The photon physics machinery in PYTHIA has been largely expanded in recent years. Historically, the model was first developed for photoproduction, i.e. a real photon on a hadron target [Sch93,Sch93a]. Thereafter physics was added in the same spirit [Sch94a,Sch97]. More recently also virtual photons have been added to the description [Fri00], including the nontrivial transition region between real photons and Deeply Inelastic Scattering (DIS). In this section we partly trace this evolution towards more complex configurations.
and cross sections can again be
parameterized in a
form like eq. (), which is not so obvious since
the photon has more complicated structure than an ordinary hadron.
In fact, the structure is still not so well understood. The model we
outline is the one studied by Schuler and Sjöstrand
[Sch93,Sch93a], and further updated in [Fri00]. In this model
the physical photon is represented by
By virtue of this superposition, one is led to a model of interactions, where three different kinds of events may be distinguished:
In order that the above classification is smooth and free of double counting, one has to introduce scales that separate the three components. The main one is , which separates the low-mass vector meson region from the high-mass one, GeV. Given this dividing line to VMD states, the anomalous parton distributions are perturbatively calculable. The total cross section of a state is not, however, since this involves aspects of soft physics and eikonalization of jet rates. Therefore an ansatz is chosen where the total cross section of a state scales like , where the adjustable parameter for light quarks. The scale is roughly equated with half the mass of the GVMD state. The spectrum of GVMD states is taken to extend over a range , where is identified with the cut-off of the perturbative jet spectrum in hadronic interactions, GeV at typical energies, see section and especially eq. (). Above that range, the states are assumed to be sufficiently weakly interacting that no eikonalization procedure is required, so that cross sections can be calculated perturbatively without any recourse to pomeron phenomenology. There is some arbitrariness in that choice, and some simplifications are required in order to obtain a manageable description.
The VMD and GVMD/anomalous events are together called resolved ones. In terms of high- jet production, the VMD and anomalous contributions can be combined into a total resolved one, and the same for parton-distribution functions. However, the two classes differ in the structure of the underlying event and possibly in the appearance of soft processes.
In terms of cross sections, eq. () corresponds
The direct cross section is, to lowest order, the perturbative cross
section for the two processes
, with a lower cut-off
, in order to
avoid double-counting with the interactions of the GVMD states.
Properly speaking, this should be multiplied by the coefficient,
The VMD factor gives the probability for the transition . The coefficients are determined from data to be (with a non-negligible amount of uncertainty) 2.20 for , 23.6 for , 18.4 for and 11.5 for . Together these numbers imply that the photon can be found in a VMD state about 0.4% of the time, dominated by the contribution. All the properties of the VMD interactions can be obtained by appropriately scaling down physics predictions. Thus the whole machinery developed in the previous section for hadron-hadron interactions is directly applicable. Also parton distributions of the VMD component inside the photon are obtained by suitable rescaling.
The contribution from the `anomalous' high-mass fluctuations to the
total cross section is obtained by a convolution of the fluctuation
As an illustration of this scenario, the phase space of events may be represented by a plane. Two transverse momentum scales are distinguished: the photon resolution scale and the hard interaction scale . Here is a measure of the virtuality of a fluctuation of the photon and corresponds to the most virtual rung of the ladder, possibly apart from . As we have discussed above, the low- region corresponds to VMD and GVMD states that encompasses both perturbative high- and nonperturbative low- interactions. Above , the region is split along the line . When the photon is resolved by the hard interaction, as described by the anomalous part of the photon distribution function. This is as in the GVMD sector, except that we should (probably) not worry about multiple parton-parton interactions. In the complementary region , the scale is just part of the traditional evolution of the parton distributions of the proton up to the scale of , and thus there is no need to introduce an internal structure of the photon. One could imagine the direct class of events as extending below and there being the low- part of the GVMD class, only appearing when a hard interaction at a larger scale would not preempt it. This possibility is implicit in the standard cross section framework.
In physics [Sch94a,Sch97], the superposition in
eq. () applies separately for each of the two
incoming photons. In total there are therefore
combinations. However, trivial symmetry reduces this to six distinct
classes, written in terms of the total cross section
(cf. eq. ()) as
The six different kinds of events are thus:
Like for photoproduction events, this can be illustrated in a parameter space, but now three-dimensional, with axes given by the , and scales. Here each is a measure of the virtuality of a fluctuation of a photon, and corresponds to the most virtual rung on the ladder between the two photons, possibly excepting the endpoint ones. So, to first approximation, the coordinates along the axes determine the characters of the interacting photons while determines the character of the interaction process. Double-counting should be avoided by trying to impose a consistent classification. Thus, for instance, with and gives a hard interaction between a VMD and a GVMD photon, while with and is a single-resolved process (directVMD; with now in the parton distribution evolution).
In much of the literature, where a coarser classification is used, our directdirect is called direct, our directVMD and directGVMD is called single-resolved since they both involve one resolved photon which gives a beam remnant, and the rest are called double-resolved since both photons are resolved and give beam remnants.
If the photon is virtual, it has a reduced probability to fluctuate into
a vector meson state, and this state has a reduced interaction probability.
This can be modelled by a traditional dipole factor
for a photon of virtuality , where
for a GVMD state. Putting it all together, the cross
section of the GVMD sector of photoproduction then scales like
For a virtual photon the DIS process
is also possible,
but by gauge invariance its cross section must vanish in the limit
. At large , the direct processes can be considered
correction to the lowest-order
DIS process, but the direct ones survive for . There is no
unique prescription for a proper combination at all , but we have
attempted an approach that gives the proper limits and minimizes
double-counting. For large , the DIS
is proportional to the structure function with the
. Since normal parton distribution
parameterizations are frozen below some scale and therefore do not
obey the gauge invariance condition, an ad hoc factor
is introduced for the conversion from
the parameterized to a
In order to avoid double-counting between DIS and direct events, a
is imposed on direct events. In the
remaining DIS ones, denoted lowest order (LO) DIS, thus .
This would suggest a subdivision
given by eq. () and
by the perturbative matrix elements. In the limit , the
DIS cross section is now constructed to vanish while the direct is not,
so this would give
here we expect the correct answer not to be a negative number but an
exponentially suppressed one, by a Sudakov form factor. This modifies
the cross section:
The overall picture, from a DIS perspective, now requires three scales to be kept track of. The traditional DIS region is the strongly ordered one, , where DGLAP-style evolution [Alt77,Gri72] is responsible for the event structure. As always, ideology wants strong ordering, while the actual classification is based on ordinary ordering . The region is also DIS, but of the direct kind. The region where is the smallest scale corresponds to non-ordered emissions, that then go beyond DGLAP validity, while the region cover the interactions of a resolved virtual photon. Comparing with the plane of real photoproduction, we conclude that the whole region involves no double-counting, since we have made no attempt at a non-DGLAP DIS description but can choose to cover this region entirely by the VMD/GVMD descriptions. Actually, it is only in the corner that an overlap can occur between the resolved and the DIS descriptions. Some further considerations show that usually either of the two is strongly suppressed in this region, except in the range of intermediate and rather small . Typically, this is the region where is not close to zero, and where is dominated by the valence-quark contribution. The latter behaves roughly , with an of the order of 3 or 4. Therefore we will introduce a corresponding damping factor to the VMD/GVMD terms.
In total, we have now arrived at our ansatz for all :
processes, finally, the parameter space is now
five-dimensional: , , , and .
As before, an effort is made to avoid double-counting, by having a
unique classification of each region in the five-dimensional space.
Remaining double-counting is dealt with as above.
In total, our ansatz for
interactions at all contains
13 components: 9 when two VMD, GVMD or direct photons interact, as is
already allowed for real photons, plus a further 4 where a `DIS photon'
from either side interacts with a VMD or GVMD one. With the label
resolved used to denote VMD and GVMD, one can write
An important note is that the dependence of the DIS and direct photon interactions is implemented in the matrix element expressions, i.e. in processes such as or the photon virtuality explicitly enters. This is different from VMD/GVMD, where dipole factors are used to reduce the total cross sections and the assumed flux of partons inside a virtual photon relative to those of a real one, but the matrix elements themselves contain no dependence on the virtuality either of the partons or of the photon itself. Typically results are obtained with the SaS 1D parton distributions for the virtual transverse photons [Sch95,Sch96], since these are well matched to our framework, e.g. allowing a separation of the VMD and GVMD/anomalous components. Parton distributions of virtual longitudinal photons are by default given by some -dependent factor times the transverse ones. The set by Chýla [Chý00] allows more precise modelling here, but indications are that many studies will not be sensitive to the detailed shape.
The photon physics machinery is of considerable complexity, and so the above is only a brief summary. Further details can be found in the literature quoted above. Some topics are also covered in other places in this manual, e.g. the flux of transverse and longitudinal photons in section , scale choices for parton density evaluation in section , and further aspects of the generation machinery and switches in section .
|+||92||single diffraction ()||[Sch94]|
|+||93||single diffraction ()||[Sch94]|
|+||281||( not )||[Daw85a]|
|+||351||( ) fusion)||[Hui97]|
|+||352||( ) fusion)||[Hui97]|