Deeply Inelastic Scattering and $\gamma^*\gamma^*$ physics

MSEL = 1, 2, 35, 36, 37, 38
ISUB =

10	$\mathrm{f}_i \mathrm{f}_j \to \mathrm{f}_k \mathrm{f}_l$
83	$\mathrm{q}_i \mathrm{f}_j \to \mathrm{Q}_k \mathrm{f}_l$
99	$\gamma^*\mathrm{q}\to \mathrm{q}$
131	$\mathrm{f}_i \gamma^*_{\mathrm{T}} \to \mathrm{f}_i \mathrm{g}$
132	$\mathrm{f}_i \gamma^*_{\mathrm{L}} \to \mathrm{f}_i \mathrm{g}$
133	$\mathrm{f}_i \gamma^*_{\mathrm{T}} \to \mathrm{f}_i \gamma$
134	$\mathrm{f}_i \gamma^*_{\mathrm{L}} \to \mathrm{f}_i \gamma$
135	$\mathrm{g}\gamma^*_{\mathrm{T}} \to \mathrm{f}_i \overline{\mathrm{f}}_i$
136	$\mathrm{g}\gamma^*_{\mathrm{L}} \to \mathrm{f}_i \overline{\mathrm{f}}_i$
137	$\gamma^*_{\mathrm{T}} \gamma^*_{\mathrm{T}} \to \mathrm{f}_i \overline{\mathrm{f}}_i$
138	$\gamma^*_{\mathrm{T}} \gamma^*_{\mathrm{L}} \to \mathrm{f}_i \overline{\mathrm{f}}_i$
139	$\gamma^*_{\mathrm{L}} \gamma^*_{\mathrm{T}} \to \mathrm{f}_i \overline{\mathrm{f}}_i$
140	$\gamma^*_{\mathrm{L}} \gamma^*_{\mathrm{L}} \to \mathrm{f}_i \overline{\mathrm{f}}_i$

Among the processes in this section, 10 and 83 are intended to stand on their own, while the rest are part of the newer machinery for $\gamma^*\mathrm{p}$ and $\gamma^*\gamma^*$ physics. We therefore separate the description in this section into these two main parts.

The Deeply Inelastic Scattering (DIS) processes, i.e. $t$-channel electroweak gauge boson exchange, are traditionally associated with interactions between a lepton or neutrino and a hadron, but processes 10 and 83 can equally well be applied for $\mathrm{q}\mathrm{q}$ scattering in hadron colliders (with a cross section much smaller than corresponding QCD processes, however). If applied to incoming $\mathrm{e}^+\mathrm{e}^-$ beams, process 10 corresponds to Bhabha scattering.

For process 10 both $\gamma$, $\mathrm{Z}^0$ and $\mathrm{W}^{\pm}$ exchange contribute, including interference between $\gamma$ and $\mathrm{Z}^0$. The switch MSTP(21) may be used to restrict to only some of these, e.g. neutral or charged current only.

The option MSTP(14)=10 (see previous section) has now been extended so that it also works for DIS of an electron off a (real) photon, i.e. process 10. What is obtained is a mixture of the photon acting as a vector meson and it acting as an anomalous state. This should therefore be the sum of what can be obtained with MSTP(14)=2 and =3. It is distinct from MSTP(14)=1 in that different sets are used for the parton distributions -- in MSTP(14)=1 all the contributions to the photon distributions are lumped together, while they are split in VMD and anomalous parts for MSTP(14)=10. Also the beam remnant treatment is different, with a simple Gaussian distribution (at least by default) for MSTP(14)=1 and the VMD part of MSTP(14)=10, but a powerlike distribution $\d k_{\perp}^2 / k_{\perp}^2$ between PARP(15) and $Q$ for the anomalous part of MSTP(14)=10.

To access this option for $\mathrm{e}$ and $\gamma$ as incoming beams, it is only necessary to set MSTP(14)=10 and keep MSEL at its default value. Unlike the corresponding option for $\gamma\mathrm{p}$ and $\gamma\gamma$, no cuts are overwritten, i.e. it is still your responsibility to set these appropriately.

Cuts especially appropriate for DIS usage include either CKIN(21)-CKIN(22) or CKIN(23)-CKIN(24) for the $x$ range (former or latter depending on which side is the incoming real photon), CKIN(35)-CKIN(36) for the $Q^2$ range, and CKIN(39)-CKIN(40) for the $W^2$ range.

In principle, the DIS $x$ variable of an event corresponds to the $x$ value stored in PARI(33) or PARI(34), depending on which side the incoming hadron is on, while the DIS $Q^2 = -\hat{t} =$-PARI(15). However, just like initial- and final-state radiation can shift jet momenta, they can modify the momentum of the scattered lepton. Therefore the DIS $x$ and $Q^2$ variables are not automatically conserved. An option, on by default, exists in MSTP(23), where the event can be `modified back' so as to conserve $x$ and $Q^2$, but this option is rather primitive and should not be taken too literally.

Process 83 is the equivalent of process 10 for $\mathrm{W}^{\pm}$ exchange only, but with the heavy-quark mass included in the matrix element. In hadron colliders it is mainly of interest for the production of very heavy flavours, where the possibility of producing just one heavy quark is kinematically favoured over pair production. The selection of the heavy flavour is already discussed in section .

Turning to the other processes, part of the $\gamma^*\mathrm{p}$ and $\gamma^*\gamma^*$ process-mixing machineries, 99 has close similarities with the above discussed 10 one. Whereas 10 would simulate the full process $\mathrm{e}\mathrm{q}\to \mathrm{e}\mathrm{q}$, 99 assumes a separate machinery for the flux of virtual photons, $\mathrm{e}\to \mathrm{e}\gamma^*$ and only covers the second half of the process, $\gamma^*\mathrm{q}\to \mathrm{q}$. One limitation of this factorization is that only virtual photons are considered in process 99, not contributions from the $\mathrm{Z}^0$ neutral current or the $\mathrm{W}^{\pm}$ charged current.

Note that 99 has no correspondence in the real-photon case, but has to vanish in this limit by gauge invariance, or indeed by simple kinematics considerations. This, plus the desire to avoid double-counting with real-photon physics processes, is why the cross section for this process is explicitly made to vanish for photon virtuality $Q^2 \to 0$, eq. (), also when parton distributions have not been constructed to fulfil this, see MSTP(19). (No such safety measures are present in 10, again illustrating how the two are intended mainly to be used at large or at small $Q^2$, respectively.)

For a virtual photon, processes 131-136 may be viewed as first-order corrections to 99. The three with a transversely polarized photon, 131, 133 and 135, smoothly reduce to the real-photon direct (single-resolved for $\gamma\gamma$) processes 33, 34 and 54. The other three, corresponding to the exchange of a longitudinal photon, vanish like $Q^2$ for $Q^2 \to 0$. The double-counting issue with process 99 is solved by requiring the latter process not to contain any shower branchings with a $p_{\perp}$ above the lower $p_{\perp}$ cut-off of processes 131-136. The cross section is then to be reduced accordingly, see eq. () and the discussion there, and again MSTP(19).

We thus see that process 99 by default is a low-$p_{\perp}$ process in about the same sense as process 95, giving `what is left' of the total cross section when jet events have been removed. Therefore, it will be switched off in event class mixes such as MSTP(14)=30 if CKIN(3) is above $p_{\perp\mathrm{min}}(W^2)$ and MSEL is not 2. There is a difference, however, in that process 99 events still are allowed to contain shower evolution (although currently only the final-state kind has been implemented), since the border to the other processes is at $p_{\perp}= Q$ for large $Q$ and thus need not be so small. The $p_{\perp}$ scale of the `hard process', stored e.g. in PARI(17) always remains 0, however. (Other PARI variables defined for normal $2 \to 2$ and $2 \to 1$ processes are not set at all, and may well contain irrelevant junk left over from previous events.)

Processes 137-140, finally, are extensions of process 58 from the real-photon limit to the virtual-photon case, and correspond to the direct process of $\gamma^*\gamma^*$ physics. The four cases correspond to either of the two photons being either transversely or longitudinally polarized. As above, the cross section of a longitudinal photon vanishes when its virtuality approaches 0.