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A new $p_{\perp}$-ordered initial-state shower

In parallel with the introduction of a new $p_{\perp}$-ordered final-state shower, also a new $p_{\perp}$-ordered initial-state shower is introduced. Thus the old PYSSPA routine is complemented with the new PYPTIS one. The advantage of a $p_{\perp}$-ordered evolution becomes especially apparent when multiple interactions are considered in the next section, and transverse momentum can be used as a common ordering variable for multiple interactions and initial-state radiation, thereby allowing `interleaved evolution'. At the level of one single interaction, much of the formalism closely resembles the virtuality-ordered one presented above. In this section we therefore only give an overview of the new main features of the algorithm, with further details found in [Sjö04a].

Also the new initial-state showers are constructed by backwards evolution, starting at the hard interaction and successively reconstructing preceding branchings. To simplify the merging with first-order matrix elements, $z$ is again defined by the ratio of $\hat{s}$ before and after an emission. For a massless parton branching into one space-like with virtuality $Q^2$ and one with mass $m$, this gives $p_{\perp}^2 = Q^2 - z (\hat{s} + Q^2)(Q^2 + m^2)/\hat{s}$, or $p_{\perp}^2 = (1-z) Q^2 - z Q^4/\hat{s}$ for $m=0$. Here $\hat{s}$ is the squared invariant mass after the emission, i.e. excluding the emitted on-mass-shell parton.

The last term, $z Q^4/\hat{s}$, while normally expected to be small, gives a nontrivial relationship between $p_{\perp}^2$ and $Q^2$, e.g. with two possible $Q^2$ solutions for a given $p_{\perp}^2$. The second solution corresponds to a parton being emitted, at very large angles, in the `backwards' direction, where emissions from the incoming parton on the other side of the event should dominate. Based on such physics considerations, and in order to avoid the resulting technical problems, the evolution variable is picked to be $p_{\perp\mathrm{evol}}^2 = (1-z) Q^2$. Also here $p_{\perp\mathrm{evol}}$ sets the scale for the running $\alpha_{\mathrm{s}}$. Once selected, the $p_{\perp\mathrm{evol}}^2$ is translated into an actual $Q^2$ by the inverse relation $Q^2 = p_{\perp\mathrm{evol}}^2/(1-z)$, with trivial Jacobian: $\mathrm{d}Q^2/Q^2 \; \mathrm{d}z = \mathrm{d}%
p_{\perp\mathrm{evol}}^2/p_{\perp\mathrm{evol}}^2 \; \mathrm{d}z$. From $Q^2$ the correct $p_{\perp}^2$, including the $z Q^4/\hat{s}$ term, can be constructed.

Emissions on the two incoming sides are interspersed to form a single falling $p_{\perp}$ sequence, $p_{\perp\mathrm{max}} >
p_{\perp 1} > p_{\perp 2} > \ldots > p_{\perp\mathrm{min}}$. That is, the $p_{\perp}$ of the latest branching considered sets the starting scale of the downwards evolution on both sides, with the next branching occurring at the side that gives the largest such evolved $p_{\perp}$.

In a branching $a \to bc$, the newly reconstructed mother $a$ is assumed to have vanishing mass -- a heavy quark would have to be virtual to exist inside a proton, so it makes no sense to put it on mass shell. The previous mother $b$, which used to be massless, now acquires the space-like virtuality $Q^2$ and the correct $p_{\perp}$ previously mentioned, and kinematics has to be adjusted accordingly.

In the old algorithm, the $b$ kinematics was not constructed until its space-like virtuality had been set, and so four-momentum was explicitly conserved at each shower branching. In the new algorithm, this is no longer the case. (A corresponding change occurs between the old and new time-like showers, as noted above.) Instead it is the set of partons produced by this mother $b$ and the current mother $d$ on the other side of the event that collectively acquire the $p_{\perp}$ of the new $a \to bc$ branching. Explicitly, when the $b$ is pushed off-shell, the $d$ four-momentum is modified accordingly, such that their invariant mass is retained. Thereafter a set of rotations and boosts of the whole $b+d$-produced system bring them to the frame where $b$ has the desired $p_{\perp}$ and $d$ is restored to its correct four-momentum.

Matrix-element corrections can be applied to the first, i.e. hardest in $p_{\perp}$, branching on both sides of the event, to improve the accuracy of the high-$p_{\perp}$ description. Also several other aspects are directly inherited from the old algorithm.

The evolution of massive quarks (charm and bottom) is more messy than in the time-like case. A sensible procedure has been worked out, however, including modified splitting kernels and kinematics [Sjö04a].

In addition to the normal sharp cutoff at a $p_{\perp\mathrm{min}}$ scale, a new option has been included to smoothly regularise the divergence at around a given regularisation scale $p_{\perp 0}$, i.e. similar to what is being done for taming the rise of the multiple interactions cross section. The physical motivation is that, since at zero momentum transfer the beam hadron behaves like a colour singlet object, then at low scales the effective coupling of partons to coloured probes should be reduced, to take into account that this `colour screening' is not included at all in the lowest-order branching and interaction matrix elements.


next up previous contents
Next: Routines and Common-Block Variables Up: Initial-State Showers Previous: Matrix-element matching   Contents
Stephen_Mrenna 2012-10-24