The notion that hadron production in inelastic electron photon scattering can be described in terms of structure functions like in electron nucleon scattering is on first sight surprising because photons are pointlike particles whereas nucleons have a radius of roughly 1 fm. Nevertheless the concept makes sense, not because the photon consists of pions, quarks, gluons etc., but because it couples to other particles and thus can fluctuate e.g. into a quark antiquark pair or a meson. These two basic processes are distinguished by the terms pointlike and hadronic throughout the paper. The idea of a photon fluctuating into a meson or other vector mesons was soon applied to estimate the inelastic scattering cross section in the vector meson dominance model [1] [2] . Calculating the structure function in the quark model [3] then opened the intriguing possibility to investigate experimentally a structure function rising towards large x and showing a distinctive scale breaking because of the proportionality to.

Excitement rose after the first calculation of the leading order QCD corrections, because Witten [4] not only calculated the markedly different x dependence of the structure function in QCD but demonstrated that the QCD parameter could in principle be determined by measuring an absolute cross section quite in contrast to lepton nucleon scattering, where small scale breaking effects in the evolution of the structure function have to be studied. This “remarkable result” [5] initiated intensive discussions between theorists and experimentalists and passed the first experimental test [6] with flying colors. QCD calculations at next to leading order [7] [8] allowed to give a precise meaning in the renormalization scheme, but also revealed a sickness of the absolute perturbative calculation, producing negative values of near.

In an invited talk at the 1983 Aachen conference on photon photon collisions [9] , the audience was warned that the implications of these discoveries for the experimental goal of a direct determination of from were not altogether positive despite “the almost incredible advances on the experimental side”. Instead it was recommended to utilize the evolution like in deep inelastic scattering, a program which was also pursued by other groups [10] .

Ten years later an algebraic error in the original calculation [7] was discovered. Correcting this error [11] [12] squeezed the negative spike near to very small x values where it is of negligible practical importance. For the same reason ad hoc attempts [13] to cure the problem (although still in principle important) proved to be unnecessary for experimental analyses at NLO accuracy.

A new approach to follow the original goal [14] showed promising results. However, based on the results of [15] [16] the structure function for virtual photons was calculated [17] in next-to-next-to-leading-order (NNLO). The findings of this investigation forces one to the conclusion that an absolute prediction for the structure function of real photons is unstable at the three loop level. The concern of the 1980’s is thus still valid, albeit at a higher order in the perturbative series.

Deep-inelastic electron-photon scattering at high energies

is characterized by a large momentum transfer Q of the scattered electron and a large invariant mass W of the hadrons. The electron energies and in initial and final state combined with the scattering angle define the (negative) momentum transfer squared, , on the electron line with

and the Bjorken scaling variable x defined as

are the essential variables for discussing the dynamics of the scattering process as can be seen from the cross section formula corresponding to Figure 1

which depends on the two structure functions and. Here is a linear combination of, describing the exchange of transversely polarized virtual photons, and

, associated with the exchange of longitudinally polarized virtual photons. The scaling variable y used in the last equation is given by

(with) and can also be directly calculated from and via

For QED processes in a region of phase space where one of the muons travels along the

direction of the incoming photon and the other is scattered at large angels (balancing the transverse momentum of the outgoing electron) the structure function has been calculated [11] [18] to be

with and M denoting the muon mass. The functions and are given by

For heavy quarks with three colors, fractional charge e_q and masses the right hand side of Equation (7) has to be multiplied by. Light quarks have current masses with (or at least). Neglecting all terms in Equations (8), (9) and respecting leads for each light quark to the expression

where is a mass parameter somehow describing the confinement of the light quarks [3] . The QCD parameter can be interpreted as an inverse confinement radius. We therefore replace by and keep in the leading log approximation only terms proportional to resulting in

as the quark model or zero order QCD expression¹ for the photon structure function if only light quarks are considered.

Using the general quark model relation

connecting structure function and quark densities one obtains for light quarks the expression

with

The factor 2 in Equation (12) accounts for the fact that the photon contains quarks and antiquarks with equal densities but the sum runs over quarks only.

Experiments measuring the photon structure function have until now only been performed at storage rings. The reaction

is dominated by the so called two photon diagram shown in Figure 7 which also includes some kinematical definitions. Originally these reactions have been considered only as a background to annihilation

but became, due to the absence of high energetic real photon beams, the only source of direct experimental information about.

The incoming leptons in Figure 7 radiate virtual photons with four momenta producing a hadronic

system X with an invariant mass The sixfold differential cross section is given

by a complicated combination of kinematical factors and six in principle unknown hadronic functions (four cross sections and two interference terms) depending on. The general formalism has been discussed in great detail in [18] , for a recent review and extension see [32] . The paper of Budnev et al. [18] served as the basis of all experimental analyses.

In the limit which is realized by very small forward scattering angles of one of the incoming leptons (e.g. the positron) the relevant formulae are greatly simplified and the cross section reads

with

and

where the definition has been used.

The two photon cross sections and depend on and. The indices represent the transverse (T) and longitudinal (L) polarization of the virtual photons. The physical interpretation of these equations is like follows: the incoming positron is replaced by a beam of quasi real photons with transverse polarization traveling along the positron direction. The number of photons in the energy interval is given by. The term denotes the number of transversely polarized photons radiated from the electron scattered into the solid angle and energy interval at angles. With the help of the polarization parameter the number of longitudinal photons is given by. A very useful feature of this formalism is the factorization of the flux factors into and where and depend on electron variables and on positron variables only. This allows for a considerable simplification calculating the photon fluxes in Monte Carlo routines simulating the experiments. It has been shown [33] that for mrad the numerical difference in evaluating the incoming photon densities from this approach or from the exact formula [18] is less than 0.5% for.

Replacing the cross sections by the structure functions

we arrive after a change of variables at

which corresponds to Equation (4) multiplied by the spectral density of the incoming photons. Under actual experimental conditions, y is quite small in general, so that is very difficult to measure. Experiments usually focus on the measurement of and neglect. This is theoretically backed further by the fact that quark model and QCD predict to be the leading component.

The standard expression (53) has first been derived by Kessler [34] . In the spirit of the leading log appro- ximation it can be replaced by

useful for rough estimates of the counting rate. One has, however, to keep in mind that neglecting the cutoff has not only numerical consequences, but quickly violates the basic assumption. It is difficult to quote unambiguous limits for the maximum allowed mean values. Detailed calculation of pair production [35] revealed that for and mrad one has and 97% of the cross section is contained in Equation (50), a result which is likely also to be valid for the quark model and QCD. It follows that the experiments need forward spectrometers with electromagnetic calorimeters very close to the beam pipe which allow to reject positrons with angles larger than about 25 mrad via the method of antitagging.

The basic experimental procedure is thus given by investigating the reaction with the electron scattered at angles larger than and the positron traveling undetected down the beam pipe (and vice versa). Due to the unknown energy of the outgoing positron is also not known and the usual relation cannot be used for calculating x. Therefore hadronic calorimeters reaching down to small forward scattering angles are needed in order to measure the invariant mass W of the produced hadronic system and calculate x from Equation (3). Unfortunately the remnants of the antiquark in Figure 1 will also be dominantly concentrated at small angles and losses in the hadronic energy are unavoidable. With sophisticated unfolding methods have to be employed in order to reconstruct x. These methods are described in the experimental publications and reviewed in [26] and [35] . A discussion of the various Monte Carlo routines used by the different experimental groups in evaluating the cross section can e.g. be found in [35] .

Following the pioneering work of the PLUTO collaboration [6] many experiments have been performed at all high energy storage rings. In order to avoid the region of small x with its mixture of correlated hadronic and pointlike contributions we exclude data with. Data where the charm component has been subtracted and all data published in conference proceedings only are also discarded. Only the most recent publication of statistically overlapping data of the same collaboration was accepted. This selection leads to 109 experimental values of with values ranging from 4.3 GeV² to 780 GeV² from the collaborations ALEPH [36] [37] , AMY [38] [39] , DELPHI [40] , JADE [41] , L3 [42] - [45] , OPAL [46] - [48] , PLUTO [31] [49] , TASSO [50] , TOPAZ [51] and TPC/2g [52] . In cases where the experimental uncertainties could only be read off the figures the tables of Nisius [35] were used. As usual the experimental x and values are obtained from an averaging procedure over the sometimes rather large x and bins. In most cases x coincides with the bin center.

After the 1980 crisis of the perturbative calculation most QCD analyses were performed like in deep inelastic scattering by comparing the data to models obtained by evolving the parton densities from a starting scale up to. For a recent extensive study see [53] . On the other hands side data at high and high x (defined by and) were fitted to the asymptotic pointlike solution (41) for, supplemented by the quark model formula (7) for a charm quark with mass 1.5 GeV [14] . The fit described the data very well and resulted in with for 20 experimental data.

Here we follow a more radical approach and fit the whole sample of 109 data sets to a model whose three components have been discussed in the previous sections:

1) The pointlike asymptotic NLO QCD prediction for 3 light flavors in the scheme with as the only free parameter using the coefficients of Table 1(a).

2) A quark model calculation of the charm and bottom quark contribution using Equation (7) multiplied by. Applying the scheme for the light quark QCD calculation it is only natural to use the masses and as quoted by the Particle Data Group [54] .

3) A detailed parameterization (VMD) for the hadronic part of the structure function [30] including the evolution. Examples for different values of are displayed in Figure 6.

Fitting the data with this model results in a value of. The quality of the fit, mea- sured in terms of the value per degree of freedom, is given by. A value slightly below unity is probably explained by the neglect of bin to bin correlations in the fitting procedure. These correlations were only given in six of the seventeen experimental publications used.

Following the method explained in [55] is converted to and therefore is obtained in NLO where the error only reflects the experimental uncertainties. In order to estimate the theoretical error we first neglected the bottom quark contribution which changes by a very small amount (0.0002). Varying within the quoted error of resulted in . The most important source of theoretical uncertainty is the treatment of the hadronic contri- bution. This error is hard to estimate. Possible interferences between hadronic and pointlike part are very likely restricted to the region which is excluded by our data selection. The authors of [29] emphasize the very good agreement of their result with pionic and photonic structure function data. Assuming a 10% normalization error for yields. Adding all errors in quadrature the final result is

which agrees nicely with the DIS average and the world average as given in [54] [56] .

In order to visualize the impressive agreement between data and theory two examples are presented. In Figure 8 the PLUTO data [31] at 4.3 GeV² (black crosses) and the OPAL data [48] at 39.7 GeV² (blue crosses) are compared with our model. The data clearly do not follow the typical mesonic dependence and also demonstrate implicitly the rather strong dependence predicted by QCD.

Next the dependence of is directly tested by selecting data with. The average x value was determined taking the mean value of the x intervals quoted in the experimental papers. The 36 data sets are grouped in bins with equal bin width in. Each value is then shifted to the center of the corresponding bin using the theoretical model and the weighted average of all data within the bin is calculated. The result is shown in Figure 9 together with the theoretical curve calculated for. The increase of the data with is clearly seen, especially in contrast to the well known slight decrease of the proton structure function for between Q² = 5 and 800 GeV². Neglecting the small dependence of the hadronic

piece the theoretical model can in LO be written as. A fit of the data in

Figure 9 according to this ansatz yields thus establishing numerically the predicted increase with beyond any doubt. This value agrees very well with the earlier analysis of [35] .

The perturbative calculations have been extended to the region [57] [58] which is experi- mentally accessible requesting also the positron being scattered into finite angles (double tagging). Regarding the theory one has in the formalism of [11] simply to replace the parameter by the variable for the cal- culation of in LO. Ueamtsu and Walsh [58] emphasized that for virtual photons also the hadronic piece is perturbatively calculable. The required additional terms are given in [17] [58] .

Gluon radiation is efficiently suppressed for virtual photons, thus moving closer to the quark

model result. Analytically this can be proven easily [57] by investigating the LO order solution, e.g., Equation (20). Because in LO is proportional to we have

Using the nominator reduces in the limit to. Since a similar relation holds for the singlet term, the final result² is

i.e. the quark model formula (34) with the log factor replaced by. As an illustration the LO prediction for, and is compared in Figure 10 with Equation (59) showing perfect agreement at small and medium x values. The parameter free QCD prediction (59) is, however, difficult to be tested experimentally because with the present value of the condition

can hardly be achieved.

²This formula also demonstrates drastically how the introduction of a second scale destroys the sensitivity to. In case of the starting scale of section III with values of 0.3 to 1.0 GeV² a reduced sensitivity is maintained.

Regarding the determination of from the measured cross section, it should be remembered

that the virtual photon photon scattering is in general described by four cross sections and two interference terms

(see Section IV). After proper integration over the interference terms the cross section formula of [18] can be written as

where the factor is approximately interpreted as describing the fluxes of transverse virtual photons from the incoming electron and positron. The polarization parameters and are for typical experimental con- ditions close to 1 and therefore

The relation between structure functions and cross sections is more complicated than discussed above for electron scattering off real photons. In the limit the general formulae given in [26] reduce to

using and. Defining an effective structure function via

one gets finally

Experimental data is scarce. The first results of the PLUTO collaboration [60] at and (black crosses in Figure 11) were only followed by data of the L3 collaboration [44] at and (blue crosses in Figure 11).

For comparison with theory is calculated in NLO for 3 flavors choosing. According to Equation (64) cannot be neglected. The longitudinal structure function of real photons is extensively discussed in the literature [59] . For virtual photons has been calculated in

LO and NLO [17] [58] . Here we combine the LO result with the NLO calculation of the pointlike and hadronic terms. The charm quark contribution for and is taken from the quark model result for real photons as recommended in [29] . Because the PLUTO data are taken at a value close to the real photon case a VMD

part was added multiplied by a form factor where is the meson mass. Using this

form factor the VMD term is reduced by a factor 2.5 and thus for the sake of simplicity the straight line model is applied improving somehow the agreement with the data at low x. Altogether the red curve (PLUTO) and the green curve (L3) in Figure 11 are in very good agreement with the data although admittedly this is not a very decisive test due to the large experimental errors. A comparison including the rather small NNLO corrections can be found in [63] .

Measurements of the photon structure function taken at colliders were confronted with theoretical models. For real photons the main component is the fixed flavor NLO asymptotic QCD result in the scheme as given in Equation (41) evaluated with the functions of Table 1. This part is complemented by charm and bottom heavy quark contributions calculated in the quark model and by a hadronic contribution taken from vector meson dominance. The model describes not only the x and distributions very well but also allows for a precise determination of the strong coupling constant, yielding. As explained above, the treatment of the hadronic and the heavy quark contribution to does not follow from first principles but is based on model assumptions. The validity of these assumptions is supported by the observation that using the standard model value of the selected data are described by the model of Section 5 with. Finally the few available data for virtual photons agree well with the QCD predictions.

New experimental input can only be expected from a new high energy collider. At the planned linear collider ILC [61] it is in principle possible to install a high intensity beam of real photons via backscattering of laser light. This would for the first time allow to study inelastic electron photon scattering in a beam of real pho- tons with a spectrum and intensity far superior to the virtual photons used until now [62] . In addition nagging doubts about the P² cutoff in some of the two photon experiments are baseless in such an environment.

First of all, I want to thank P.M. Zerwas for his constant support and the many discussions concerning the theoretical basis. I am very grateful for the help I got from R. Nisius. Useful conversations with M. Klasen are also gratefully acknowledged.

ChristophBerger, (2015) Photon Structure Function Revisited. Journal of Modern Physics,06,1023-1043. doi: 10.4236/jmp.2015.68107