to the photon pdf , at large @xmath141 and small @xmath142 . thus , the shapes of these distributions can give information about the nonperturbative contribution to the photon pdf . finally , we note that the scale dependence of the @xmath86 component is still large , being only lo in @xmath177 , and it dominates the overall scale uncertainty of the theoretical calculation .    in fig . [ fig : isodep ] we plot the total predictions of the same two distributions , again for zero initial inelastic photon pdf , but now comparing the two different isolation prescriptions in the theoretical calculation . in these plots we show the predictions with the smooth isolation prescription ( blue hashed band ) and the sharp isolation prescription ( red hashed band ) , again varying the factorization scale within @xmath176 . the first thing to note here is that the difference between the predictions is about the same size as the scale uncertainty , with similar dependence on the kinematic variables . another striking feature is that the prediction using the smooth isolation prescription is uniformly larger than that using the sharp isolation , in contrast to expectations . this is probably due to incomplete cancellations in the sharp isolation calculation between the large negative collinear fragmentation contribution and the positive real contribution , due to indirect constraints on the emitted final - state quark in the real emission contribution . presumably higher - order qcd corrections will affect the predictions for both isolation predictions , to resolve this issue . as noted previously , our calculation is only lo in @xmath177 ; we expect both the factorization scale uncertainty and the isolation prescription discrepancy to be reduced at nlo in @xmath177 . in any event , we will use the two isolation prescriptions , as well as the factorization scale dependence , as a measure of the theoretical uncertainty of our calculation .    . the blue hashed band is calculated using the smooth isolation prescription and the red hashed band is calculated using the sharp isolation prescription . also shown are the zeus data points with combined statistical and systematic errors . [ fig : isodep],title="fig:",scaledwidth=47.0% ] . the blue hashed band is calculated using the smooth isolation prescription and the red hashed band is calculated using the sharp isolation prescription . also shown are the zeus data points with combined statistical and systematic errors . [ fig : isodep],title="fig:",scaledwidth=47.0% ]    with this understanding of the theoretical uncertainties of the calculation we can now compare the zeus data against predictions for the differential distributions , while varying the initial inelastic photon momentum fraction @xmath0 of the photon pdf , described in sec . [ sec : photonpdf ] . for this analysis , the initial quark and gluon pdfs are just the ct14nlo pdfs , except that the sea - quark normalizations are rescaled in order to maintain a total momentum fraction of 1 . this rescaling has little effect in our analysis , because the photon momentum fractions considered here are small . for instance , a photon momentum fraction of @xmath178 induces a reduction of the sea - quark momentum by only 0.9% , while the cm photon pdf induces a reduction of the sea - quark momentum by 1.6% . at this stage of the analysis we have not refit the quark and gluon pdfs , since the @xmath179 process is dominantly sensitive to the photon pdf directly , whereas the indirect sensitivity through changes in the quark and gluon pdfs is negligible .     and @xmath142 , with the smooth isolation prescription , with factorization scale @xmath180 . the curves , from bottom to top are with initial inelastic photon momentum fractions of @xmath181 ( black dashed ) , 0.1% ( blue dotted ) , 0.2% ( red dot - dashed ) , and for the cm photon ( green solid ) . also shown are the zeus data points with combined statistical and systematic errors . [ fig : eteta],title="fig:",scaledwidth=47.0% ]   and @xmath142 , with the smooth isolation prescription , with factorization scale @xmath180 . the curves , from bottom to top are with initial inelastic photon momentum fractions of @xmath181 ( black dashed ) , 0.1% ( blue dotted ) , 0.2% ( red dot - dashed ) , and for the cm photon ( green solid ) . also shown are the zeus data points with combined statistical and systematic errors . [ fig : eteta],title="fig:",scaledwidth=47.0% ]    in fig . [ fig : eteta ] we plot the differential cross sections for @xmath174 and @xmath175 as a function of the photon variables , @xmath141 and @xmath142 , using the smooth isolation prescription , with a factorization scale of @xmath180 . the curves , from bottom to top are with initial inelastic photon momentum fractions of @xmath181 ( black dashed ) , 0.1% ( blue dotted ) , 0.2% ( red dot - dashed ) , and for the cm initial photon ( green solid ) , which has initial momentum fraction 0.26% . also shown are the zeus data points with combined statistical and systematic errors . with these choices of @xmath72 and the isolation prescription , we see that the theory can fit the data well for @xmath182% . on the other hand , the theory fits poorly for the cm initial photon , overshooting the data at large @xmath141 and small @xmath142 . of course , the best fit for @xmath183 is correlated with the choice of @xmath72 and the isolation prescription . however , since these choices tend to move the curves up or down uniformly , it is still possible to constrain the initial photon pdf by the shape of the distributions . in particular , it is impossible to get a good fit to the prediction using the cm initial photon pdf , regardless of the choices of @xmath72 and the isolation prescription .    in fig . [ fig : q2x ] we plot the differential cross sections for @xmath184 and @xmath185 as a function of the lepton variables , @xmath186 and @xmath31 against the zeus data , using the exact same theoretical choices and initial inelastic photon pdfs as for the previous plot . in this case we see that it is impossible to fit the data , regardless of the initial photon pdf or the choices of factorization scale and isolation prescription . in particular the theory fits the data very poorly at small @xmath31 and @xmath186 . in fact , we note that prediction for the smallest bin in @xmath31 is far from the zeus data point , and is essentially independent of the initial photon pdf . we expect that the predictions in these bins are highly sensitive to higher - order qcd radiation , so that it is difficult to fit the full lepton distributions with a fixed - order calculation .     and @xmath31 , with the smooth isolation prescription , with factorization scale @xmath187 . the curves , from bottom to top are with initial inelastic photon momentum fractions of @xmath181 ( black dashed ) , 0.1% ( blue dotted ) , 0.2% ( red dot - dashed ) , and for the cm photon ( green solid ) . also shown are the zeus data points with combined statistical and systematic errors . [ fig : q2x],title="fig:",scaledwidth=47.0% ]   and @xmath31 , with the smooth isolation prescription , with factorization scale @xmath187 . the curves , from bottom to top are with initial inelastic photon momentum fractions of @xmath181 ( black dashed ) , 0.1% ( blue dotted ) , 0.2% ( red dot - dashed ) , and for the cm photon ( green solid ) . also shown are the zeus data points with combined statistical and systematic errors . [ fig : q2x],title="fig:",scaledwidth=47.0% ]    the fact that these fixed - order calculations are more reliable for the photon distributions than for the lepton distributions can be seen further by looking at the phase - space constraints for the two sets of variables . in fig . [ fig : limits ] we show plots of the constraints on the photon variables @xmath141 and @xmath142 and on the lepton variables @xmath186 and @xmath31 . in these figures , the dashed lines indicate the bins that are plotted by the zeus data . the combined dark red@xmath5light blue regions indicate the regions of phase space allowed by the zeus kinematic constraints of eq . ( [ eq : zeuscuts ] ) , for the fully inclusive event , whereas the dark red region only is allowed for the lo photon - initiated subprocess . for the photon distributions , the constraints are dominated by the photon cuts on @xmath141 and @xmath142 , with only a small cut on the photon - initiated contribution in the upper - left corner due to the requirement of @xmath188 . thus , all of the bins in @xmath141 and @xmath142 have a large photon - initiated contribution and can be considered very inclusive . in contrast , for the lepton distributions the additional photon constraints have a large effect in many of the bins . for instance , the photon - initiated contribution to the smallest @xmath186 bin is largely removed by the requirement of @xmath189 gev , and the photon - initiated contribution to the smallest @xmath31 bin is completely removed by the requirement of @xmath190 . these bins are dominated by events with additional particles in order to satisfy the kinematics , so we would not expect our fixed - order calculation to do well at predicting the lepton distributions .     and @xmath142 ( left ) and @xmath186 and @xmath31 ( right ) . the dashed lines indicate the bins that are plotted by the zeus experiment . the dark red region only is kinematically accessible by the photon - initiated contribution , while the dark red@xmath5light blue regions are kinematically allowed in general . [ fig : limits],title="fig:",scaledwidth=46.0% ]   and @xmath142 ( left ) and @xmath186 and @xmath31 ( right ) . the dashed lines indicate the bins that are plotted by the zeus experiment . the dark red region only is kinematically accessible by the photon - initiated contribution , while the dark red@xmath5light blue regions are kinematically allowed in general . [ fig : limits],title="fig:",scaledwidth=47.2% ]    based on these arguments we will use only the distributions in the photon variables @xmath141 and @xmath142 to constrain the initial inelastic photon pdf , for a total of eight data points . we also reiterate that the constraints due to the remaining ct14 experimental data set are much weaker than these data , and are neglected in the present analysis . we define the chi - squared function for these data points by @xmath191 where @xmath192 , @xmath193 , and @xmath194 are the theory prediction , the experimental measurement , and its combined statistical and systematic error for the data point @xmath24 . in fig . [ fig : chi2 ] we plot @xmath75 versus the initial inelastic photon momentum fraction @xmath0 for both the smooth and sharp isolation prescriptions and for several values of the factorization scale @xmath72 . note that the value of @xmath0 determined by the minimum of @xmath75 depends significantly on the isolation prescription and on the factorization scale , giving best fits for the initial momentum fraction that can vary from less than 0 to above 0.1% . in addition , due to this theoretical uncertainty in the current calculation , it is not possible to unambiguously determine an error band on @xmath0 , using the standard ct approach of applying some tolerance criterion on the rise in the @xmath75 around the best fit . however , from fig .  [ fig : chi2 ] we do see that not all choices of theoretical parameters are able to fit the shape of the data points equally well . therefore , we can determine a conservative limit on the value of @xmath0 by requiring that the data and theory not disagree beyond some level . a @xmath75 distribution with eight data points will have @xmath195 at the 90% confidence level . for the remaining 2947 data points used in the ct14 analysis is @xmath196 in going from @xmath63% to @xmath59% . ] therefore , we define that any theoretical prediction with @xmath197 is ruled out as a bad fit to the data at the 90% confidence level . it is impossible to satisfy this criterion for @xmath198% for either choice of the isolation prescription and for any value of @xmath72 . furthermore we find that the cm choice of the photon pdf has @xmath199 for any choice of isolation and factorization scale and so is ruled out by this data .     versus initial inelastic photon momentum fraction @xmath0 using the smooth isolation prescription ( left ) and the sharp isolation prescription ( right ) for factorization scales @xmath200 , @xmath141 , @xmath201 , and @xmath202 . the horizontal line at @xmath203 is the 90% confidence level limit for eight data points . [ fig : chi2],title="fig:",scaledwidth=47.2% ]   versus initial inelastic photon momentum fraction @xmath0 using the smooth isolation prescription ( left ) and the sharp isolation prescription ( right ) for factorization scales @xmath200 , @xmath141 , @xmath201 , and @xmath202 . the horizontal line at @xmath203 is the 90% confidence level limit for eight data points . [ fig : chi2],title="fig:",scaledwidth=47.2% ]    thus , we find our maximal initial inelastic photon pdf to have @xmath59% at the 90% confidence level . of course , the exact value of the momentum fraction is correlated with the shape of the initial photon pdf . from fig . [ fig : limits ] we see that the zeus dis - plus - isolated - photon data constrains the photon pdf in the kinematic region given roughly by @xmath35 for @xmath36 gev@xmath37 . outside of this region the photon pdf is very weakly constrained , but we believe that the radiative ansatz gives a reasonable expectation for its overall shape . as for the minimal possible value of the initial inelastic photon momentum fraction , it could , in principle , be negative , which is not ruled out by the analysis of this section . for instance , one could begin the evolution with zero initial photon pdf at a lower value of the scale @xmath4 . however , we take the reasonable assumption that it should be nonzero at the low scale of @xmath1 gev . we thus propose the initial pdfs with @xmath63% and @xmath59% as our 90% c.l . photon pdfs . a similar analysis gives @xmath204% at the 68% confidence level , but the data are still consistent with @xmath63% at the 68% c.l .    in fig . [ fig : comp1 ] we compare , at the scale @xmath205 gev , the ct14qed photon pdfs with @xmath63% and @xmath59% against the nlo mrst2004qed photon pdfs , using the current quark masses ( labeled mrst0 ) and using the constituent quark masses ( labeled mrst1 ) , and against the nlo nnpdf2.3qed average photon pdf with @xmath206 . we should emphasize that the ct14qed photon pdfs only contain the inelastic contribution in these plots . the nnpdf2.3 average photon pdf has more structure in its shape at large and small values of @xmath31 than do the other pdfs , but it is still consistent with the zeus data in the @xmath31 range that is probed by the experiment . gev : ct14qed with @xmath181 ( solid green ) , ct14qed with @xmath178 ( short - dashed black ) , mrst2004qed0 using current quark masses ( dotted orange ) , mrst2004qed1 using constituent quark masses ( dot - dashed brown ) , and nnpdf2.3qed with @xmath206 and average photon ( long - dashed blue ) . [ fig : comp1],scaledwidth=46.0% ]    in fig . [ fig : comp2 ] we compare the same set of photon pdfs at the higher scale of 85 gev and the very high scale of 1 tev . here we can make some very interesting observations . the most obvious thing in these figures is that the ct and mrst photon pdfs become very similar at large @xmath186 , whereas the nnpdf photon pdf is distinctly different and much smaller at small values of @xmath31 . this difference is due to the different approaches to the evolution of the pdfs taken by the different groups . whereas in the mrst and cteq - tea approaches , the qcd and qed scales are chosen to be identical and evolved together , in the nnpdf2.3qed pdfs that are included in the lhapdf library  @xcite , the qcd and qed scales are separate and the two scales are evolved successively ; first the qed scale is evolved from @xmath4 to @xmath140 and then the qcd scale is evolved from @xmath4 to @xmath140 . as discussed in ref . @xcite , the successive evolution of qed and qcd differs from the combined evolution by terms that are subleading by @xmath207 and can induce large unresummed logarithms between the two scales . this difference in the evolution at small @xmath31 is also seen to be consistent with the behavior seen in the right panel of fig .  2 in ref . @xcite , where the nnpdf photon pdf also is smaller at small @xmath31 and large @xmath186 than when it is evolved using the code partonevolution  @xcite . we expect that the difference between the nnpdf2.3qed photon pdf and the other photon pdfs at high @xmath140 would be less significant if the nnpdf2.3qed pdfs were evolved from the low scale simultaneously in qed and qcd . another observation from fig . [ fig : comp2 ] , concerning the ct14qed and mrst2004qed photon pdfs is that the impact of the initial photon distribution becomes less significant as @xmath186 increases and more photons are produced through radiation off the quarks . from these plots we see that the fractional deviation between the different photon pdfs decreases with increasing @xmath186 . in fact at very small @xmath31 and large @xmath186 the differences in the sea - quark distributions of the pdfs presumably have more impact on the photon pdf than does the initial photon distribution . gev ( left ) and @xmath208 tev ( right ) : ct14qed with @xmath181 ( solid green ) , ct14qed with @xmath178 ( short - dashed black ) , mrst2004qed0 using current quark masses ( dotted orange ) , mrst2004qed1 using constituent quark masses ( dot - dashed brown ) , and nnpdf2.3qed with @xmath206 and average photon ( long - dashed blue ) . [ fig : comp2],title="fig:",scaledwidth=46.0% ]   gev ( left ) and @xmath208 tev ( right ) : ct14qed with @xmath181 ( solid green ) , ct14qed with @xmath178 ( short - dashed black ) , mrst2004qed0 using current quark masses ( dotted orange ) , mrst2004qed1 using constituent quark masses ( dot - dashed brown ) , and nnpdf2.3qed with @xmath206 and average photon ( long - dashed blue ) . [ fig : comp2],title="fig:",scaledwidth=46.0% ] in this paper , we have presented ct14qed , which is the first set of ct14 parton distribution functions obtained by including qed evolution at leading order ( lo ) with next - to - leading - order ( nlo ) qcd evolution in the global analysis by the cteq - tea group . this development will provide better theory predictions to compare with the precision data , such as drell - yan pair production , measured at the lhc . the ct14qed pdfs are based on the ct14 nlo initial distributions with the addition of an initial photon pdf . ( there is also an inconsequential rescaling of the quark sea pdfs , in order to maintain the momentum sum rule . ) the inelastic contribution to the photon pdf is parametrized at the initial scale @xmath4 using a generalization of the radiative ansatz introduced by the mrst group in their previous study . the initial photon pdf then depends on two independent parameters [ cf . ( [ eq : photonpdfintro ] ) ] , which are related to the scales at which the radiation off the up and down valence quarks is cut off . however , given the weak constraints on the initial photon pdf we find it convenient at this time to set the scales equal , so that the initial photon pdf is parametrized by a single parameter , which we take to be the momentum fraction carried by the inelastic photon at the initial scale @xmath4 . for comparison purposes we have also defined a `` current mass '' ( cm ) photon pdf , comparable to the mrst current mass pdf , for which the initial photon momentum fraction is @xmath209 . a set of neutron pdfs can also be obtained with a small amount of isospin breaking , suggested by the radiative ansatz applied to first order in @xmath49 , and which automatically ensures that the number and momentum sum rules are satisfied . however , as previously seen by both the mrst and nnpdf groups , we find that the constraints from isospin violation effects ( generally small and most important at large @xmath31 ) in nuclear scattering and from the requirement of the momentum sum rule , imposed by the dis and tevatron data in the ct14 , are relatively weak . thus , in order to constrain the photon pdf , we focused on the scattering process @xmath210 , which was measured by the zeus experiment at hera . this process is dominantly sensitive to the inelastic photon pdf directly , with negligible indirect sensitivity through the modification of the quark and gluon pdfs by qed effects . it also has the advantage that the initial - state photon subprocess contribution occurs at leading order , so that it does not compete with other much larger contributions . in this paper we have produced for the first time a consistent and systematic calculation for this cross section that combines both the photon- and quark - initiated subprocesses , and simultaneously reduces the factorization scale dependence of either calculation . details of this calculation were presented in sec . [ sec : theory ] . the photon isolation cut , which required that the final - state photon must contain at least 90% of the energy in the jet to which it belongs ( where jets are formed with the @xmath156 cluster algorithm with parameter @xmath157 ) , was modeled using two different models of photon isolation . we used the two different isolation prescriptions , as well as the factorization scale dependence as a measure of the theoretical uncertainty of our calculation .    by comparing the zeus data for the distributions of transverse energy and pseudorapidity of the final - state photon against our calculation of the differential distributions , we were able to constrain the initial inelastic photon momentum fraction inside the proton to be @xmath211 at the 90% confidence level hence , the cm choice of photon pdf has been ruled out by this data . for completeness , we also compared the ct14qed pdfs to some of the nlo ( in @xmath177 ) photon pdfs published by the mrst and the nnpdf groups . phenomenological applications of the ct14qed pdfs will be discussed in future publications . as shown in fig . [ fig : isodep ] , the theoretical uncertainties due to the factorization scale dependence and the isolation prescription are currently larger than the experimental uncertainties of the zeus data . thus , extending our calculation to nlo in @xmath177 should be able to further constrain the initial photon pdf . this is a project that we are currently undertaking . parametrizations for the ( inelastic photon ) ct14qed pdf sets ( both proton and neutron versions ) will be distributed in a standalone form via the cteq - tea web site @xcite , or as a part of the lhapdf6 library @xcite . for backward compatibility with version 5.9.x of lhapdf , our web site also provides ct14 grids in the lhapdf5 format , as well as an update for the cteq - tea module of the lhapdf5 library , which must be included during compilation to support calls of all eigenvector sets included with ct14 @xcite . we will also distribute sets with the inclusive photon pdfs , ct14qedinc . for the proton , ct14qedinc at the initial scale @xmath4 is the sum of the ( inelastic ) ct14qed and the elastic component of the photon pdf , given by the equivalent photon approximation . the proton ct14qedinc pdfs are then evolved from @xmath4 to @xmath140 as discussed in sec . [ sec : qcdplusqed ] . for the neutron , ct14qedinc is equal to ct14qed , since the neutron has zero electric charge , and therefore it has no elastic component of the photon pdf . this work was supported by the national science foundation under grant no . we also thank sayipjamal dulat , tie - jiun hou , joey huston , pavel nadolsky and jianwei qiu for helpful discussions . gev to @xmath212 gev . [ fig : apfelcomp],scaledwidth=60.0% ]      we have checked our code against the public evolution code partonevolution  @xcite , which also solves the evolution equations at lo in qed and nlo in qcd . the main difference ( other than technicalities of implementation ) in the partonevolution code is that it also includes the charged leptons as partons in the proton . using the toy model of ref . @xcite with zero initial photon pdf , ensuring all input parameters agree , using the same formulation for the running of @xmath177 and @xmath49 , and removing the lepton pdfs and their contribution to @xmath213 from partonevolution , we find excellent agreement between the two programs . evolving from @xmath214 gev to @xmath212 gev , we find differences of less than @xmath215% over most of the range of @xmath31 for all of the pdfs , including the photon . reinstating the lepton pdf contribution to the evolution equations in partonevolution , which in principle should be included for consistency , we find their effects on the quark and gluon pdfs to be negligible , with changes of less than @xmath216% . the only noticeable effect is the reduction of the photon pdf by about @xmath217% with mild @xmath31 dependence , due to splitting of the photons into lepton - antilepton pairs . this is presumably much less than other uncertainties in our analysis , so it is reasonable to leave out the lepton pdf contribution in our code . the percent difference in our prediction for the photon pdf , relative to the partonevolution prediction with and without the inclusion of the lepton pdfs , is shown in fig . [ fig : pecomp ] .    we have also checked our code against the program apfel  @xcite , which includes qed at lo and qcd at up to nnlo . the main difference in the apfel program is that the qcd and qed factorization scales can be taken to be independent , and the evolution with respect to each scale can be done successively . however , using the setting `` qavdp '' in apfel averages the two possible orderings for performing the evolutions , which should agree with our approach to @xmath81 . in addition , the apfel code has been recently updated to allow the simultaneous evolution of the qed and qcd scales , using the `` qunid '' setting . we have compared our code with apfel 2.6.0 , starting with the ct10nlo pdfs  @xcite with zero initial photon pdf at @xmath214 gev and evolving in qed at lo and qcd at nlo to @xmath212 gev . we have done the comparison using both the `` qavdp '' and the `` qunid '' settings for apfel . we obtain excellent agreement for the quark and gluon pdfs , with differences of less than @xmath215% over most of the range of @xmath31 for both apfel settings . in fig . [ fig : apfelcomp ] we show the results for the photon pdf . we obtain pretty good agreement with apfel with the `` qavdp '' setting , with differences of less than @xmath218% except at large @xmath219 . this is consistent with the @xmath81 differences expected in the different evolution procedures . we obtain excellent agreement with apfel with the `` qunid '' setting , with differences of less than @xmath220% over the full range of @xmath221 shown . we note that in fig . [ fig : apfelcomp ] we replace the evolution subroutine for @xmath49 in the apfel program with the code used in the ct global analysis code ; however , using the original @xmath49 subroutine in apfel still gives differences of less than 1% for the evolved photon pdf over the full range of @xmath31 when using the `` qunid '' setting . this is certainly much smaller than the uncertainties in the initial photon pdf itself . we have not checked our code directly against the mrst evolution code or the recently developed qcdnum+qed evolution code  @xcite , but we do note that the comparison between qcdnum+qed and apfel `` qavdp '' for the evolution of the photon pdf in ref . @xcite looks qualitatively similar to the results that we have found in fig . [ fig : apfelcomp ] . in addition benchmarking studies between apfel and these two evolution codes in ref . @xcite show agreement at a similar level to that which we have found with our code here . s.  dulat _ et al . _ , phys . d * 93 * , no . 3 , 033006 ( 2016 ) doi:10.1103/physrevd.93.033006 [ arxiv:1506.07443 [ hep - ph ] ] . r.  d.  ball _ et al . _ [ nnpdf collaboration ] , jhep * 1504 * , 040 ( 2015 ) [ arxiv:1410.8849 [ hep - ph ] ] . l.  a.  harland - lang , a.  d.  martin , p.  motylinski and r.  s.  thorne , eur . j.  c * 75 * , no . 5 , 204 ( 2015 ) [ arxiv:1412.3989 [ hep - ph ] ] . s.  alekhin , j.  blumlein and s.  moch , phys .  rev . d * 89 * , no . 5 , 054028 ( 2014 ) [ arxiv:1310.3059 [ hep - ph ] ] . a.  d.  martin , r.  g.  roberts , w.  j.  stirling and r.  s.  thorne , eur . j.  c * 39 * , 155 ( 2005 ) [ hep - ph/0411040 ] . a.  d.  martin and m.  g.  ryskin , eur . j.  c * 74 * , 3040 ( 2014 ) doi:10.1140/epjc / s10052 - 014 - 3040-y [ arxiv:1406.2118 [ hep - ph ] ] . s.  chekanov _ et al . _ [ zeus collaboration ] , phys . b * 687 * , 16 ( 2010 ) [ arxiv:0909.4223 [ hep - ex ] ] . s.  moch , j.  a.  m.  vermaseren and a.  vogt , nucl . b * 688 * , 101 ( 2004 ) [ hep - ph/0403192 ] . a.  vogt , s.  moch and j.  a.  m.  vermaseren , nucl . b * 691 * , 129 ( 2004 ) [ hep - ph/0404111 ] . j.  pumplin , d.  r.  stump , j.  huston , h.  l.  lai , p.  m.  nadolsky and w.  k.  tung , jhep * 0207 * , 012 ( 2002 ) . 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we describe the implementation of quantum electrodynamic ( qed ) evolution at leading order ( lo ) along with quantum chromodynamic ( qcd ) evolution at next - to - leading order ( nlo ) in the cteq - tea global analysis package . 
 the inelastic contribution to the photon parton distribution function ( pdf ) is described by a two - parameter ansatz , coming from radiation off the valence quarks , and based on the ct14 nlo pdfs . 
 setting the two parameters to be equal allows us to completely specify the inelastic photon pdf in terms of the inelastic momentum fraction carried by the photon , @xmath0 , at the initial scale @xmath1 gev . 
 we obtain constraints on the photon pdf by comparing with zeus data  @xcite on the production of isolated photons in deep inelastic scattering , @xmath2 . for this comparison we present a new perturbative calculation of the process that consistently combines the photon - initiated contribution with the quark - initiated contribution . 
 comparison with the data allows us to put a constraint at the 90% confidence level of @xmath3 for the inelastic photon pdf at the initial scale of @xmath1 gev in the one - parameter radiative ansatz . 
 the resulting inelastic ct14qed pdfs will be made available to the public . 
 in addition , we also provide ct14qedinc pdfs , in which the _ inclusive _ photon pdf at the scale @xmath4 is defined by the sum of the inelastic photon pdf and the elastic photon distribution obtained from the equivalent photon approximation .