Equivalent photon flux in leptons

With the 'gamma/lepton' option of a PYINIT call, an $\mathrm{e}\mathrm{p}$ or $\mathrm{e}^+\mathrm{e}^-$ event (or corresponding processes with muons) is factorized into the flux of virtual photons and the subsequent interactions of such photons. While real photons always are transverse (T), the virtual photons also allow a longitudinal (L) component. This corresponds to cross sections

$$\d\sigma(\mathrm{e}\mathrm{p}\rightarrow\mathrm{e}\mathbf{X}... ...^2) \;\d\sigma(\gamma^*_{\xi}\mathrm{p}\rightarrow\mathbf{X})$$ (55)

and

$$\d\sigma(\mathrm{e}\mathrm{e}\rightarrow\mathrm{e}\mathrm{e}... ...igma(\gamma^*_{\xi_1}\gamma^*_{\xi_2}\rightarrow\mathbf{X})\;.$$ (56)

For $\mathrm{e}\mathrm{p}$ events, this factorized ansatz is perfectly general, so long as azimuthal distributions in the final state are not studied in detail. In $\mathrm{e}^+\mathrm{e}^-$ events, it is not a good approximation when the virtualities $Q_1^2$ and $Q_2^2$ of both photons become of the order of the squared invariant mass $W^2$ of the colliding photons [Sch98]. In this region the cross section have terms that depend on the relative azimuthal angle of the scattered leptons, and the transverse and longitudinal polarizations are non-trivially mixed. However, these terms are of order $Q_1^2Q_2^2/W^2$ and can be neglected whenever at least one of the photons has low virtuality compared to $W^2$.

When $Q^2/W^2$ is small, one can derive [Bon73,Bud75,Sch98]

$$f_{\gamma/l}^{\mathrm{T}}(y,Q^2) = \frac{\alpha_{\mathrm{em}}}{2\pi} \left( \frac{(1+(1-y)^2}{y} \frac{1}{Q^2}-\frac{2m_{l}^2y}{Q^4} \right)\;,$$ (57)

$$f_{\gamma/l}^{\mathrm{L}}(y,Q^2) = \frac{\alpha_{\mathrm{em}}}{2\pi} \frac{2(1-y)}{y} \frac{1}{Q^2}\;,$$ (58)

where $l=\mathrm{e}^{\pm},~\mu^{\pm}$ or $\tau^{\pm}$. In $f_{\gamma/l}^{\mathrm{T}}$ the second term, proportional to $m_{l}^2/Q^4$, is not leading log and is therefore often omitted. Clearly it is irrelevant at large $Q^2$, but around the lower cut-off $Q^2_{\mathrm{min}}$ it significantly dampens the small-$y$ rise of the first term. (Note that $Q^2_{\mathrm{min}}$ is $y$-dependent, so properly the dampening is in a region of the $(y,Q^2)$ plane.) Overall, under realistic conditions, it reduces event rates by 5-10% [Sch98,Fri93].

The $y$ variable is defined as the light-cone fraction the photon takes of the incoming lepton momentum. For instance, for $l^+l^-$ events,

$$y_i = \frac{q_i k_j}{k_i k_j} ~, \qquad j=2 (1)~\mathrm{for}~i=1 (2) ~,$$ (59)

where the $k_i$ are the incoming lepton four-momenta and the $q_i$ the four-momenta of the virtual photons.

Alternatively, the energy fraction the photon takes in the rest frame of the collision can be used,

$$x_i = \frac{q_i (k_1 + k_2)}{k_i (k_1 + k_2)} ~, \qquad i=1,2 ~.$$ (60)

The two are simply related,

$$y_i = x_i + \frac{Q_i^2}{s} ~,$$ (61)

with $s=(k_1 + k_2)^2$. (Here and in the following formulae we have omitted the lepton and hadron mass terms when it is not of importance for the argumentation.) Since the Jacobian $\d (y_i, Q_i^2) / \d (x_i, Q_i^2) = 1$, either variable would be an equally valid choice for covering the phase space. Small $x_i$ values will be of less interest for us, since they lead to small $W^2$, so $y_i/x_i \approx 1$ except in the high-$Q^2$ tail, and often the two are used interchangeably. Unless special $Q^2$ cuts are imposed, cross sections obtained with $f_{\gamma/l}^{\mathrm{T,L}}(x,Q^2) \, \d x$ rather than $f_{\gamma/l}^{\mathrm{T,L}}(y,Q^2) \, \d y$ differ only at the per mil level. For comparisons with experimental cuts, it is sometimes relevant to know which of the two is being used in an analysis.

In the $\mathrm{e}\mathrm{p}$ kinematics, the $x$ and $y$ definitions give that

$$W^2 = x s = y s - Q^2 ~.$$ (62)

The $W^2$ expression for $l^+l^-$ is more complicated, especially because of the dependence on the relative azimuthal angle of the scattered leptons, $\varphi_{12} = \varphi_1 - \varphi_2$:

$$W^2 = x_1 x_2 s + \frac{2 Q_1^2 Q_2^2}{s} - 2 \sqrt{1 - x_1 - \frac{Q_1^2}{s}} \sqrt{1 - x_2 - \frac{Q_2^2}{s}} Q_1 Q_2 \cos\varphi_{12}$$

$$= y_1 y_2 s - Q_1^2 - Q_2^2 + \frac{Q_1^2 Q_2^2}{s} - 2 \sqrt{1 - y_1} \sqrt{1 - y_2} Q_1 Q_2 \cos\varphi_{12} ~.$$ (63)

The lepton scattering angle $\theta_i$ is related to $Q_i^2$ as

$$Q_i^2 = \frac{x_i^2}{1-x_i} m_i^2 + (1-x_i) \left( s - \frac{2}{(1-x_i)^2} m_i^2 - 2 m_j^2 \right) \sin^2(\theta_i/2) ~,$$ (64)

with $m_i^2 = k_i^2 = {k'}^2_i$ and terms of $O(m^4)$ neglected. The kinematical limits thus are

$$(Q_i^2)_{\mathrm{min}} \approx \frac{x_i^2}{1 - x_i} m_i^2 ~,$$ (65)

$$(Q_i^2)_{\mathrm{max}} \approx (1 - x_i) s ~,$$ (66)

unless experimental conditions reduce the $\theta_i$ ranges.

In summary, we will allow the possibility of experimental cuts in the $x_i$, $y_i$, $Q_i^2$, $\theta_i$ and $W^2$ variables. Within the allowed region, the phase space is Monte Carlo sampled according to $\prod_i (\d Q_i^2/Q_i^2) \, (\d x_i / x_i) \, \d\varphi_i$, with the remaining flux factors combined with the cross section factors to give the event weight used for eventual acceptance or rejection. This cross section in its turn can contain the parton densities of a resolved virtual photon, thus offering an effective convolution that gives partons inside photons inside electrons.