Photoproduction and $\gamma\gamma$ physics

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 $\gamma\gamma$ physics was added in the same spirit [Sch94a,Sch97]. Only 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.

The total $\gamma\mathrm{p}$ and $\gamma\gamma$ 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

$$\vert \gamma \rangle = \sqrt{Z_3} \, \vert \gamma_B \rangle ... ...tau} \frac{e}{f_{\ell\ell}} \, \vert \ell^+ \ell^- \rangle ~.$$ (119)

By virtue of this superposition, one is led to a model of $\gamma\mathrm{p}$ interactions, where three different kinds of events may be distinguished:

$\bullet$

Direct events, wherein the bare photon $\vert \gamma_B \rangle$ interacts directly with a parton from the proton. The process is perturbatively calculable, and no parton distributions of the photon are involved. The typical event structure is two high-$p_{\perp}$ jets and a proton remnant, while the photon does not leave behind any remnant.

$\bullet$

VMD events, in which the photon fluctuates into a vector meson, predominantly a $\rho^0$. All the event classes known from ordinary hadron-hadron interactions may thus occur here, such as elastic, diffractive, low-$p_{\perp}$ and high-$p_{\perp}$ events. For the latter, one may define (VMD) parton distributions of the photon, and the photon also leaves behind a beam remnant. This remnant is smeared in transverse momentum by a typical `primordial $k_{\perp}$' of a few hundred MeV.

$\bullet$

Anomalous or GVMD (Generalized VMD) events, in which the photon fluctuates into a $\mathrm{q}\overline{\mathrm{q}}$ pair of larger virtuality than in the VMD class. The initial parton distribution is perturbatively calculable, as is the subsequent QCD evolution. It gives rise to the so-called anomalous part of the parton distributions of the photon, whence one name for the class. As long as only real photons were considered, it made sense to define the cross section of this event class to be completely perturbatively calculable, given some lower $p_{\perp}$ cut-off. Thus only high-$p_{\perp}$ events could occur. However, alternatively, one may view these states as excited higher resonances ($\rho'$ etc.), thus the GVMD name. In this case one is lead to a picture which also allows a low-$p_{\perp}$ cross section, uncalculable in perturbation theory. The reality may well interpolate between these two extreme alternatives, but the current framework more leans towards the latter point of view. Either the $\mathrm{q}$ or the $\overline{\mathrm{q}}$ plays the rôle of a beam remnant, but this remnant has a larger $p_{\perp}$ than in the VMD case, related to the virtuality of the $\gamma \leftrightarrow \mathrm{q}\overline{\mathrm{q}}$ fluctuation.

The $\vert \ell^+ \ell^- \rangle$ states can only interact strongly with partons inside the hadron at higher orders, and can therefore be neglected in the study of hadronic final states.

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 $k_0$, which separates the low-mass vector meson region from the high-mass $\vert \mathrm{q}\overline{\mathrm{q}}\rangle$ one, $k_0 \approx m_{\phi}/2 \approx 0.5$ 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 $k_V^2/k_{\perp}^2$, where the adjustable parameter $k_V \approx m_{\rho}/2$ for light quarks. The $k_{\perp}$ scale is roughly equated with half the mass of the GVMD state. The spectrum of GVMD states is taken to extend over a range $k_0 < k_{\perp}< k_1$, where $k_1$ is identified with the $p_{\perp\mathrm{min}}(s)$ cut-off of the perturbative jet spectrum in hadronic interactions, $p_{\perp\mathrm{min}}(s) \approx 1.5$ 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-$p_{\perp}$ 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 to

$$\sigma_{\mathrm{tot}}^{\gamma\mathrm{p}}(s) = \sigma_{\mathr... ...athrm{p}}(s) + \sigma_{\mathrm{anom}}^{\gamma\mathrm{p}}(s) ~.$$ (120)

The direct cross section is, to lowest order, the perturbative cross section for the two processes $\gamma\mathrm{q}\to \mathrm{q}\mathrm{g}$ and $\gamma \mathrm{g}\to \mathrm{q}\overline{\mathrm{q}}$, with a lower cut-off $p_{\perp}> k_1$, in order to avoid double-counting with the interactions of the GVMD states. Properly speaking, this should be multiplied by the $Z_3$ coefficient,

$$Z_3 = 1 - \sum_{V=\rho^0,\omega,\phi,\mathrm{J}/\psi } \lef... ...mathrm{e},\mu,\tau} \left( \frac{e}{f_{\ell\ell}} \right)^2 ~,$$ (121)

but normally $Z_3$ is so close to unity as to make no difference.

The VMD factor $(e/f_V)^2 = 4\pi\alpha_{\mathrm{em}}/f_V^2$ gives the probability for the transition $\gamma \to V$. The coefficients $f_V^2/4\pi$ are determined from data to be (with a non-negligible amount of uncertainty) 2.20 for $\rho^0$, 23.6 for $\omega$, 18.4 for $\phi$ and 11.5 for $\mathrm{J}/\psi$. Together these numbers imply that the photon can be found in a VMD state about 0.4% of the time, dominated by the $\rho^0$ contribution. All the properties of the VMD interactions can be obtained by appropriately scaling down $V\mathrm{p}$ 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 rate

$$\sum_{\mathrm{q}} \left( \frac{e}{f_{\mathrm{q}\overline{\ma... ...right) \int_{k_0}^{k_1} \frac{\d k_{\perp}^2}{k_{\perp}^2} ~,$$ (122)

which is to be multiplied by the abovementioned reduction factor $k_V^2/k_{\perp}^2$ for the total cross section, and all scaled by the assumed real vector meson cross section.

As an illustration of this scenario, the phase space of $\gamma\mathrm{p}$ events may be represented by a $(k_{\perp},p_{\perp})$ plane. Two transverse momentum scales are distinguished: the photon resolution scale $k_{\perp}$ and the hard interaction scale $p_{\perp}$. Here $k_{\perp}$ is a measure of the virtuality of a fluctuation of the photon and $p_{\perp}$ corresponds to the most virtual rung of the ladder, possibly apart from $k_{\perp}$. As we have discussed above, the low-$k_{\perp}$ region corresponds to VMD and GVMD states that encompasses both perturbative high-$p_{\perp}$ and nonperturbative low-$p_{\perp}$ interactions. Above $k_1$, the region is split along the line $k_{\perp}= p_{\perp}$. When $p_{\perp}> k_{\perp}$ 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 $k_{\perp}> p_{\perp}$, the $p_{\perp}$ scale is just part of the traditional evolution of the parton distributions of the proton up to the scale of $k_{\perp}$, 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 $k_1$ and there being the low-$p_{\perp}$ part of the GVMD class, only appearing when a hard interaction at a larger $p_{\perp}$ scale would not preempt it. This possibility is implicit in the standard cross section framework.

In $\gamma\gamma$ physics [Sch94a,Sch97], the superposition in eq. () applies separately for each of the two incoming photons. In total there are therefore $3 \times 3 = 9$ combinations. However, trivial symmetry reduces this to six distinct classes, written in terms of the total cross section (cf. eq. ()) as

$$\sigma_{\mathrm{tot}}^{\gamma\gamma}(s) = \sigma_{\mathrm{dir}\times\mathrm{dir}}^{\gamma\gamma}(s) + \sigm... ...{\gamma\gamma}(s) + \sigma_{\mathrm{GVMD}\times\mathrm{GVMD}}^{\gamma\gamma}(s)$$

$$+ 2 \sigma_{\mathrm{dir}\times\mathrm{VMD}}^{\gamma\gamma}(s) + 2 \... ...mma\gamma}(s) + 2 \sigma_{\mathrm{VMD}\times\mathrm{GVMD}}^{\gamma\gamma}(s) ~.$$ (123)

A parameterization of the total $\gamma\gamma$ cross section is found in [Sch94a,Sch97].

The six different kinds of $\gamma\gamma$ events are thus:

$\bullet$

The direct$\times$direct events, which correspond to the subprocess $\gamma \gamma \to \mathrm{q}\overline{\mathrm{q}}$ (or $\ell^+\ell^-$). The typical event structure is two high-$p_{\perp}$ jets and no beam remnants.

$\bullet$

The VMD$\times$VMD events, which have the same properties as the VMD $\gamma\mathrm{p}$ events. There are four by four combinations of the two incoming vector mesons, with one VMD factor for each meson.

$\bullet$

The GVMD$\times$GVMD events, wherein each photon fluctuates into a $\mathrm{q}\overline{\mathrm{q}}$ pair of larger virtuality than in the VMD class. The `anomalous' classification assumes that one parton of each pair gives a beam remnant, whereas the other (or a daughter parton thereof) participates in a high-$p_{\perp}$ scattering. The GVMD concept implies the presence also of low-$p_{\perp}$ events, like for VMD.

$\bullet$

The direct$\times$VMD events, which have the same properties as the direct $\gamma\mathrm{p}$ events.

$\bullet$

The direct$\times$GVMD events, in which a bare photon interacts with a parton from the anomalous photon. The typical structure is then two high-$p_{\perp}$ jets and a beam remnant.

$\bullet$

The VMD$\times$GVMD events, which have the same properties as the GVMD $\gamma\mathrm{p}$ events.

Like for photoproduction events, this can be illustrated in a parameter space, but now three-dimensional, with axes given by the $k_{\perp 1}$, $k_{\perp 2}$ and $p_{\perp}$ scales. Here each $k_{\perp i}$ is a measure of the virtuality of a fluctuation of a photon, and $p_{\perp}$ corresponds to the most virtual rung on the ladder between the two photons, possibly excepting the endpoint $k_{\perp i}$ ones. So, to first approximation, the coordinates along the $k_{\perp i}$ axes determine the characters of the interacting photons while $p_{\perp}$ determines the character of the interaction process. Double counting should be avoided by trying to impose a consistent classification. Thus, for instance, $p_{\perp} > k_{\perp i}$ with $k_{\perp 1} < k_0$ and $k_0 <k_{\perp 2} < k_1$ gives a hard interaction between a VMD and a GVMD photon, while $k_{\perp 1} > p_{\perp} > k_{\perp 2}$ with $k_{\perp 1} > k_1$ and $k_{\perp 2} < k_0$ is a single-resolved process (direct$\times$VMD; with $p_{\perp}$ now in the parton distribution evolution).

In much of the literature, where a coarser classification is used, our direct$\times$direct is called direct, our direct$\times$VMD and direct$\times$GVMD 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 $(m_V^2/(m_V^2 + Q^2))^2$ for a photon of virtuality $Q^2$, where $m_V \to 2 k_{\perp}$ for a GVMD state. Putting it all together, the cross section of the GVMD sector of photoproduction then scales like

$$\int_{k_0^2}^{k_1^2} \frac{\d k_{\perp}^2}{k_{\perp}^2} \, \... ...\, \left( \frac{4k_{\perp}^2}{4k_{\perp}^2 + Q^2} \right)^2 ~.$$ (124)

For a virtual photon the DIS process $\gamma^*\mathrm{q}\to \mathrm{q}$ is also possible, but by gauge invariance its cross section must vanish in the limit $Q^2 \to 0$. At large $Q^2$, the direct processes can be considered as the $\mathcal{O}(\alpha_{\mathrm{s}})$ correction to the lowest-order DIS process, but the direct ones survive for $Q^2 \to 0$. There is no unique prescription for a proper combination at all $Q^2$, but we have attempted an approach that gives the proper limits and minimizes double-counting. For large $Q^2$, the DIS $\gamma^*\mathrm{p}$ cross section is proportional to the structure function $F_2 (x, Q^2)$ with the Bjorken $x = Q^2/(Q^2 + W^2)$. Since normal parton distribution parameterizations are frozen below some $Q_0$ scale and therefore do not obey the gauge invariance condition, an ad hoc factor $(Q^2/(Q^2 + m_{\rho}^2))^2$ is introduced for the conversion from the parameterized $F_2 (x, Q^2)$ to a $\sigma_{\mathrm{DIS}}^{\gamma^*\mathrm{p}}$:

$$\sigma_{\mathrm{DIS}}^{\gamma^*\mathrm{p}} \simeq \left( \f... ...^2 \, \left\{ x q(x, Q^2) + x \overline{q}(x,Q^2) \right\} ~.$$ (125)

Here $m_{\rho}$ is some nonperturbative hadronic mass parameter, for simplicity identified with the $\rho$ mass. One of the $Q^2/(Q^2+m_{\rho}^2)$ factors is required already to give finite $\sigma_{\mathrm{tot}}^{\gamma\mathrm{p}}$ for conventional parton distributions, and could be viewed as a screening of the individual partons at small $Q^2$. The second factor is chosen to give not only a finite but actually a vanishing $\sigma_{\mathrm{DIS}}^{\gamma^*\mathrm{p}}$ for $Q^2 \to 0$ in order to retain the pure photoproduction description there. This latter factor thus is more a matter of convenience, and other approaches could have been pursued.

In order to avoid double-counting between DIS and direct events, a requirement $p_{\perp}> \max(k_1, Q)$ is imposed on direct events. In the remaining DIS ones, denoted lowest order (LO) DIS, thus $p_{\perp}< Q$. This would suggest a subdivision $\sigma_{\mathrm{LO\,DIS}}^{\gamma^*\mathrm{p}} = \sigma_{\mathrm{DIS}}^{\gamma^*\mathrm{p}} - \sigma_{\mathrm{direct}}^{\gamma^*\mathrm{p}}$, with $\sigma_{\mathrm{DIS}}^{\gamma^*\mathrm{p}}$ given by eq. () and $\sigma_{\mathrm{direct}}^{\gamma^*\mathrm{p}}$ by the perturbative matrix elements. In the limit $Q^2 \to 0$, the DIS cross section is now constructed to vanish while the direct is not, so this would give $\sigma_{\mathrm{LO\,DIS}}^{\gamma^*\mathrm{p}} < 0$. However, 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:

$$\sigma_{\mathrm{LO\,DIS}}^{\gamma^*\mathrm{p}} = \sigma_{\ma... ...m{p}}}{\sigma_{\mathrm{DIS}}^{\gamma^*\mathrm{p}}} \right) \;.$$ (126)

Since we here are in a region where the DIS cross section is no longer the dominant one, this change of the total DIS cross section is not essential.

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, $Q^2 \gg k_{\perp}^2 \gg p_{\perp}^2$, 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 $Q^2 > k_{\perp}^2 > p_{\perp}^2$. The region $k_{\perp}^2 > \max(Q^2,p_{\perp}^2)$ is also DIS, but of the $\mathcal{O}(\alpha_{\mathrm{s}})$ direct kind. The region where $k_{\perp}$ is the smallest scale corresponds to non-ordered emissions, that then go beyond DGLAP validity, while the region $p_{\perp}^2 > k_{\perp}^2 > Q^2$ cover the interactions of a resolved virtual photon. Comparing with the plane of real photoproduction, we conclude that the whole region $p_{\perp}> k_{\perp}$ 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 $p_{\perp}< k_{\perp}< \min(k_1, Q)$ 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 $Q^2$ and rather small $W^2$. Typically, this is the region where $x \approx Q^2/(Q^2 + W^2)$ is not close to zero, and where $F_2$ is dominated by the valence-quark contribution. The latter behaves roughly $\propto (1-x)^n$, with an $n$ 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 $Q^2$:

$$\sigma_{\mathrm{tot}}^{\gamma^*\mathrm{p}} = \sigma_{\mathr... ...p}} + \sigma_{\mathrm{GVMD}}^{\gamma^*\mathrm{p}} \right) \;,$$ (127)

with four main components. Most of these in their turn have a complicated internal structure, as we have seen.

Turning to $\gamma^*\gamma^*$ processes, finally, the parameter space is now five-dimensional: $Q_1$, $Q_2$, $k_{\perp 1}$, $k_{\perp 2}$ and $p_{\perp}$. 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 $\gamma^*\gamma^*$ interactions at all $Q^2$ 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

$$\sigma_{\mathrm{tot}}^{\gamma^*\gamma^*} (W^2, Q_1^2, Q_2^2) = \sigma_{\mathrm{DIS}\times\mathrm{res}}^{\gamma^*\gamma^*} \; \ex... ...\gamma^*}} \right) + \sigma_{\mathrm{dir}\times\mathrm{res}}^{\gamma^*\gamma^*}$$

$$+ \sigma_{\mathrm{res}\times\mathrm{DIS}}^{\gamma^*\gamma^*} \; \ex... ...\gamma^*}} \right) + \sigma_{\mathrm{res}\times\mathrm{dir}}^{\gamma^*\gamma^*}$$ (128)

$$+ \sigma_{\mathrm{dir}\times\mathrm{dir}}^{\gamma^*\gamma^*} + \lef... ... + W^2} \right)^3 \; \sigma_{\mathrm{res}\times\mathrm{res}}^{\gamma^*\gamma^*}$$

Most of the 13 components in their turn have a complicated internal structure, as we have seen.

An important note is that the $Q^2$ dependence of the DIS and direct photon interactions is implemented in the matrix element expressions, i.e. in processes such as $\gamma^*\gamma^*\to \mathrm{q}\overline{\mathrm{q}}$ or $\gamma^*\mathrm{q}\to \mathrm{q}\mathrm{g}$ 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 $Q^2$-dependent factor times the transverse ones. The new set by Chýla [Chý00] allows more precise modelling here, but first 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 subsection , scale choices for parton density evaluation in subsection , and further aspects of the generation machinery and switches in subsection .