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Joined Interactions

When the backwards evolution of initial-state radiation traces the prehistory to hard interactions, two partons participating in two separate hard scatterings may turn out to have a common ancestor, joined interactions (JI).

The joined interactions are well-known in the context of the evolution of multiparton densities [Kon79], but have not been applied to a multiple-interactions framework. A full implementation of the complete kinematics, intertwining MI, ISR and JI all possible ways, is a major undertaking, worth the effort only if the expected effects are non-negligible. Given the many uncertainties in all the other processes at play, one would otherwise expect that the general tuning of MI/ISR/FSR/ data would hide the effects of JI.

The current program can simulate the joining term in the evolution equations, and thereby estimate how often and at what $p_{\perp}$ values joinings should occur. However, the actual kinematics has not been worked out, so the suggested joinings are never performed. Instead the evolution is continued as if nothing had happened. Therefore this facility is more for general guidance than for detailed studies.

To see how it works, define the two-parton density $f^{(2)}_{bc}(x_b, x_c, Q^2)$ as the probability to have a parton $b$ at energy fraction $x_b$ and a parton $c$ at energy fraction $x_c$ when the proton is probed at a scale $Q^2$. The evolution equation for this distribution is

$\displaystyle \d f^{(2)}_{bc}(x_b, x_c, Q^2)$ $\textstyle =$ $\displaystyle \frac{\d Q^2}{Q^2} \, \frac{\alpha_{\mathrm{s}}}{2\pi}
\int \!\! ...
f^{(2)}_{ac}(x_a, x_c, Q^2) \, P_{a \to b d}(z) \,
\delta(x_b - z x_a) \right.$  
    $\displaystyle + \left. f^{(2)}_{ba}(x_b, x_a, Q^2) \, P_{a \to c d}(z) \,
\delta(x_c - z x_a) \right.$  
    $\displaystyle + \left. f_a(x_a, Q^2) \, P_{a \to bc}(z) \,
\delta(x_b - z x_a) \, \delta(x_c - (1-z) x_a) \right\}~.$ (229)

As usual, we assume implicit summation over the allowed flavour combinations. The first two terms in the above expression are the standard ones, where $b$ and $c$ evolve independently, up to flavour and momentum conservation constraints, and are already taken into account in the ISR framework. It is the last term that describes the new possibility of two evolution chains having a common ancestry.

Rewriting this into a backwards-evolution probability, the last term gives

$\displaystyle \d\mathcal{P}_{bc}(x_b, x_c, Q^2)$ $\textstyle =$ $\displaystyle \left\vert \frac{\d Q^2}{Q^2} \right\vert \, \frac{\alpha_{\mathr...
... f_a(x_a, Q^2)}{x_b x_c f^{(2)}_{bc}(x_b, x_c, Q^2)} \,
z (1-z) P_{a \to bc}(z)$  
  $\textstyle \simeq$ $\displaystyle \left\vert \frac{\d Q^2}{Q^2} \right\vert \, \frac{\alpha_{\mathr...
..._a, Q^2)}{x_b f_b(x_b, Q^2) \, x_c f_c(x_c, Q^2)} \,
z (1-z) P_{a \to bc}(z) ~,$ (230)

introducing the approximation $f^{(2)}_{bc}(x_b, x_c, Q^2) \simeq f_b(x_b, Q^2) \, f_c(x_c, Q^2)$ to put the equation in terms of more familiar quantities. Just like for MI and normal ISR, a form factor is given by integration over the relevant $Q^2$ range and exponentiation. In practice, eq. ([*]) is complemented by one more term given by the above probability.

next up previous contents
Next: Pile-up Events Up: Multiple Interactions (and Beam Previous: Multiple Interactions (and Beam   Contents
Stephen Mrenna 2007-10-30