_{1}

^{*}

The flux of papers from electron positron colliders containing data on the photon structure function ended naturally around 2005. It is thus timely to review the theoretical basis and confront the predictions with a summary of the experimental results. The discussion will focus on the increase of the structure function with x (for x away from the boundaries) and its rise with , both characteristics being dramatically different from hadronic structure functions. The agreement of the experimental observations with the theoretical calculations is a striking success of QCD. It also allows a new determination of the QCD coupling constant which very well corresponds to the values quoted in the literature.

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

Excitement rose after the first calculation of the leading order QCD corrections, because Witten [

In an invited talk at the 1983 Aachen conference on photon photon collisions [

Ten years later an algebraic error in the original calculation [

A new approach to follow the original goal [

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

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

which depends on the two structure functions

(with

For QED processes

direction of the incoming photon and the other is scattered at large angels (balancing the transverse momentum of the outgoing electron) the structure function

with

For heavy quarks with three colors, fractional charge e_{q} and masses

where

as the quark model or zero order QCD expression^{1} for the photon structure function if only light quarks are considered.

Using the general quark model relation

connecting structure function and quark densities

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.

The first QCD analysis of the photon structure function [

The nth Mellin moment of a function

For example, the quark model function

Like in DIS the quark densities are grouped into two classes described by different evolution equations, flavor non-singlet ( NS) and singlet (S):

where

analogous but not identical to the convention defined by Equation (12). Because of the factor x in Equation (18) the moments of the structure function are related to the moments of the quark densities by

or

We start with a discussion of the LO result for the pointlike

with

and

with

and

Similar (albeit more complicated) relations hold for

with

and

The dependence of

All splitting terms

with

A very useful combination of the results obtained so far is given by

where

Setting

with

The splitting functions

where m is now a continuous complex variable and the contour c has to lie on the right hand side of the rightmost singularity in

In practice, instead of inverting

were chosen because the shape of the corresponding valence and sea distributions in x-space is quite different. The pointlike LO solution in x-space is then given by

We focus on the asymptotic solution

pressions for

for

The results of both inversion methods agree very well, which is demonstrated in

The coefficients necessary for calculating the asymptotic functions

Obviously the LO QCD calculation (37) preserves the

0.7147 | −0.70394 | 0.1046 | −1.45262 | |

0.1723 | 0 | 0.1036 | 0 | |

0.0167 | 0.00011 | −0.1199 | −0.00024 | |

−0.0277 | −0.00031 | −0.1759 | −0.00138 | |

0.0361 | 0.00028 | 1.2996 | −0.00300 | |

−0.0010 | 0.00009 | −1.6261 | 0.01201 | |

0 | 0 | 0.2478 | −0.00740 |

0.6953 | −0.71529 | 0.0858 | −1.72537 | |

0.1761 | 0 | 0.1080 | 0 | |

0.0328 | 0.00033 | −0.1535 | −0.00061 | |

−0.0534 | −0.00093 | −0.4943 | 0.00153 | |

0.0695 | 0.00085 | 2.6800 | −0.02831 | |

−0.0193 | −0.00026 | −3.2036 | 0.05486 | |

0 | 0 | 0.5097 | −0.02808 |

is compared to the evaluation of Equation (37) in units of

The fact that the

To complete the picture,

Finally a careful inspection of the LO QCD result in

any finite value of L the quotient

In next to leading order the moments of the parton densities are changed, for example Equation (20) reads now

All NLO effects are contained in

(22). A similar relation holds for

densities, in NLO the quark model like relation Equation (19) between structure function and quark densities is also changed. Depending on the factorization scheme used, products of quark densities and the so called Wilson terms have to be added to the right hand side. The lengthy expressions needed to calculate the moments of the structure function in the

containing all NLO contributions in the last three terms on the right hand side.

For the numerical evaluation we prefer again to regroup all terms according to the valence and sea scheme. After inverting the moments the final equation describing the pointlike solution

is obtained. The strong coupling constant now has to be evaluated in NLO

with

Like in the LO case the structure of Equation (41) does not change if non asymptotic solutions are considered. One has then, however, for each pair of

Due to the negative correction

A further example is studied in

by the green curve. Due to an unfortunate algebraic error [

1992 were not correct and resulted in a strongly negative sea term (black curve in

As already shown by Witten [

piece an additional term which in lowest order is written as

determines the moments of the structure function for^{2} (see next section). In any case (43) is free of singularities but with

The pointlike terms can be rewritten as

The terms proportional to

be combined with the first term in (43) into a new hadronic contribution

with the

In this sum of hadronic terms and the asymptotic solution

We have shown this assumption to be valid for the LO and the NLO calculations. However in NNLO a completely different situation is to be faced. The most dangerous singularities originate now from NNLO terms

which leads in x-space to a divergent term

The principal problem of the poles of

luation of

The coupling of the photon to the final-state hadrons is mediated by quarks and antiquarks. If the transverse momentum

which is identical to the vector meson dominance (VMD) ansatz describing the hadronic nature of the photon

if the photon vector meson couplings

The result for ^{2} whereas below

Experiments measuring the photon structure function have until now only been performed at

is dominated by the so called two photon diagram shown in

The incoming leptons in

system X with an invariant mass

by a complicated combination of kinematical factors and six in principle unknown hadronic functions (four cross sections and two interference terms) depending on

In the limit

with

and

where the definition

The two photon cross sections

Replacing the cross sections

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

The standard expression (53) has first been derived by Kessler [

useful for rough estimates of the counting rate. One has, however, to keep in mind that neglecting the cutoff

The basic experimental procedure is thus given by investigating the reaction

Following the pioneering work of the PLUTO collaboration [^{2} to 780 GeV^{2} from the collaborations ALEPH [

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

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

2) A quark model calculation of the charm and bottom quark contribution using Equation (7) multiplied by

3) A detailed parameterization (VMD) for the hadronic part of the structure function [

Fitting the data with this model results in a value of

Following the method explained in [

which agrees nicely with the DIS average

In order to visualize the impressive agreement between data and theory two examples are presented. In ^{2} (black crosses) and the OPAL data [^{2} (blue crosses) are compared with our model. The data clearly do not follow the typical mesonic

Next the ^{2} = 5 and 800 GeV^{2}. Neglecting the small

piece the theoretical model can in LO be written as

The perturbative calculations have been extended to the region

Gluon radiation is efficiently suppressed for virtual photons, thus moving

model result. Analytically this can be proven easily [

Using ^{2} is

i.e. the quark model formula (34) with the log factor replaced by

^{2}This formula also demonstrates drastically how the introduction of a second scale destroys the sensitivity to^{2} a reduced sensitivity is maintained.

Regarding the determination of

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 [

where the factor

The relation between structure functions and cross sections is more complicated than discussed above for electron scattering off real photons. In the limit

using

one gets finally

Experimental data is scarce. The first results of the PLUTO collaboration [

For comparison with theory

LO and NLO [

part was added multiplied by a form factor

form factor the VMD term is reduced by a factor 2.5 and thus for the sake of simplicity the straight line model

Measurements of the photon structure function

New experimental input can only be expected from a new high energy ^{2} 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