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Photon Structure Functions

A hadron can be described by its parton contents, and as photons fluctuate into partonic states a similar description can also be adopted for the photon. The partonic content of a photon is described by the photon structure functions, which is closely related to the PDF's described previously. If one of the photons in a $\gamma$$\gamma$-collision is almost on-shell with a Q² $\approx$0, the whole process can be viewed as deep inelastic scattering of the tagged electron off the quasi-real target photon. The cross section can then be expressed as:

$${\frac{d\theta_{e\gamma \rightarrow eX}}{dxdQ^2}}$$ = $${\frac{2\pi\alpha^2}{xQ^4}}$$[(1 + (1 - y)²F^$\gamma$₂(x, Q²) - y²F^$\gamma$_L(x, Q²)]

, where

x = $${\frac{Q^2}{2q_2\cdot q_1}}$$ $$\approx$$ $${\frac{Q^2}{Q^2+W^2}}$$        ,        y = $${\frac{q_2 \cdot q_1}{kq_2}}$$ $$\approx$$ 1 - $${\frac{E_{tag}}{E_{beam}}}$$cos²($${\frac{\theta_{tag}}{2}}$$).

y is normally very small in the region studied, so it is only possible to measure F^$\gamma$₂. In the simple parton model F^$\gamma$₂ is taken as a sum over the quark and antiquark density functions. The Q² evolution of these PDF's are described by the Altarelli-Parisi equations. In case of the photon, there is an extra term corresponding to a gamma going into a q$\bar{q}$ pair. This renders the equations inhomogeneous and a linear rise of F^$\gamma$₂ with ln(Q²) is expected. This was seen by the LEP experiments, and is presented in Fig. 6.5.

One of the more important results from HERA was the observed rise of the proton structure function (F^p₂) at low x-values. A similar rise is also expected for the F^$\gamma$₂ structure function at very low x. Evidence for this is piling up from the latest results (Fig. 6.6) of the LEP experiments [27].

Figure 6.5: F₂^$\gamma$ as a function of log Q².
Figure 6.6: The low-x behavior of F^$\gamma$₂.