HERA at low-x

Nomenclature

HERA High Energy Ring Accelerator

DIS Deep Inelastic Scattering

QCD Quantum ChromoDynamics

QED Quantum ElectroDynamics

pQCD perturbative Quantum ChromoDynamics

PDF Parton Distribution Function

DGLAP Dokshitzer, Gribov, Lipatov, Altarelli, Parisi QCD evolution equations

BFKL Balitsky, Fadin, Kuraev, Lipatov QCD evolution equations

(N(N))LO (Next to (Next to)) Leading Order

BK Balitsky, Kovchegov non-linear QCD evolution equations

JIMWLK Jalilian-marian, Iancu, McLerran, Weigert, Leonidov, Kovner non-linear QCD evolution equations

CGC Colour Glass Condensate

DVCS Deeply Virtual Compton Scattering

GPD Generalised Parton Distribution

DDIS Diffractive Deep Inelastic Scattering

LRG Large Rapidity Gap

DPDF Diffractive Parton Distribution Function

Introduction

Deep Inelastic Scattering (DIS) is the scattering of point-like particles, leptons, such as electrons, muons and neutrinos, from larger particles, hadrons, such as protons, neutrons and nuclei, with the aim of determining the nature of the constituents of the former and of mapping the kinematic behaviour of these constituents. These constituents are called partons; quarks, anti-quarks and gluons. The DIS experiments allow us to measure the fractional momentum of the hadron taken by the struck parton, $x$, and thus to measure the momentum distributions of the partons. These distributions, for all flavours of quark and for the gluons, are known as Parton Distribution Functions (PDFs). These distributions evolve with the $4$-momentum involved in the interaction (generally termed the scale of the interaction) as predicted by Quantum Chromodynamics (QCD). Over the course of the last 50 years, these measurements first established QCD as the correct theory of the strong interaction, which binds the partons into the proton and then were used to make accurate measurements of the PDFs, which are essential input to the predictions of all scattering cross sections at the Large Hadron Collider. However, the measurement of the parton distributions and the features of QCD are of interest in their own right. QCD is a rich theory, its field carrier, the gluon, is self-interacting, unlike the photon of QED and this gives rise to novel features. Firstly, to the running of the strong coupling, which decreases as scale increases, such that we may make successful perturbative calculations (pQCD), which have served us well for more than 40 years. Secondly, to the possibility of dense gluon population in the proton, where we need to go beyond the conventional successful calculations of QCD into a new regime. The reality of this new regime was first glimpsed at HERA in the kinematic region where the fractional momentum of the struck quark, $x$, is very small – the low-$x$ region. This is the subject of the current article. We begin by outlining the role of DIS in the development of QCD. A more detailed account can be found in Devenish and Cooper-Sarkar [124].

DIS experiments

HERA was an electron-proton collider which ran at the DESY Laboratory in Hamburg from 1992-2007. Two experiments, ZEUS and H1, with somewhat different detector designs, measured the data. A total of 500 pb$^{-1}$ of data were taken per experiment and this was split roughly equally between electron $e^{-}$ and positron $e^{+}$ beams of energy $E_e = 27$ GeV. The proton beam was run at 4 different energies, $E_p = 460, 575, 800$ and 920 GeV, giving centre of mass energies, $\sqrt{s} = 225, 251, 300$ and 318 GeV, respectively. Early data were run at $E_p = 800$ GeV, the bulk of the data were run at 920 GeV and two short runs were made at the lower energies. After the experiments had finished taking data in 2007 and completed most of their individual experimental analyses, the data from H1 and ZEUS were combined taking full account of their systematic uncertainties, such that the combination represents a ’best of both worlds’, with each experiment calibrating the other, reflecting the differing strengths of each. Most of the data presented here uses these combined data, except when a physics point is best presented using early data.

Deep Inelastic Scattering can be pictured as the scattering of a lepton (in the case of HERA electron or positron) on a proton by exchange of a virtual vector boson. The commonly used Lorentz invariants will be defined for the basic process

where $l, l'$ represent leptons (lepton is taken to include antileptons, unless it is necessary to distinguish them), $N$ represents a nucleon and $X$ represents the hadronic final state particles. The associated four vectors are $k, k'$ for the incoming and outgoing leptons respectively, and $P$ for the target (or incoming) nucleon. The process is mediated by the exchange of a virtual vector boson, $V^{*}$ ($\gamma$, $W$ or $Z$), see Figure 1. This boson has four momentum given by

and the four vector $P_X$ of the hadronic final state system $X$ is given by

Various Lorentz invariants are useful in the description of the kinematics of the process:

the centre of mass energy squared for the $lp$ interaction;

the (negative of) the invariant mass squared of the virtual exchanged boson;

the Bjorken $x$ variable, which will be interpreted in the quark-parton model as the fraction of the momentum of the incoming nucleon taken by the struck quark, and

which gives a measure of the amount of energy transferred between the lepton and the hadron systems. It is also related to the scattering angle in the lepton-quark centre-of-mass frame. Note that (ignoring masses)

so that only two of these quantities are independent.

The general form for the differential cross section for $e-p$ scattering by exchange of neutral current (NC) vector bosons ($\gamma^*$ or $Z^*$) is

where $Y_{\pm}=1\pm(1-y)^2$, $F_2$, $xF_3$ and $F_L$ are the structure functions of the hadron and the notation $lN$ indicates lepton scattering from a nucleon. The term $xF_3$ is only non-negligible if $Z$ exchange is important and this only happens for $Q^2$ values, $Q^2\gtrsim5000\,\mathrm{GeV}^2$. Similarly the exchange of charged current (CC) vector bosons ($W^+$, $W^-$) is only non-negligible at such high $Q^2$. Figure 2 shows the kinematic reach of HERA. One can see that low $x$, $x\lesssim0.01$, is only probed at low $Q^2$, $Q^2\lesssim500\,\mathrm{GeV}^2$, and so we do not consider these cases further and we adopt the simplified form of the differential cross section due to virtual photon exchange.

Now let us consider how the variables $x$, $Q^2$, $y$ are determined from quantities measured in the HERA experiments. At HERA the signature of a DIS NC event is one in which the scattered beam electron is recorded in the EM section of the calorimeter and the total transverse momentum of the final state is approximately zero. An example of a high $Q^2$ NC event in the H1 detector is shown in Figure 3.

Looking at this diagram we can easily see the scattered electron and the hadron energy deposits. Both of these can be measured and the kinematic variables can be determined in various different ways. Only the method using electron variables will be described here, but other methods involving the hadron variables allow cross calibration and access to kinematic regions where the resolution on the electron variables is poor.

We measure the electron kinematic variables as follows. The proton axis is conventionally taken to define the positive $z$ direction from which all angles are measured. If $E_e$, $E_p$ are the electron and proton beam energies, $E'$ and $\theta_e$ the energy and angle of the scattered electron to the beam-line, then:

Note the main point that the fractional momentum of the struck quark $x$ can actually be measured simply by measuring the 4-vector of the outgoing electron in an NC scatter.

The parton model

We will now briefly review the interpretation of the structure functions in terms of the parton model and QCD, details are given in the contribution of D Soper to this encyclopedia.

This parton picture grew out of the attempt by Feynman to provide a simple physical picture of the scaling that had been predicted by Bjorken and observed in the first high energy deep inelastic electron scattering experiments at SLAC, where $F_2$ was observed to be independent of $Q^2$ for $x$ values around $x \sim0.25$. The quark-parton model (QPM) states that the nucleon is full of point-like non-interacting scattering centres known as partons. The lepton-hadron reaction cross-section is then approximated by an incoherent sum of elastic lepton-parton scatters via an exchanged vector boson, see Figure 4. Only spin $\frac{1}{2}$ partons (quarks or antiquarks) interact directly with the leptonic

probe particles. Thus the electromagnetic scattering of two spin $\frac{1}{2}$ point-like particles is the underlying fundamental process and this can be calculated in QED.

The result of the calculation is

for fermions of charge $e$, the electron and $e'$, the quark, with $s'$ being the centre of mass energy of the electron-quark scatter. Since $e'=e_i e$ (where $e_i=2/3,-1/3$) this may be written in the form

Assuming that the quark takes a fraction $x'$ of the proton’s 4-momentum then $s'=x's$. Considering Figure 4, the 4-momentum squared of the outgoing quark is

because quarks are assumed massless. But $P^2$ is also $\sim0$ since the mass of the Nucleon is negligible compared to the scale $Q^2$, and we get

so that

That is that the measurable kinematic quantity $x$ can be identified with the fractional momentum of the struck quark.

To obtain a cross-section for electron-hadron scattering from this result a distribution function $q_i(x)$ which gives the probability that the struck quark $i$ carries a fraction $x$ of the hadron’s momentum $p$ must be defined. This gives the parton number, or density, distribution. It is also useful to define the momentum distribution $xq(x)$ and this is what is meant by the term parton distribution function or PDF.

Thus the double differential cross-section for incoherent scattering off all possible types of quark with a specific value of $x$ is

or equivalently

Now we compare this QPM prediction to the general form of the lepton hadron cross section in Eq. (2) and we see that the parton model predicts

where we have generalised the result to include antiquarks, which are also spin-1/2 point-like particles. Thus the parton model predicts Bjorken scaling: $F_2$ depends only on $x$ not on $Q^2$.

The parton model also predicts,

as a consequence of scattering from spin $\frac{1}{2}$ partons, and this was also verified approximately in early data.

These were not the only observations which led to the QPM becoming established. In lepton-hadron scattering we may use hydrogen or deuterium targets so we may scatter from proton and neutron targets. This will give a different mixture of $u$- and $d$-type quarks. One can also use neutrinos and antineutrinos as the probe and these interactions give us different $F_2^{\nu N}$ and $xF_3^{\nu N}$ structure functions with different flavour dependencies. One can derive that for scattering of isoscalar targets

and the early data on charged lepton scattering and the neutrino scattering structure functions are indeed related by $\sim5/18$. The (anti-)neutrino scattering data also allowed us to established that there are antiquarks in the nucleon and to establish the picture that there are in fact 3 valence-quarks in the nucleon, which give it is flavour properties, and a ’sea’ of quark-antiquark pairs which are flavourless. These data also allowed us to measure (roughly) the shapes of the valence and sea quark momentum distributions.

However, there were remaining issues. Firstly, one would expect that the sum over the momentum quarks and antiquarks in the nucleon

integrated over $x$ would be equal to unity

This may be checked experimentally, since

and the result

implies that there is more momentum in the nucleon than that carried by the quarks and antiquarks. Secondly, the partons are assumed to be essentially free particles in the QPM, but at the same time are confined inside the hadron. Thirdly, although Bjorken scaling is approximately satisfied, there is clear evidence for scale breaking at large and small $x$. These problems led to the development of the theory of QCD.

Development of the QCD improved parton model

The simple QPM diagram of Figure 4 is augmented by the QCD Compton process, as shown in Figure 5, and the BGF process, as shown in

$z = Q^2/(2p_i.q) = x/y$, the analogue of Bjorken $x$ for the $\gamma^*$-parton scattering. The cross-section for the hard interaction is calculable in pQCD and the overall $\gamma^*p$ cross-section is then given by a convolution of this cross section with the probability of finding the parton with momentum fraction $y$ in the proton. The results for the QCDC and BGF processes each add an extra term to the point-like scatter of Figure 4. This can be framed as a redefinition of the parton density such that it includes the parton splittings and thus gains a scale dependence $q(x) \to q(x,Q^2)$. QCD cannot predict the shape of the parton distribution at the starting scale $Q_0^2$ (which should be chosen large enough that perturbative calculations are valid, i.e. that the strong coupling $\alpha_s(Q_0^2)$ is small) but it can predict how it will evolve with $Q^2$.

In this equation there are two ’splitting functions’ $P_{qq}$ and $P_{qg}$ which can be interpreted as the probability for $q \to q(z)g(1-z)$ splitting (for the QCDC process) and $g \to q(z)\bar{q}(1-z)$ splitting (for the BGF process), respectively.

Just as the quark density evolves, so will the gluon density

where we have introduced two more splitting functions which describe the probabilities for $q \to g(z)q(1-z)$ and $g \to g(z)g(1-z)$ splitting. The splitting functions are summarised graphically in Figure 7.

The two equations Eqs. (14) and (15) above are known as the DGLAP evolution equations, Gribov and Lipatov [161]; Lipatov [184];

Dokshitzer [126]; Altarelli and Parisi [54]. At leading order ($\alpha_s^0$) the structure function $F_2$ can be related to the evolved parton distributions as

and the structure function $F_L$ is still zero. But there is one more sophistication of this result needed even at leading order.

The leading order QCD corrections to the parton densities are all weighted by $\alpha_s\ln(Q^2/Q_0^2)$ which could diverge at high $Q^2$. However, the QCD coupling $\alpha_s$ runs $\alpha_s(Q^2)\propto1/\ln(Q^2)$. Thus divergences at high $Q^2$ are avoided. The final step in setting up the DGLAP equations with a correct asymptotic behaviour is to replace $\alpha_s$ with $\alpha_s(Q^2)$. The ‘leading logs’ that are now being summed in the DGLAP equations correspond to diagrams with multiple gluon emission as shown in Figure 8, with an important restriction. To pick out the leading $\alpha_s\ln(Q^2/Q_0^2)$

terms requires that the transverse momenta of the successive steps shown in the diagram must be ordered in the transverse momentum of the gluons

This approximation is often known as the leading log approximation (LLA).

Beyond leading order the splitting functions are expanded as power series in $\alpha_s(Q^2)$ e.g.

The relationship between the parton distributions and the measurable structure functions becomes more complex e.g.

where $\bar{e}^2=\sum_i e_i^2$ and the sums run over all active quark and antiquark flavours. $C_q$ and $C_g$ are the coefficient functions, which are also be expanded as power series in $\alpha_s$ beyond leading order. The $F_L$ structure function is non-zero, e.g. at NLO

At small $x$ the gluon term becomes dominant and controls the behaviour of $F_L$ as well as the evolution of $F_2$.

Complete QCD calculations for the DIS process are now available to Next-to-Next-to-Next to leading order (NNNLO or N3LO).

Although the formalism has become more complex, it remains true that given the shapes of the parton distributions at a given input scale, QCD evolution can predict them at any higher scale and can thus predict the measurable structure functions for DIS experiments. But how do we determine the initial distributions? This has been done by parameterising the $x$ dependence of the parton density at $Q_0^2$ and determining the parameters by fitting to the DIS data. A function of the form

is used for each flavour of PDF, where $P(x)$ is a polynomials in $x$ or $\sqrt{x}$. These could be Bernstein or Chebyshev polynomials, or one can use a neural net. Such PDF fits are done by three major global fitting groups: CT, Hou et al. [163], MSHT, Bailey et al. [59], and NNPDF, Ball et al. [64]. This is the topic of the contribution of F Olness to this encyclopedia.

There were some expectations for the values of the exponents $\alpha$ and $\beta$. In particular there were predictions from Regge theory, see Sec. 4.4, for the exponent $\alpha$ which controls the behaviour at low $x$. It turned out that the data defied our simple expectations. This will be discussed in § 5.

HERA data

Here we collect together some relevant results from HERA. We will begin by showing the measurements of the reduced cross section from the final combination of the H1 and ZEUS data of both run 1 and run 2 of HERA, Abramowicz et al. [21]. The reduced cross section

is obviously related to the differential cross section of Eq. (2) which is the directly measurable quantity. For most of the kinematic plane the kinematic variable $y$ is relatively small and the reduced cross section gives $F_2$. Recalling, that to leading order in QCD,

Thus $F_2$ shows us the behaviour of the quark and antiquark distributions as a function of $Q^2$.

In Figure 9 we show the reduced cross section as a function of $Q^2$ for various $x$ values. An important feature is the consistency of the data from fixed target experiments and from the HERA collider.

To illustrate one of the most striking features of HERA data we plot the data versus $x$ at fixed $Q^2$. This is done in Figure 10 for both the final HERA data and for early data, and the quantity plotted is $F_2$ rather than $\sigma_r$. The figures show that $F_2$ rises steeply as $x$ decreases and that the strength of the rise increases as $Q^2$ increases. The rise in $x$ is strong, at $Q^2 = 120\,\mathrm{GeV}^2$ $F_2$ increases by almost a factor of 3 as $x$ decreases from 0.08 to 0.003. It is worth remarking that while a range of possible behaviours had been predicted for $F_2$ at small $x$, including a strong rise, when the first preliminary HERA data showing a rising $F_2$ was presented at the Durham Workshop in 1993 it caused great excitement. The early fixed target data for $Q^2 = 1.5\,\mathrm{GeV}^2$ and $x > 10^{-3}$ (triangles), shown in Figure 10 left hand side, explains why – these data showed a fairly flat behaviour as $x$ decreased with only the merest hint of a possible rise. At very low $Q^2 \sim0.2\,\mathrm{GeV}^2$ $F_2$ no longer rises steeply. The behaviour of $F_2$ for such very small $Q^2$ is interesting and will be covered in some detail at HERA. This is covered in more detail in section 4.4.

In order to extract $F_2$ we must correct $\sigma_r$ for the contribution of $F_L$. For modern data this is done by using the PDFs extracted from a full QCD fit to the reduced cross section data and using the relationship 19 to obtain $F_L$ from the fitted parton distributions.

We now show the reduced cross sections as a function of $x$ in different $Q^2$ bins for a very wide range of $Q^2$ values. Figs 11 and 12 show the reduced cross section as a function of $x$ in high $Q^2$ and low $Q^2$ bins respectively, with the NNLO QCD fit overlaid. Whereas the high $Q^2$ data are very well fitted, in the low $Q^2$ bins of Figure 11 we can see that the NNLO QCD fit does not describe the data so well at low $x$ values, where the impact of $F_L$ on the reduced cross section can be seen clearly as a turn over due to the $-y^2/Y_+ \cdot F_L$ term (recall that high $y$ occurs at low $x$ for a given $Q^2$ because of the relationship $y = Q^2/(sx)$). In fact it seems that the data would like a larger $F_L$ than that predicted by the theory. The $\chi^2$ scan of the NNLO QCD fit vs the minimum $Q^2$ value of data input to the fit, $Q^2_{\min}$, is shown in Figure 13. One can see that, beyond the leading order, the data are considerably less well fitted for $Q^2 < 10\,\mathrm{GeV}^2$. This will be discussed further in Section 4 where modifications to the theory at low $x$ and $Q^2$ are discussed.

$F_L$

$F_L$ is an important quantity because of its rather direct relation to $xg$ – the momentum distribution of the gluon in the proton. One may express the cross-section as

To measure $F_L$ directly at a fixed value of $(x,Q^{2})$ requires data at two different values of $y$ and hence at two different centre of mass energies (because of the constraint $Q^{2}=sxy$). Such a measurement was made by both ZEUS and H1 by varying the proton beam energy down to 575 and 460 GeV down from 920 such that $\sqrt{s}=251$ and 225 GeV not 318 GeV.

The reduced cross section at three different beam energies for H1 data, Andreev et al. [56], is shown in Figure 14.

The method to extract $F_L$ is illustrated in Figure 15, where the reduced cross section is plotted against $y^2/Y_+$ for different beam energies, in $x$, $Q^2$ bins. $F_L$ is extracted from the slope of the plot and $F_2$ from its intercept.

Figure 16 shows the $F_L$ and corresponding $F_2$ values at the same $x$, $Q^2$. Superficially the NLO DGLAP QCD fit shown describes the data on $F_L$ reasonably well, but the but errors are large. When these data are averaged over $x$ and displayed as a function $Q^2$ one can see that the DGLAP predictions for $F_L$ lie low. This is shown and discussed in the context of theoretical models in § 5.

$F_{2}^{c\bar{c}}$

The structure functions studied so far are inclusive objects. However, it is possible to study the contribution of heavy flavours to the reduced cross section. This is particularly interesting since heavy flavours are dominantly produced through the boson-gluon fusion process (BGF), which involves the gluon density $xg$ directly.

The advent of HERA, with a reach to much smaller $x$ values, together with detectors capable of measuring details of the DIS final state gives the prospect of accurate data on $F_{2}^{c\bar{c}}$ over a wide range of $x$ and $Q^{2}$. We will concentrate on charm production here because the data on beauty production from HERA are not very discriminating. Most of the data on charm production in DIS from HERA has come observation of $D^{*}$ and $D$ decays.

To get from the measured $ep \to eD^{*}X$ cross-section to the inclusive charm production cross-section one needs to extrapolate to the full $D^{*}$ phase space, to know the probability that the charm quark will fragment into a $D^{*}$ meson ($P(c \to D^{*})$) and to correct for the small fraction of indirect $D$ mesons produced through $B$ meson decay or fragmentation. All of these steps involve extra uncertainty in comparison to the inclusive measurements. Finally $F_{2}^{c\bar{c}}$ is related to the $ep \to ec\bar{c}X$ cross-section by

ZEUS and H1 have combined their final data on charm and on beauty reduced cross sections, see Abramowicz et al. [22]. The plots in

Figure 17 show the ZEUS and H1 combined reduced cross sections for charm data (left) and their ratio to various calculations in perturbative QCD (right). When considering heavy flavour production it is necessary to think about the treatment of heavy quarks in QCD. There have been two approaches: the Fixed Flavour Number (FFN) scheme in which only 3 flavours are considered in the proton and heavy flavours are always calculated explicitly by $g \to q\bar{q}$ splitting, taking into account the mass of the heavy quarks; and: ii) the Zero-Mass Variable Flavour Number (ZMVFN) scheme in which the number of quarks in the proton is considered to change as the scale increases, such that at $Q^{2}$ values much above $4m_{c}^{2}$ the charm quark is considered massless (similarly the bottom quark for $Q^{2} > 4m_{b}^{2}$) and all log terms are resummed just as for the light quarks. In the former case mass terms are treated properly but $\ln(Q^{2}/m_{c}^{2})$ diagrams are not resummed, in the latter resummations are treated properly but at lower $Q^{2}$ mass terms are not. There are also General Mass Variable Flavour Number schemes (GMVFN) which combine the features of the ZMVFN and FFN schemes. Looking at Figure 17 it is evident that the theoretical calculations do not match the data, particularly for $Q^{2} \sim12\mathrm{GeV}^{2}$. This is true whether a FFN (FF3A) calculation or a GMVFN (RTOPT) calculation is made.

These results will be discussed further in Section 4, which discusses discrepant behaviour at low $x$.

$F_{2}^{p}$ at very low $Q^{2}$

A remarkable feature of the proton structure function $F_{2}$ is its rapid rise at low $x$ – as already shown in Figure 10. The persistence of this rise down to $Q^{2}$ values as small as 0.65 GeV$^{2}$ challenges our understanding of QCD. For smaller values of $Q^{2}$, the rise of the structure function at small $x$ diminishes, as shown in terms of the reduced cross section in Figure 18 and terms of the $F_{2}$ extracted from it in Figure 19. These very low $Q^{2}$ data allow us to examine the transition from the perturbative to non-perturbative regions.

The data can be characterised in terms of two slopes, with respect to $\ln Q^{2}$ and $\ln(1/x)$ respectively. The latter is often written as

such that $F_{2}(x,Q^{2}) \sim x^{-\lambda(Q^{2})}$, and is shown in Figure 20.

This left hand side of this figure shows that there is a region at low $Q^{2}$, $Q^{2} \lesssim0.65\mathrm{GeV}^{2}$ for which this slope is flat in accordance with Regge predictions, see Sec. 4.4. For higher $Q^{2}$ it rises in accordance with the predictions of perturbative QCD, see § 5.

The slope $\frac{\partial F_{2}}{\partial\ln Q^{2}}$ is shown in the right-hand-side of Figure 20 as a function of $x$. This is another result from HERA that caused a lot of interest and discussion because the turn over at low $x$ was interpreted as evidence for saturation. At its simplest this is the idea that when gluon densities are becoming very large the gluons may recombine and the evolution equations must become non-linear. We will discuss this further in section 4.4. However, note that in Figure 20, because of the strong correlation between $x$ and $Q^{2}$ inherent in deep inelastic kinematics, the slope $\frac{\partial F_2}{\partial\ln Q^2}$ is not evaluated at a fixed values of $Q^2$, but rather at the mean values shown along the top of the plot for each data point. The plot shows that the rate of rise of $F_2$ as a function of $Q^2$ is starting to abate as $x < 10^{-3}$ and $Q^2 < 2\,\mathrm{GeV}^2$. This could be because we are entering the non-perturbative region, rather than evidence for saturation. The interpretation of the low $Q^2$ data is discussed further in Section 4.4.

Low $x$ physics at HERA

Now we will discuss all of the above data in terms of their theoretical interpretations.

We have seen the striking rise of $F_{2}$ at low-$x$ in Figure 10, but to get some feeling for why this caused some excitement we can look at the very early data in Figure 21. The data are compared to contemporary predictions. The predictions labelled CTEQ1 and MRSD$_{0}$ are the conventional predictions from QCD fits to PDFs based on previous data, which did not probe lower than $x \sim0.01$. For these predictions the input parton distributions to DGLAP evolution had flat gluon and seaquark distributions at a starting scale of $Q_{0}^{2} \sim4$ GeV$^{2}$. The curves labelled GRV and MRSD$_{-}$ are also from QCD PDF fits but with unconventional input. In particular the MRSD$_{-}$ curve tries to account for QCD evolution according to the BFKL equation, which will be discussed later, by starting with a steep input gluon distribution and the GRV curve is based on an idea to start the QCD evolution at very low $Q^{2} \sim0.3$ GeV$^{2}$, where perturbative QCD is not expected to work, but a steep rise of $F_{2}$ can be generated at higher $Q^{2}$ by evolution from this low scale. It was a major surprise that the non-conventional pictures appeared to fit the data better.

With hindsight it is hard to understand why we were so sure that a flat input was expected. Reasons why this was conventional are discussed in Section 4.4. However, rising of $F_{2}$ at small-$x$ is actually predicted by pQCD, but the predictions do not give a scale in either $x$ or $Q^{2}$ at which one might expect to see such effects.

In the DGLAP formalism at leading order¹ the gluon splitting functions

are singular as $z \to0$. Thus the gluon distribution will become large as $x \to0$, and its contribution to the evolution of the parton distribution becomes dominant. In particular the gluon will ‘drive’ the quark singlet distribution, and hence the structure function $F_{2}$, to become large as well, the rise increasing in steepness as $Q^2$ increases. Quantitatively,

may be solved subject to the nature of the boundary function $xg(x,Q_0^2)$. Inputing a flat gluon at $Q_0^2$, the solution is (from De Rujula et al. [117]; Ball and Forte [62])

where

Given a long enough evolution length from $Q_0^2$ to $Q^2$, this will generate a steeply rising gluon distribution at small $x$, starting from a flattish behaviour of $xg(x,Q^2)$ at $Q^2=Q_0^2$.

Over the $x,Q^2$ range of HERA data this solution mimics a power behaviour, $xg(x,Q^2)\sim x^{-\lambda_g}$, with

where $t=\ln(Q^2/\Lambda^2)$, $t_0=\ln(Q_0^2/\Lambda^2)$. This steep behaviour of the gluon generates a similarly steep behaviour of $F_2$ at small $x$, $F_2\sim x^{-\lambda}$, where $\lambda\sim\lambda_g$.

So why was the observed rise of $F_2$ unexpected? Because, before the advent of HERA data, the starting scale for perturbative evolution had always been taken to be $Q_0^2 \gtrsim4\ \mathrm{GeV}^2$, to be sure that a perturbative calculation would be valid. The evolution length from $Q^2$ of 4 to 15 GeV$^2$ is not large enough to generate a steep slope from a flat input.

There are two assumptions which could be challenged here: i) the value $Q_0^2\sim4\ \mathrm{GeV}^2$ is not set in stone, in fact modern PDF analyses start from $Q_0^2\sim1.5\ \mathrm{GeV}^2$ and; ii) the input itself need not necessarily be flat. First considering i) there was a further surprise when HERA data was extended to lower $Q^2$ and we could see that the rise of $F_2$ continues down to $Q^2\sim1\mathrm{GeV}^2$, we have seen this in Figure 12 and Figure 19. This would seem to imply that we need a steep input even at rather low $Q^2$. One may worry that the value of $\alpha_s(Q_0^2)$ is becoming too high for perturbative QCD to be applicable at such low $Q^2$. However, the inputs are not necessarily flat. This brings us back to ii) should one consider a non flat input. The motivation for a steeply rising input came from the BFKL equation, Fadin et al. [138]; Kuraev et al. [179, 180]; Balitsky and Lipatov [61].

At small $x$ terms in $\ln(1/x)$ are becoming large and the conventional $\ln Q^2$ summation of the DGLAP equations does not account for this. It may also be necessary to sum $\ln(1/x)$ terms. Such a summation is performed by the BFKL equation. To leading order in $\ln(1/x)$ with fixed $\alpha_s$, this predicts a steep power law behaviour

where

(for $\alpha_s\simeq0.2$, as appropriate for $Q^2\sim4\mathrm{GeV}^2$). Thus the observation of a steep behaviour in $F_2$ at a relatively low $Q^2$ was seized upon by those who saw this as evidence for BFKL. Higher statistics data showed that an effective power $\lambda_g\simeq0.5$ was actually too steep, but the calculation above is only to leading order and more data compatible lower slopes of $\lambda_g\simeq0.3$ are predicted by more sophisticated calculations.

The BFKL equation concerns the behaviour of the gluon. Referring back to the gluon emissions from the quark line of Figure 8, we know that the DGLAP equations also involve gluon emissions from a gluon line, with a quark–antiquark emission connecting to the $\gamma^*$, as for the BGF process. The corresponding diagram is shown in Figure 22, left-hand-side. In the DGLAP formalism these emissions $p_t$ ordered just as they were in Eqn (17). The difference in the BFKL formalism is that these emissions need not be ordered. Then using the optical theorem to related the total cross section to the imaginary part of the forward elastic scattering amplitude, we get the gluon ladder diagram of Figure 22 right-hand-side. Since the transverse momenta in the BFKL ladder are not ordered this allows of some physics processes not allowed in DGLAP, for example emission high $p_t$ jets in the forward direction, see subsection 4.3.

BFKL phenomenology has advanced a great deal in more recent years and NLL $\ln(1/x)$ resummation may now be incorporated with NNLO DGLAP $\ln Q^2$ resummation in the QCD fits, see subsection 4.1.1.

A steep input distribution can be input to pure DGLAP fits, in order to avoid the need for use of the BFKL evolution, but is this successful? This begs the question, which input is steep? is it the initial distributions of both the sea-quarks and gluons? As more HERA data were collected in the early years of HERA running, DGLAP NLO QCD global fits gave good $\chi^2$ fits to the HERA data, starting with a conventional flat gluon and sea input, provided that a low starting scale $Q_0^2\sim2\mathrm{GeV}^2$ was used. But as we have seen in Figure 19, the steep slope of $F_2$ extends further down in $Q^2$ to $Q^2\sim1\mathrm{GeV}^2$. By the end of the HERA-I run in 1999, the precision of the data had increased sufficiently that the low-$x$ behaviour of the sea and the gluon distributions could be fitted separately. In Figure 23 we show the total sea and gluon PDFs for various $Q^2$ values down to $Q^2 = 1\,\mathrm{GeV}^2$, below the starting scale for evolution. The data are still reasonably well described even at this low scale, see Figure 24, but these results led to a new kind of surprise since it became clear that the flat input which had been used for both the sea and the gluon ($\lambda_s = \lambda_g \sim0$) was actually a compromise between a sea distribution which remains steep, $\lambda_s \sim0.2$ down to $Q^2 \sim1\,\mathrm{GeV}^2$, and a gluon distribution which is becoming valence-like, $\lambda_g < 0$ at low $Q^2$. Figure 23 shows the result from an NLO QCD fit made by Chekanov et al. [96], but it remains true for the full HERA combination, for NNLO analysis and for all the conventional DGLAP based QCD fits of CT, MSHT and NNPDF. This behaviour contradicts the original argument that the steep behaviour of the gluon distribution is driving the steep behaviour of the sea which is measured in $F_2$.

Figure 24 also shows the limits on the kinematic range of applicability of the DGLAP formalism. The fit can no longer describe the data for $Q^2 \lesssim1\,\mathrm{GeV}^2$. Furthermore, even though the DGLAP formalism works for $Q^2 \gtrsim1\,\mathrm{GeV}^2$, doubts remain about its applicability when $\lambda_g < \lambda_s$. Does the data interpretation beyond the DGLAP formalism? At at low $x$ the form of the DGLAP equation is such that, at LO, one has

$F_2$ and its $Q^2$ evolution are essentially the experimental variables that we are using to determine the sea and the gluon PDFs respectively. But the behaviour of the slope of $\partial F_2/\partial\ln Q^2$ depends on the splitting function as well as on the gluon density. If it could be established that unconventional behaviour of this slope is attributable to $P_{qg}$, rather than to the gluon PDF then this would indicate the need for an alternative evolution equation. This question can be investigated by the measurement of other quantities that depend on the gluon differently than $\frac{\partial F_2(x,Q^2)}{\partial\ln Q^2}$, for example $F_L$ and $F_2^{c\bar c}$. We have seen these measurements in Secs 3.1 and 3.2, we discuss the insights that have been gained from $F_{2}^{c\bar{c}}$ in § 5.2. The role of $F_{L}$ enters into the fitting of the reduced cross section of the inclusive data and is discussed in that context in the next section.

alternative fits to $\sigma_{r}$

As more HERA data were collected it became evident that the DGLAP QCD fits to these data at low $Q^{2}$, which for HERA also implies low $x$, are not all that good, see Figure 13, which shows the $\chi^{2}$ of the HERAPDF2.0 NNLO, NLO and LO QCD fits to the full HERA combined cross section data as a function of the minimum $Q^{2}$ of data entering the fit. It is evident that as $Q_{\min}^{2}$ decreases the $\chi^{2}$ deteriorates. Examination of the plots of $\sigma_{r}$ vs $x$ at low $Q^{2}$, see Figure 12, showed us that the turn over of the reduced cross section for low $x$, $x \lesssim10^{-4}$ is not well described. Recall that this turn over is due to the contribution of $F_{L}$ in $\sigma_{r}=F_{2}-y^{2}/Y_{+}F_{L}$, so that it seems that the fit wants a larger $F_{L}$ than that predicted by the DGLAP formalism. Given that the $F_{L}$ is strongly related to the gluon (see Eq. (19)) and the gluon of the DGLAP formalism is becoming valence like, ie smaller at low-$x$ and $Q^{2}$, this could be interpreted as meaning that the DGLAP formalism needs some modification. We consider two alternatives: i) the use of NLL $\ln(1/x)$ resummation in addition to NNLO DGLAP, Abdolmaleki et al. [15], and the need for higher twist terms at low $Q^{2}$, Abt et al. [23], in subsections 4.1.1, 4.1.2 respectively. However, note that both these approaches are still limited to $Q^{2}\gtrsim1\mathrm{GeV}^{2}$. It is informative to consider the transition to very low $Q^{2}$ as we will do in Section 4.4.

$\ln(1/x)$ resummation

In recent years the programm HELL has become available, Bonvini et al. [76], to perform next-to-leading $\ln(1/x)$ resummations together with next-to-next-to leading order NNLO $\ln(Q^{2})$ resummations. This has been used by the NNPDF collaboration to produce the NNPDF3.1sx PDF, Ball et al. [63], set and by the HERA collaboration, Abdolmaleki et al. [15]. In Figure 25 (left-hand-side) we show the difference in the $\chi^2$ vs $Q^2_{\min}$ plot for NNLO vs NNLO+NLLX treatment using the HERAPDF formalism. The NNLx treatment leads to a decrease in $\chi^2$ for the fit of $\sim70$ points. This is essentially because it provides a much better description of the turn over of the reduced cross section at small $x$, see Figure 25 (right-hand-side), since it predicts a somewhat larger $F_L$ than the conventional DGLAP NNLO approach, see Figure 26.

The PDFs with and without the $\ln(1/x)$ resummation are shown in Figure 27. Here we can see that the inclusion of this low-$x$ resummation results in a gluon which is unsuppressed at low $x$, contrary to the somewhat counter-intuitive behaviour of the pure DGLAP fit, for which the gluon density is falling, and it is thus not possible to state that the gluon is ’driving’ the sea by $g \to q\bar{q}$ splitting. However, once the BFKL-type $\ln(1/x)$ resummation is included, we recover the rising gluon density and we may thus also have to consider gluon-gluon recombination and saturation, see Sec 4.4.

higher twist

There are other approaches to improving the fit to the HERA reduced cross section at low $x$ and $Q^2$. One of the simplest its to introduced higher twist terms into the formalism. Higher-twist effects have a $1/Q^{2n}$ dependence and are thus important at low $Q^2$. They are usually considered at high $x$, which is not accessible at low $Q^2$ for HERA data, but it is possible that there could be low $x$ higher twist effects due to gluon-gluon interactions and recombination at high gluon density, Bartels et al. [69]. Such effects are visualised as gluon ladders with recombining gluons. The leading-twist perturbative QCD forms of the structure functions $F_2$ and $F_L$ are augmented by simple twist-4 terms

where $A_2^{\mathrm{HT}}$ and $A_L^{\mathrm{HT}}$ are free parameters in the fits. It is found that only the $A_L^{\mathrm{HT}}$ term is needed to produce fits with a better $\chi^2$, see Figure 28. This is also in accordance with the predictions of Bartels et al. [69].

Figure 29 shows the corresponding NNLO fits to the reduced cross section, with and without the higher twist term included in $F_L$. The turn over of the reduced cross section at low-$x$ is much better described by the higher twist fit (HHT) because $F_L$ has been increased by the higher twist contribution. Figure 30 shows the improved fit to $F_L$ data for both NLO and NNLO fits. It also shows that this picture is not successful for very low $Q^2$, $Q^2 \lesssim2\,\mathrm{GeV}^2$. Furthermore, the PDFs of HHT are very similar to the PDFs of HERAPDF2.0, even though the values of $A_L^{\mathrm{HT}}$ extracted are quite high: e.g. $A_L^{\mathrm{HT}} = 5.5 \pm0.6\,\mathrm{GeV}^2$ from the NNLO fit. Thus the gluon PDF in this picture remains suppressed, which is somewhat contradictory to the idea that it is a high gluon density responsible for the gluon-gluon interactions which generate higher twist terms.

Interpretation of $F_2^{c\bar{c}}$

In Figure 31 we show the data on the charm reduced cross section, as a function of $x$ for various values of $Q^2$, in ratio to the predictions of the HERAPDF FFN analysis (as the central line) but also to two different GMVFN calculations (FONLL) of the NNPDF collaboration both with and without $\ln(1/x)$ resummation. In both cases the PDFs used are the NNPDF3.1sx set, which included $\ln(1/x)$ resummation calculations for the inclusive DIS HERA data to which it was fitted. Although neither of these calculations is fully successful they do describe the shape of the data better than the standard FFN calculation.

This could be regarded as additional evidence for $\ln(1/x)$ resummation in the HERA data, but one should beware that the path from the observation of a $D$ or $D^*$ to the reduced charm cross section involves much more uncertainty than is involved in inclusive cross section measurements.

Searching for BFKL in forward jets

Further evidence for BFKL type resummation from semi-inclusive processes comes from the observation of forward jets. Considering the gluon emission diagrams of Figure 22 we can see that for a $p_T$ ordered gluon ladder such as required by the DGLAP framework (at leading order) there can be no possibility of a hard (ie high-$p_T$) jet in the forward direction (ie collinear with the proton), because the gluons are assumed to have zero $p_T$ at the start of the ladder gaining $p_T$ at each splitting in an ordered manner. However for a disordered BFKL ladder forward jet production is possible. Hence to look for hard jets emitted in the forward direction the experiments making cuts on jets as follows: $E_{T,\mathrm{jet}} > 3.5, 5$ GeV, $0.5 < E_{T,\mathrm{jet}}^2/Q^2 < 2$, to ensure hard jets of comparable scale to $Q^2$; $x_{\mathrm{jet}} > 0.035$, $1.7 < \eta_{\mathrm{jet}} < 2.8$ to ensure that they are forward jets.

In Figure 32 the data from H1 in the forward region is shown for two cuts $E_{T,\mathrm{jet}} > 3.5$ GeV and 5 GeV, together with predictions from various Monte-Carlo models, on the left hand-side, and from an LO BFKL calculation and an NLO (DGLAP) calculation (DISENT), on the right-hand-side.

The Monte-Carlo models ARIADNE, Lönnblad [185], LEPTO, Ingelman et al. [166], RAPGAP, Jung [168], and LDC, Kharraziha and Lönnblad [172], have been used for predictions. ARIADNE includes one of the main features present in the BFKL approach, which is the absence of the strong $k_T$ ordering. LDC is based on the CCFM approach, Ciafaloni [112]; Catani et al. [92]; Marchesini [190], (see Sec. 4.4). LEPTO is based on conventional leading log DGLAP evolution and RAPGAP is also based on DGLAP evolution but contains an additional resolved virtual photon component. LDC and LEPTO lie below the data by a factor of 2, whereas ARIADNE gives a reasonable good description. As an interesting result, RAPGAP also gives a good description of the H1 data.

It is clear that the conventional DGLAP calculation of DISENT undershoots the data. However, is also clear that the BFKL calculations at LO overshoot the data. These early BFKL calculations suffer, however, from several deficiencies. They are asymptotic and do not contain the correct kinematic constraints of the produced jets. Furthermore they do not allow the implementation of a jet algorithm as used in the experimental analysis. Also NLO $\ln(1/x)$ terms in the BFKL kernel predict large negative corrections which are expected to reduce the forward cross section as well. Recently the BFKL calculations have been improved by taking into account higher order consistency conditions as a way of including sub-leading corrections to the BFKL equation, Kwieciński et al. [181]. The consistency constraint requires that the virtuality of the emitted gluons along the chain should arise predominantly from the transverse components of momentum.

Including the consistency conditions, good agreement between the predictions and the forward jet data is found, for both the H1 and the ZEUS data, as shown in Figure 33.

Thus the BFKL picture appears successful, but this is not definitive because the conventional picture also describes the data IF a resolved photon component is included. This involves the idea that the virtual photon exchanged in the DIS scattering can have a ’resolved’ as well as a ’point-like’ component. The resolved component arises from the idea that the virtual photon can split into a $q-\bar{q}$ pair and thence into more partons, i.e. that it has a structure. This phenomenon is established in the lower $Q^{2}$, $Q^{2}\lesssim5\,\mathrm{GeV}^{2}$ data at HERA. Unfortunately, since BFKL effects are expected at low $x$, the kinematics of HERA implies low $x$ is only probed at relatively low $Q^{2}$ and thus it is not possible to distinguish between these explanations cleanly. We simply note that the idea of the photon having a $q\bar{q}$ component has something in common with the dipole picture which we will discuss later in the context of saturation, see Sec. 4.4.

Very low $Q^2$ interpretation

$\sigma(\gamma^*p)$

We have seen a change of behaviour of $F_2$ at low $Q^2$ such that it stops rising steeply. To understand this it is helpful to look at the HERA data from another point of view.

At small $x$ $F_2$ is related to the $\gamma^*p$ total cross-section

where $W^2 \simeq Q^2/x$ is the centre of mass energy for the $\gamma^*p$ subprocess. The above equation shows that $F_2(x,Q^2)\rightarrow\operatorname{const.}Q^2$ as $Q^2\rightarrow0$, since the real photon proton scattering cross section, $\sigma_{\mathrm{tot}}^{\gamma p}$ is finite. The surprising features of the HERA data are that perturbative QCD appears to describe the data to really quite low values of $Q^2$ and that the strong rise of $F_2$ at small $x$ persists to even smaller $Q^2$ values. How can we describe the data for $Q^2\lesssim1\mathrm{GeV}^2$?

The relation 31 implies that a rise of $F_2$ at small-$x$ corresponds to a rising $\gamma^*p$ cross-section with $W^2$. This suggests another approach to understanding the behaviour of $F_2$ at low $x$ and that is the framework of Regge theory that was used to explain the high energy behaviour of hadron-hadron total cross-sections. Regge theory predicts that the high energy behaviour of hadronic scattering amplitudes is $\operatorname{Im}A(ab\rightarrow cd)\sim\sum_i\beta_i s^{\alpha_i}$, where $\alpha_i$ is the intercept of a Regge trajectory which has the right quantum numbers for an exchange in the crossed channel $a\bar{c}\rightarrow b\bar{d}$. Using the optical theorem, the corresponding total cross-section will vary as $\sigma_{\mathrm{tot}}(ab)\sim\sum_i\beta_i s^{\alpha_i-1}$. For the corresponding forward elastic scattering amplitude, the leading Regge trajectory has the quantum numbers of the vacuum and is known as the Pomeron, for which the value $\alpha_P\approx1.08$ has been determined from hadron-hadron data. This prediction also describes the high energy behaviour of real-photon hadron cross-section measurements successfully. The model was extended to describe virtual-photon proton scattering by Donnachie and Landshoff [131] (DL) who assumed that any $Q^2$ dependence would reside in the functions $\beta_i(Q^2)$ only and that the intercepts, $\alpha_i$ would be independent of $Q^2$. Using Eq. 31 this approach then predicts a flattish input for the gluon and sea PDFs, since $\sigma(\gamma^*p)\sim s^{\alpha_i-1}$ implies $F_2\sim x^{1-\alpha_i}=x^{-0.08}$. This prediction had been so successful in hadronic physics that it led to our expectation that the $x$ dependence of the low $Q^2$ input, for the momentum distributions of sea quarks and gluons, to the DGLAP evolution should be flat. Indeed this was successful in the early DIS data which did not probe lower than $x\sim0.01$.²

While the DL approach successfully described the pre-HERA low $x$ data for $Q^2$ values up to about $10\mathrm{GeV}^2$, it cannot describe the steeply rising $F_2$ data measured at HERA. This has already been shown in Figure 20 which shows the slope $\lambda(Q^2)=\frac{\partial\ln F_2}{\partial\ln(1/x)}$ or $F_2\sim x^{-\lambda(Q^2)}$, at low $x$. For $Q^2$ values less than $1\mathrm{GeV}^2$ or so, the value of $\lambda$ is consistent with that from hadronic Regge theory, whereas for $Q^2>1\mathrm{GeV}^2$ the slope rises steadily to reach a value greater than 0.3 by $Q^2\sim100\mathrm{GeV}^2$. This larger value of $\lambda$ is close to that expected from using BFKL ideas. Indeed the BFKL equation could be viewed as a method for calculating the slope of the Pomeron trajectory using perturbative QCD. However, the result does not correspond to the measured Pomeron slope in low energy hadron-hadron collisions. Thus the idea of two Pomerons arose: a hard or perturbative Pomeron trajectory with intercept $\sim1.3-1.5$ and the soft or non-perturbative Pomeron of hadronic physics with intercept $\sim1.08$. DL, Donnachie and Landshoff (2001), extended their Regge model by the addition of a hard Pomeron with intercept 1.44, which allowed them to describe the low $x$ HERA data up to $Q^2$ values of a few hundred $\mathrm{GeV}^2$. Theirs is one of many attempts, using ideas from Regge theory, to model the ‘transition region’ from real photoproduction through to deep inelastic scattering with $Q^2\gg1\mathrm{GeV}^2$.

Figs. 34 and 35 show the data in terms of $\sigma_{\mathrm{tot}}^{\gamma^*p}$ as a function of $W^2$ and $Q^2$ respectively. Note that this cross section is derived from $F_2$ and thus the contribution of $F_L$ to the DIS reduced cross section must be accounted for in its extraction. This is done in two ways: i) using a parametrisation of $F_L$ data (BKS), Badelek et al. (1997), and ii) using a pQCD fit with higher twist (HHT), Abt et al. [23].

Fig. 34 shows the rise of the $\sigma_{\mathrm{tot}}^{\gamma^*p}$ with $W^2$, for $Q^2$ values from $0.045\mathrm{GeV}^2$ to $30,000\mathrm{GeV}^2$. At the lowest $Q^2$, the data show a slow rise compatible with the original Regge prediction of $F_2(x)\sim0.08$. As $Q^2$ rises, the slope also rises towards the hard Pomeron value $\sim0.3$, as already summarise in terms of the slope of $F_2$ with $\ln(1/x)$ in Figure 20. Note that beyond $Q^2\sim200\mathrm{GeV}^2$ the data can no longer be described in terms of a simple slope. This is because these data are no longer at low $x$, $x\lesssim0.01$, and Eqn. 31 does not apply. The data for all $Q^2$ values are parametrised by the ALLM parametrisation, Abramowicz et al. [19]; Abramowicz and Levy [18], which used early HERA data, and the HHT-ALLM, Abt et al. [24], which used the final combined HERA I+II data.

Fig. 35 shows the data as a function of $Q^2$ for data which are all in the low $x$ region, for different $W$ values. For all $W$, the data is compatible with a smooth transition in the $Q^2$ range $0.1-1\mathrm{GeV}^2$ from an approximately constant behaviour to $\sim1/Q^2$ at higher $Q^2$. From Eqn 31 this implies approximate Bjorken scaling. In fact, as we have already seen, these higher $Q^2$ data are fully compatible with the slow logarithmic scale breaking of pQCD. This figure also makes clear the limitations of the Regge descriptions which only give a good fit at low $Q^2$

The advantages of looking at the data in terms of $\sigma^{\gamma^*p}$ will become evident in the next subsection.

Non-linear evolution equations, saturation

Either considering the rise of $F_2$ as $x \to0$ or the steep energy dependence of $\sigma(\gamma^*p)$ at large $Q^2$, such behaviour will eventually violate the Froissart bound. If the origin of the rise arises from a high gluon density then this problem may be avoided because the gluons can shadow each other from the $\gamma^*$. At even higher densities the gluons will ‘recombine’ via the process $gg \to g$ – the inverse of gluon splitting – and the gluon density, and hence $F_2$, will saturate. These ideas were formalized in the simplest terms by Gribov, Levin & Ryskin by the addition of a non-linear term to the equation for gluon evolution

where $R$ is the proton radius. When $xg(x,Q^2) \sim\pi Q^2R^2/\alpha_s(Q^2)$ the non-linear term cancels the linear term and evolution stops, this is saturation. For a nucleon radius of 1fm = 5GeV$^{-1}$ and scale $Q^2 = 1$GeV$^2$, taking $\alpha_s = 0.5$, this is a very high gluon density $\sim150$, but it has been suggested that there could be hot-spots within the nucleon of R=2GeV$^{-2}$, yielding a gluon density of $\sim24$.

The various evolution equations applicable across the $x$, $Q^2$ plane are summarised schematically in Figure 36. At small $Q^2$ non-perturbative effects must be accounted for no matter what the value of $x$ because the strong coupling is large. For higher $Q^2$, but not too low $x$, the DGLAP $\ln(Q^2)$ resummation applies. At low $x$, for $Q^2$ above the non perturbative region but not very high, the BFKL $\ln(1/x)$ resummation applies. At low-$x$ and higher $Q^2$ some combination of these resummations must be made. Such a ’modified’ DGLAP approach was discussed in Section 4.1.1. A somewhat different picture, considering the $p_t$ of the gluon emissions, is the CCFM equation. Note the appearance of the ‘critical line’ above which the gluon density is in principle high enough for non-linear effects to be important. In this region, for large enough $Q^2$, one has the possibility that $\alpha_s$ will be small enough for weak-coupling but non-perturbative methods to be applicable, of which the GLR equation would be just a first approximation. No scales are given on the diagram of Figure 36 because it is a matter of debate where each region begins. Some argue that the non-perturbative region should extend to higher $Q^2$ as $x$ decreases subsuming the high density region, others that the high density region starts lower in $x$ and subsumes the BFKL region. One of the many hotly argued questions raised by the HERA data is does HERA have the reach to see high density effects and have they been seen? We will first consider this in terms of inclusive variables.

The most successful approach to the high density regime is that of the Colour Glass Condensate (CGC), which is an effective field theory for saturated gluons. In this theory the usual parton model description is replaced by classical strong gluon fields. However, quantum fluctuations around the classical solution are enhanced by terms proportional to $\alpha_s\ln(1/x)$ and these require resummation by non-linear evolution equations. The Balitsky-Kovchegov (BK), Balitsky [60], Kovchegov [174, 175], equation is such an equation, valid at high $N_c$, which reduces to the BFKL equation at low gluon density. The Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner (JIMWLK) equation is a more exact alternative. The phenomenology of these approaches can be described in terms of the dipole models, to which we now turn. The CGC is discussed in detail in the contribution of E. Iancu to this encyclopedia.

Dipole models have proved to be a very fruitful approach in exploring low-$x$ physics by modelling the behaviour of $F_2$ or $\sigma(\gamma^*p)$ through the transition region towards $Q^2 = 0$. The idea is that the virtual photon splits into a $q\bar{q}$ pair³ (a colour dipole) with a transverse size $r \sim1/Q$, the splitting occurring a ‘long’ time $\sim1/(mx)$ before the dipole interacts with the proton. The dipole then scatters coherently from the proton in a time which is short in comparison to $\sim1/(mx)$. The $\gamma^* \to q\bar{q}$ process is described by QED and the strong interaction physics is then contained in the cross-section, $\hat{\sigma}$, for the dipole-proton interaction.

Many authors have worked on these models in the context of DIS processes both inclusive and diffractive (see § 6 for a discussion of diffractive processes). Here we outline the idea using the simplest possible dipole model of Golec-Biernat and Wusthoff (GBW), Golec-Biernat and Wusthoff [155]. The original GBW model provides a simple parameterization of $\hat{\sigma}$ with explicit saturation, in the sense that $\hat{\sigma} \to\sigma_0$ (a constant) as $r$ becomes large, but in a such a way that the approach to saturation is $x$ dependent, controlled by the ‘saturation radius’, $R_0$. Explicitly the dipole cross-section is given by

where $\sigma_0$ is a constant, and

$R_0(x)^2$ is understood to be inversely proportional to the gluon density ($\lambda\sim\lambda_g$), so that $R_0$ is a measure of the transverse separation of the gluons in the target. Thus when the dipole separation is large compared to the gluon separation (small $Q^2$ and small $x$) the dipole cross-section saturates, $\hat{\sigma} \sim\sigma_0$, and $\sigma(\gamma^*p)$ also tends to a constant, giving $F_2 \propto1/Q^2$ from Eq. (31). When the dipole separation is small compared to the gluon separation (high $Q^2$ and large $x$), $\hat{\sigma} \sim\sigma_0/(Q^2R_0^2)$ and $\sigma(\gamma^*p)$ varies as $1/Q^2$, so that $F_2$ exhibits Bjorken scaling. Because of the $x$ dependence of $R_0$, $\hat{\sigma}$ saturates for smaller dipole sizes as $x$ decreases.

We have seen this behaviour already in Figure 35. Let us look at it again in Figure 37 in terms of the behaviour of $F_2$ as a function of $Q^2$ at fixed $y=Q^2/sx$. In this figure the data are compared to the GBW dipole model. We can see that the data do exhibit the features $F_2 \propto1/Q^2$ at low $Q^2$ and $F_2 \sim$ constant at high $Q^2$, as described above. There is a clear change in behaviour around $Q^2 \approx1\,\mathrm{GeV}^2$, which might then be taken as a rough estimate of the saturation scale for HERA data. However, one needs to be somewhat careful before jumping to the conclusion that the GBW dipole model shows that saturation effects have been seen at HERA. Given Eq. (31), $F_2 \sim1/Q^2$ at low $Q^2$ is required in any model since the photoproduction cross-section is finite, and at high $Q^2$ slow, logarithmic scale breaking is required by pQCD. Thus the real challenge is not in describing the general behaviour of $F_2$ and/or $\sigma(\gamma^*p)$ but in describing the exact shape of the data across the transition, from low to high $Q^2$, now that high precision data are available.

The original GBW model did not include QCD evolution in $Q^2$ and this limited its success at larger $Q^2$. This deficiency was corrected in a second version, Bartels et al. [70], in which $\hat{\sigma}$ was related to the gluon density explicitly, such that gluon evolution with the scale $\mu^2 = C/r^2 + \mu_0^2$ is included.

Results from both versions are shown in Figure 37.

Another feature of saturation models which is illustrated in the GBW model is the dependence of the dipole cross-section on a scaled variable, here $r/R_0(x)$. Stasto, Golec-Biernat and Kwiecinski, Stasto et al. [218], have made the interesting observation that this leads to a new scaling property of $\sigma(\gamma^*p)$. At low $x$, $\sigma(\gamma^*p)$ should depend only on the dimensionless variable $\tau= Q^2R_0^2$. What is more, such a scaling property seems to be satisfied by HERA data. One expects

Figure 38 shows HERA data with $x < 0.01$ as a function of $\tau$. Not only do the data show this scaling very clearly, but also the expected change in behaviour from small to large $\tau$ values seems to occur around $\tau\sim1$. This new scaling – geometrical scaling – holds for $Q^2$ values up to about 400 GeV$^2$ but only for $x < 0.01$. Again, though it is not a proof, the remarkable demonstration of geometrical scaling by the low $x$ HERA data does show that they have many of the attributes of a saturated system.

More recent successful dipole models include the Impact-Parameter dependent Saturation (IPSat) model, Rezaeian et al. [214], in which a Gaussian impact parameter, $b$, dependence, $T_G(b)$ is given to the dipole amplitude

This model is very successful in describing $F_2$, $F_2^{c\bar{c}}$ and $F_L$ at low-$x$, and it also predicts a gluon which is not decreasing at low-$x$ and $Q^2 in contrast to the predictions of DGLAP evolution, see Figure 39. This figure also shows the value of the saturation scale, $Q_s$ as a function of $x$, for various values of the impact parameter, $b$. We notice that the values of the saturation scale are rather low, $Q_s^2 \lesssim2\,\mathrm{GeV}^2$, even for the lowest $x$ values reached at HERA $\sim10^{-5}$. This makes it hard to claim definitive evidence for saturation at HERA, since these low $Q^2$ values require the consideration of further non-perturbative effects such as higher twist contributions, and most of the HERA data are not in the saturation region. However, more compelling evidence can be found in semi-inclusive data sets on vector meson production and Deeply Virtual Compton Scattering (DVCS), see § 6.

Two further points can be made. Firstly, it should be noted that there are other dipole models that fit the HERA reduced cross section data as successfully without an explicit $x$ dependent saturation scale. Figure 40 compares the IPsat model with a linearised version of the same model, without saturation, IPnonsat. The fits are almost indistinguishable such that the good fit of the IPsat model cannot be claimed as definitive evidence for saturation.

Secondly, saturation can also be achieved without the dipole picture. The operator product expansion beyond leading twist has diagrams in which two, three or four gluons may be exchanged in the $t$-channel such that these gluons may be viewed as recombining. This recombination could lead to gluon saturation.

Diffraction

Introduction

We now turn to diffractive processes, which many models predict may be a productive hunting ground for novel low $x$ dynamics such as saturation. Diffractive processes in proton-(anti)proton scattering, and to a lesser extent, other projectiles scattering diffractively on proton targets, have been the subject of sustained and intense study since well before HERA, Goulianos [160]. High energy data now also exist from the LHC, [1, 171, 2]. Although it was not widely expected prior to data taking, the study of quasi-elastic and diffractive processes has been one of the most successful areas of study at HERA, and certainly one of the most prolific in terms of publications. The narrative here closely follows that of Newman and Wing [204].

The kinematics of HERA, with a strong forward boost of the hadronic centre of mass relative to the lab frame for non-extreme $y$ values, were particularly favourable for the study of diffractive excitations of the real and virtual photon. Two different diffractive processes are of interest: quasi elastic scattering and diffractive dissociation. In the (quasi-)elastic case, $\gamma^{(*)}p \to Vp$, the photon coupling to the beam lepton either remains intact (Deeply Virtual Compton Scattering, DVCS) or converts to a vector meson $V$. In the diffractive dissociation case, $\gamma^{(*)}p \to Xp$ the system $X$ produced at the proton vertex is a multi-particle state covering a continuum of invariant masses, $M_X$. Most of the interest has centred around the case where the proton remains intact and is scattered through a small angle (implying a small absolute value of the squared four-momentum transfer at the proton vertex, $|t|$ (see Figure 42(a) and Figure 48(a))).

A QCD-based treatment of exclusive vector meson production, [216, 86, 145], by quasi-elastic scattering predicts that cross sections should be $Q^2$-suppressed relative to inclusive DIS and relatable to the parton densities of the proton. The lowest order partonic exchange would be a pair of gluons with compensating colour charges, [187, 209].

QCD models of single diffractive photon dissociation need to be broken down into two categories. First ‘hard’ diffractive processes, similar to exclusive vector meson production, are expected. In these the colour singlet exchange couples fully to the hard interaction with the photon. Such an exchange may again be perturbatively calculable starting from a knowledge of the proton structure. Second, ‘soft’ diffraction, in which the colour singlet exchange is a composite virtual object, similar to the pomeron of peripheral hadronic scattering, in which a single parton participates in the hard scattering, the remainder forming a low transverse momentum remnant system, cleanly separated in rapidity from the oppositely-travelling intact proton. This second type of configuration, Ingelman and Schlein [165], Donnachie and Landshoff [129], produces a leading twist contribution with similar $Q^2$ dependence to the total cross section, limited only by kinematic constraints at large $Q^2$. In the DIS regime, it can be related to a concept of diffractive parton densities.

A major issue in inclusive diffraction at HERA has been determining in which kinematic regions each of these ‘soft’ and ‘hard’ processes are dominant and understanding the transition between the two.

The detailed dependence on centre-of-mass energy of elastic and total hadron–hadron cross sections (via the optical theorem) has historically been remarkably well described in a large kinematic domain by Regge phenomenology, Regge [212, 213]. In this framework, interactions take place via the $t$-channel exchange of reggeons, related to mesons, Chew et al. [111], and the pomeron (IP), which is the colourless, flavourless leading vacuum singularity, Chew and Frautschi [110]. The pomeron is the mediator of diffractive scattering. Although the pomeron is essentially a phenomenological object, associated mainly with soft processes, and now often called the ‘soft’ pomeron, it has also been calculated perturbatively. For example, it has been calculated by connecting two exchanged gluons in the $t$-channel (the lowest order partonic process that can describe a flavourless, colourless exchange) by adding further gluon ‘rungs’. This leading-logarithmic $1/x$ approach leads to the ‘BFKL pomeron’ or ’hard’ pomeron discussed elsewhere in this contribution.

At asymptotically large energies, pomeron exchange dominates the elastic channel, such that both elastic and total cross sections display a slow increase with centre-of-mass energy. Interactions in which one or both of the hadrons dissociate to higher mass states, Feinberg and Pomerančuk [139]; Good and Walker [157], also occur naturally in this approach. Such processes are characterised by the presence of large regions of rapidity space in which no hadrons are produced. They are dominated by diffractive exchange at large centre-of-mass energy $\sqrt{s}$ and small dissociation masses $M_X$. The inclusive dissociation mass distribution may be treated via Mueller’s generalisation of the optical theorem, Mueller [198], such that an appropriate Regge description involves diagrams that contain three-reggeon couplings.

Pomeron language has been adopted to a large extent at HERA, as a matter of convenience. This should not be taken to imply the existence of a universal $t$-channel pomeron exchange, since the detailed properties of the exchange and its Regge trajectory clearly vary between processes and with kinematic variables at HERA, as discussed in detail below.

As an alternative to Regge theory, an $s$-channel picture, Miettinen and Pumplin [197]; Dederichs and Faessler [118], can be developed around the original Good & Walker approach, Good and Walker [157]. In this approach different projectile Fock states are absorbed differently on the target. The application of this $s$-channel approach to HERA data involves the elastic and quasi-elastic scattering of colour dipoles, corresponding to $q\bar{q}$ (or higher multiplicity) fluctuations of real and virtual photons on the proton, Mueller [199]; Nikolaev and Zakharov [205]. An attractive feature of this approach is the possible presence of ‘saturation’ phenomena, whereby the low $x$ growth of parton densities is tamed as ultimately required to satisfy unitarity. This effect is easily incorporated at the parametric level, Golec-Biernat and Wusthoff [154], while more sophisticated treatments are based for example on the Color Glass Condensate model, Gelis et al. [151].

Whilst such models were originally applied at HERA to describe the total $\gamma^* p$ cross section (see Figure 41), vector meson production and other exclusive processes are easily incorporated with no further free parameters except those that describe the $t$ dependence, Kowalski and Teaney [176]; Kowalski et al. [177].

Diffractive dissociation can also be included, Nikolaev and Zakharov [206], in this approach although it has proved more difficult to describe due to the need to incorporate higher terms, corresponding for example to a dipole formed from a $q\bar{q}$ pair and a gluon, Bartels et al. [67]; Golec-Biernat and Wusthoff [156]. The dipole approach to diffraction at HERA is discussed further in Sections 5.2.5 and 5.3.8.

We now discuss experimental results on quasi-elastic scattering and diffractive dissociation in in Section 5.2 and Section 5.3, respectively.

Exclusive Production of Vector Mesons and Photons

Kinematics and Experimental Selection

Quasi-elastic vector meson production and DVCS (Figure 42(a)) are the simplest diffractive processes that have been studied at HERA. For a fixed final state vector meson or photon, they are usually described in terms of the kinematic variables, $Q^2$, $W$ and $t$. Distributions in all three of these variables have been measured in analyses covering the vector meson species, $\rho$, $\omega$, $\phi$, $\rho^0$, $J/\psi$, $\psi'$ and $\Upsilon$ as well as in DVCS. No evidence has been found for the exclusive production at the photon vertex of particles with non-$1^{--}$ quantum numbers such as the $\pi^0$, Adloff et al. [37], as would be expected for the exchange of the postulated negative $C$-parity partner of the pomeron, the odderon, Lukaszuk and Nicolescu [188].

The most precise vector meson data are obtained by reconstructing two-prong decays via charged decay products (notably $\rho^0 \to\pi^+\pi^-$, $\phi\to K^+K^-$ and $J/\psi\to e^+e^-$ or $\mu^+\mu^-$) and requiring no further activity beyond the noise levels in the detector, except that associated with the scattered beam electron. DVCS selections require a similar lack of activity in the detector beyond the final state electron and photon, the main experimental complication being separating the signal from the competing purely electromagnetic Bethe–Heitler process, which generates identical final states. In contrast to inclusive diffractive studies, the simplicity of these final states and the high precision of the tracking detectors have allowed precise $t$ measurements without the need to tag the intact outgoing proton. This reconstruction method uses $|t-t_{\min}|=p_T^2(V)$, where the vector meson transverse momentum $p_T(V)$ is obtained from its decay products and $|t_{\min}|$, the minimum kinematically accessible value of $|t|$, is usually negligible.

General Characteristics of Vector Meson Production

Vector meson production has emerged as a sensitive probe of the transition from the soft diffractive dynamics which are familiar from hadronic scattering experiments to a harder regime which may be calculated perturbatively. The former regime is encountered wherever no hard scale is present, the classic example being $\rho^0$ photoproduction, Aid et al. [42]; Breitweg et al. [84, 78]. Under such circumstances, the energy dependence of the photon–proton process is in good agreement with the form predicted by Regge asymptotics:

where the pomeron trajectory, $\alpha_{\mathbb{P}}(t)=\alpha_{\mathbb{P}}(0)+\alpha'_{\mathbb{P}}t$, is assumed to be linear and its intercept $\alpha_{\mathbb{P}}(0)\simeq1.085$, Donnachie and Landshoff [130]; Cudell et al. [116]. This is the ’soft’ pomeron. This has been found to work well for $\rho^0$ photoproduction data, though interestingly, the slope of the pomeron trajectory has been found, Breitweg et al. [84], to be significantly smaller than the value of $\alpha'_{\mathbb{P}}\sim0.25$, Donnachie and Landshoff [128]; Abe et al. [16], obtained from soft $pp$ and $p\bar{p}$ scattering. A possible explanation for this may be found in process-dependent absorptive corrections, which are absent in DIS, present to some extent in photoproduction and to a larger extent in fully hadronic scattering. Detailed models of these effects can be found for example in Kaidalov et al. [169, 170]; Gotsman et al. [159]. Further characteristics of this soft regime are, Aid et al. [42]; Breitweg et al. [78], a skewed lineshape for the $\rho$ meson due to its interference with non-resonant $\pi^{+}\pi^{-}$ production and a large value, $B \sim10\ \mathrm{GeV}^{-2}$, of the slope parameter describing the $t$ dependence according to

These $B$ values suggest that the interaction takes place over a transverse distance of typically 1 – 1.5 fm, when interpreted in a simple optical model as a measurement of the mean impact parameter between the incoming photon and proton,

Wherever hard scales are present, usually provided either by heavy quarks in the vector meson or by large $Q^{2}$, but sometimes also by large $\lvert t\rvert$, the qualitative picture changes. The energy dependence becomes progressively steeper, such that the $W$ dependence, parametrised similarly to Eq. (37), yields an increased effective value of $\alpha_{\mathbb{P}}(0)$. This effect is illustrated in Figure 43, where a compilation of photoproduction data on the total cross section and different vector meson production cross sections is shown as a function of $W$. The steepening of the dependence on $W$ for the heaviest vector mesons can be interpreted in terms of the presence of a ’hard’ pomeron exchange or alternatively in terms of the scale dependence of the proton gluon density at low $x$, as discussed further below.

The transition from soft hadronic to perturbative behaviour is neatly mapped out in a single process in $\rho^{0}$ electroproduction data. As $Q^{2}$ increases, the $t$ slope parameter $B$ decreases, the $W$ dependence becomes steeper and the lineshape skewing disappears, all in a manner which has been measured with good precision, [32, 8, 102, 80]. An example, also including comparisons with other exclusive vector meson production processes, is shown in Figure 44. Here, the effective pomeron intercept (Eq. (37)) is shown as a function of a scale, which is chosen to be $(Q^{2} + M_{V}^{2})/4$ (see § 7.4). As the scale increases, the effective pomeron intercept shifts from values typical of soft hadronic scattering to values which are compatible with results for the equivalent quantity $\alpha_{\mathbb{P}}(0) = 1 + \lambda$ in fits of inclusive low-$x$ HERA data to the form $F_{2}(x,Q^{2}) \propto x^{-\lambda(Q^{2})}$, [34, 104], as also discussed here in Sec. 3.3 (Figure 20a).

The exponential $t$ slopes of vector meson production processes are also found to vary systematically with scale and are approximately invariant in $Q^{2} + M_{V}^{2}$. A compendium of results is shown in Figure 45. Although the uncertainties are often large and there is some scatter, the data suggest a convergence towards an asymptotic value of $B \sim5\ \mathrm{GeV}^{-2}$. In optical models, this can be interpreted as the point at which the physics is entirely short-distance in nature, the size of the probe becomes negligible and the slope parameter measures the size of the proton. Quantitatively, this indicates an effective proton size of around $0.6\ \mathrm{fm}$, which is interestingly smaller than the value of $\sim0.8\ \mathrm{fm}$ which is well measured using electromagnetic probes. Interpreting vector meson production in terms of gluon exchange, this suggests that the gluon radius of the proton may be smaller than its quark radius.

Vector Meson Production in QCD

The observed relationship between the $W$ dependences of inclusive DIS and vector meson production in the presence of a sufficiently large scale encourages a perturbative approach. This has evolved considerably in the HERA era, to the point where most observables can be successfully predicted. Although the basic quark charge counting SU(4) prediction for the ratio of cross sections $\rho^{0}:\phi:J/\psi= 9:1:2:8$ holds approximately, when viewed as a function of the scale $(Q^{2} + M_{V}^{2})/4$, Aaron et al. [8], vector meson wavefunction effects remain a significant source of uncertainty.

A QCD collinear factorisation theorem, Collins et al. [114], valid for the leading power of $Q^{2}$ where $\lvert t\rvert\ll\Lambda_{\mathrm{QCD}}^{2}$, relates cross sections for heavy vector meson production from longitudinally polarised photons to the generalised parton densities (GPDs) of the proton, Diehl [125], (see § 7.3). Neglecting skewing effects, in which the two exchanged partons carry different fractions of the proton longitudinal momentum, the GPDs reduce to the square of the gluon density of the proton. The process is then driven to first approximation by the exchange of a pair of gluons from the proton structure, as illustrated in Figure 42(b). Later approaches, Martin et al. [192], incorporated transversely polarised photon cross sections at sufficiently large $Q^2$ and light vector mesons in a similar framework, with some degree of success.

Since it is relatively uncomplicated theoretically, has a reasonably large scale, probes small $x$ values and is experimentally clean, $J/\psi$ photoproduction is an ideal testing ground for these ideas and, a potentially competitive means of extracting the gluon density of the proton. The many measurements at HERA, [48, 94, 53], have now been supplemented by measurements from ultra-peripheral processes at the LHC, extending the $W$ range to beyond 1 TeV, [25] (Aaij et al. 2013). The $W$ dependence of the cross section for $J/\psi$ photoproduction has been calculated using the proton gluon density by a number of groups, [193, 195, 148]. An example comparison with data is shown in Figure 46. From comparisons between the predictions with different gluon densities, it is clear that there is high sensitivity. However, the normalisation is not well predicted and there remains considerable model dependence.

More detailed comparisons with QCD models require an understanding of the helicity structure of vector meson production. The five independent helicity amplitudes, describing transitions from either longitudinally or transversely polarised photons to either longitudinally or transversely polarised vector mesons, can be extracted from the production and decay angular distributions via spin density matrix elements as parametrised in Schilling and Wolf [217]. Whilst helicity amplitudes for $J/\psi$ production at modest $|t|$ are consistent with $s$-channel helicity conservation, Aktas et al. [48], significant ‘helicity flip’ amplitudes, for which the vector meson helicity differs from that of the photon, are observed for the case of $\phi$ production, Adloff et al. [35]; ); Breitweg et al. [85]; ); Aaron et al. [8]; ); Chekanov et al. [102], and for $J/\psi$ production at large $|t|$, Aktas et al. [45]; ); Chekanov et al. [109]. This reflects the often unequal sharing of the longitudinal momentum between the two quarks, corresponding to values of $z$ close to 0 or 1 in Figure 41, particularly for the transverse polarisation case with $\rho$ mesons, This allows the helicity to flip at sufficiently large $t$, that perturbative QCD-based models give good predictions.

Probably the most important feature of angular distribution analyses is that they allow an unfolding of the cross sections $\sigma_L$ and $\sigma_T$, corresponding to vector meson production from longitudinally and transversely polarised vector mesons, respectively. The $Q^2$ dependences of $\sigma_L$ and $\sigma_T$ are separately predicted by QCD models. The basic leading order dependences at $t = 0$ are $\sigma_L \propto1/Q^6$ and $\sigma_T \propto1/Q^8$, Brodsky et al. [86]. These are expected to be strongly violated, in particular due to the $Q^2$ dependence of the proton gluon density, but also due to the quark virtuality and the running of $\alpha_s$, Frankfurt et al. [145]. The HERA data for all vector meson species exhibit approximate scaling with power law dependences of $\sigma_L \propto(Q^2 + M_V^2)^{-2.1}$ and $\sigma_T \propto(Q^2 + M_V^2)^{-2.9}$, which are reasonably well reproduced by both collinear factorisation and dipole-based models, Martin et al. [192]; Kowalski et al. [177].

As discussed in § 7.2 the variation of the slope parameter $B$ with centre-of-mass energy $W$ is usually parametrised in terms of the slope of the pomeron trajectory $\alpha'_P$. Values close to zero are measured with small uncertainties in some hard exclusive processes, notably $J/\psi$ production at large $|t|$, Aktas et al. [48]; Chekanov et al. [94, 99]. This is consistent with expectations in the BFKL approach, Nikolaev et al. [207]; Brodsky et al. [87], where $\alpha'_P$ is related to the average transverse momentum of partons along the gluon ladder. However, it is hard to draw quantitative conclusions in light of the smaller-than-expected $\alpha'_P$ values for soft processes such as $\rho^0$ photoproduction (§ 7.2).

Deeply Virtual Compton Scattering and Generalised Parton Densities

Due to the underlying exchange of a pair of gluons, which in general differ in both longitudinal and transverse momentum, hard exclusive DIS processes have emerged in recent years as candidates with which to understand correlations between partons in the proton and thus the transverse spatial, momentum and spin distributions of partons. This information is encoded in GPDs, Diehl [125], which were introduced in § 7.2.1. HERA data offer unique sensitivity to GPDs at low $x$. This topic is most commonly associated with the DVCS process, though it is also relevant to vector meson production.

The dominant underlying parton level process for DVCS at low $x$ is similar to that shown in Figure 42(b), though there is also a ‘handbag’ contribution in which the incoming (virtual) and outgoing (real) photons couple directly to quarks from the GPDs. DVCS has the advantage of avoiding the complication of the vector meson wavefunction, but cross sections are suppressed due to the final state photon coupling. Indeed, DVCS studies emerged relatively late in the lifetime of HERA, due to the smaller cross sections compared with their vector meson counterparts. For the HERA kinematic region, integrated over azimuthal degrees of freedom, interference between DVCS and the competing Bethe–Heitler process is small and DVCS cross sections can be extracted by statistically subtracting the Bethe–Heitler component using Monte Carlo models tuned in regions in which the DVCS contribution can be neglected.

By the end of HERA operation, $Q^2$, $W$ and $t$ distributions for DVCS had all been measured with good precision, Chekanov et al. [106]; Aaron et al. [6, 4]. As shown in Figure 44 and Figure 45, the exponential $t$ slope is $B \simeq5$ GeV$^{-2}$ and parametrisations of the cross section in the usual $W^\delta$ form yield $\delta\simeq0.5 - 0.6$. Example $Q^2$ dependence data are shown in Figure 47. The DVCS cross section falls more slowly (roughly as $(Q^2)^{-1.5}$) with photon virtuality than is the case for vector meson production. This observation is in line with theoretical predictions, Frankfurt et al. [146], and is partially explained by the presence of the quark-driven handbag contribution, which is not relevant to vector meson production. Together, these observations indicate that DVCS can be considered as a hard process, even at relatively small $Q^2$.

Asymmetries in the azimuthal degree of freedom between the lepton and the (virtual-to-real) photon scattering planes in DVCS give access to the real part of the amplitude, and are most sensitive to the non-diagonal GPDs. The beam charge asymmetry at HERA is clearly non-zero, Aaron et al. [6], and its azimuthal dependence has been used to confirm that the DVCS amplitude is dominantly imaginary for HERA kinematics. It has not yet been possible to determine beam polarisation asymmetries in the manner successfully pursued in the fixed target configuration by HERMES, Airapetian et al. [44]. Nonetheless, the H1 and ZEUS data provide important low $x$ ingredients in models of GPDs, the principal sensitivity being to the GPD $H$, Kumericki and Mueller [178].

Exclusive Processes in Dipole Models and Low $x$ Saturation

Viewed in the proton rest frame, vector meson production naturally factorises into three separate processes, occurring on distinct time-scales (c.f. Figure 41 and Figure 42(b)). The incoming real or virtual photon first fluctuates into a $q\bar{q}$ pair, which creates a colour dipole of transverse size $r_{\perp} \sim1/\sqrt{Q^{2}}$, Wusthoff [221]. The dipole then scatters elastically from the proton, a process which is generically described in terms of a dipole cross section. Finally, the outgoing colour dipole combines back to a photon in the DVCS case, or else hadronises to a final state vector meson in a manner which depends on the non-perturbative vector meson wavefunction. The scale choice of $(Q^{2} + M_{V}^{2})/4$, already used in the previous sections, corresponds to the inverse of the scanning radius of the dipole–proton interaction, for the case where the incoming photon longitudinal momentum is shared equally between the quark and antiquark forming the dipole, Frankfurt et al. [145]; Ivanov et al. [167]. It is appropriate for heavy vector mesons and for light vector mesons produced by longitudinally polarised photons.

At small enough $r_{\perp}$, the dipole cross section can be described perturbatively in the manner discussed in § 7.2.1. At larger $r_{\perp}$, the dipole cross section is not known a priori, but is sensitive to any low $x$ saturation of the proton parton densities. There are various parametrisations of the dipole cross section which include saturation phenomena, some of which are based on particular approximations to QCD with non-linear evolution such as the Colour Glass Condensate, Iancu et al. [164], and some of which are purely phenomenological, Golec-Biernat and Wusthoff [154, 156]; Forshaw et al. [143, 144].

In the most sophisticated dipole models of vector meson production, the target matter density, and hence any saturation effects, depend on the distance from the centre of the proton, such that the dipole cross section varies with impact parameter, Kowalski and Teaney [176]; Kowalski et al. [177]; Rezaeian et al. [214], which in turn is closely related to $1/t$ (see also Marquet et al. [191] for a commentary approach). When viewed differentially in $t$, the influence of the saturation effects thus becomes stronger as $t$ increases, Kowalski et al. [177]; Abelleira Fernandez et al. [17]. Such models adequately describe almost all aspects of vector meson production and DVCS at HERA, though the energy frontier for such processes has now moved on to the Ultraperipheral Collision programme at the LHC.

Inclusive Diffractive Dissociation

Kinematics and Experimental Selection

The kinematic variables describing the inclusive diffractive DIS (DDIS) process $ep \to eXp$ are illustrated in Figure 48(a). In addition to $x$, $Q^{2}$ and $t(<0)$, there is one further non-trivial invariant, corresponding to the mass $M_{X}$ of the diffractive final state. In practice, the variable $M_{X}$ is usually expressed in terms of

Small values of $\beta$ thus refer to events large diffractive masses, larger than $Q^{2}$, whereas exclusive processes have values of $\beta$ close to unity. In models based on a factorisable pomeron (§ 7.6.1), $\beta$ may be interpreted as the fraction of the pomeron longitudinal momentum which is carried by the struck parton. The longitudinal momentum fraction of the colourless exchange with respect to the incoming proton is denoted $x_{\mathbb{P}}$, such that $\beta x_{\mathbb{P}} = x$.

Experimentally, diffractive $ep$ scattering is characterised by the presence of a leading proton in the final state, retaining most of the initial state proton energy, and by a lack of any other hadronic activity in the forward (outgoing proton) direction, such that the system $X$ is cleanly separated and $M_{X}$ may be measured in the central detector components. These signatures have been widely exploited at HERA to select diffractive events by tagging the outgoing proton in the H1 Forward Proton Spectrometer or the ZEUS Leading Proton Spectrometer (proton-tagging method) or by requiring the presence of a large gap in the rapidity distribution of hadronic final state particles in the forward region (LRG method). In a third approach ($M_X$ method, Breitweg et al. [82]; Chekanov et al. [100, 104]), the inclusive DIS sample is decomposed into diffractive and non-diffractive contributions based on their characteristic dependences on $M_X$. Whilst the LRG and $M_X$-based techniques yield better statistics than the LPS method, they suffer from systematic uncertainties associated with an admixture of proton dissociation to baryon states with small masses, typically $M_Y \sim1.6$ GeV, which is irreducible due to the limited forward detector acceptance. Detailed comparisons between cross sections obtained by the LRG and proton tagging methods can be found in Chekanov et al. [107]; Aaron et al. [10]. Neither collaboration observes any evidence for deviations from a constant ratio of cross sections measured by the two methods, indicating that the ratio of probabilities for the proton to scatter elastically and to undergo a low mass excitation is independent of the inclusive kinematic quantities.

Inclusive Diffraction Data

Observations of diffractive dissociation in DIS, Derrick et al. [120]; Ahmed et al. [39], and of jet production in diffractive photoproduction, Ahmed et al. [41]; Derrick et al. [122], and DIS, Derrick et al. [121], were among the earliest HERA publications. These were based on rapidity gaps which were much larger than could conceivably occur at significant rates in standard fragmentation models, Andersson et al. [55]. As increasingly larger datasets became available, the precision on the inclusive diffraction cross sections improved correspondingly, until the final measurements covered three-fold differential cross sections with a precision of a few percent, Aaron et al. [12]; Chekanov et al. [107], over large phase space regions.

In analyses of data with protons tagged in the LPS and FPS, the $t$ dependence of DDIS has been measured and, as in the case of exclusive vector meson production, found to be compatible with an exponential parametrisation of the form of Eq (38). Example H1 data on the slope parameter $B$ are shown in Figure 49, Aaron et al. [10]. Similar results, with globally slightly larger $B$ values, are obtained by ZEUS, Chekanov et al. [107]. Beyond the low $M_X$ resonance region, $B$ is typically around $6 - 7$ GeV$^{-2}$, independently of $\beta$ and $Q^2$, but with some dependence on $x_{\mathbb{P}}$, which is further discussed in Section 5.3.4. This is indicative of an approximate factorisation of the proton vertex from the virtual photon vertex in DDIS (Section 5.3.3).

The semi-inclusive DDIS cross section is usually presented in the form of a diffractive reduced cross section $\sigma_r^{D(3)}$, which is related to the experimentally measured differential cross section analogously to the inclusive case (Sec. 2), such that

where $Y_{+} = 1 + (1-y)^{2}$ and $y$ is the usual Bjorken variable. The reduced cross section depends at moderate scales, $Q^{2}$, on two diffractive structure functions $F_{2}^{D(3)}$ and $F_{L}^{D(3)}$ according to

For $y$ not too close to unity, $\sigma_{r}^{D(3)} = F_{2}^{D(3)}$ holds to very good approximation. Measurements have also been made of the full four dimensional reduced cross section, dependent also on $t$, Chekanov et al. [107]; Aaron et al. [10]. However, since the $t$ dependence factorises to good approximation, this degree of freedom is usually integrated out. A more detailed exposition of the formalism of diffractive DIS can be found for example in Aktas et al. [49].

A combination of data from the H1 and ZEUS experiments using the proton tagging method is published in Aaron et al. [11]. The cross-calibration of the systematic uncertainties between the two experiments implicit in the averaging procedure leads to an improvement in precision well beyond that expected from statistical considerations alone. Full combinations of data from the two collaborations for the LRG method have not been performed, mainly due to some non-trivial discrepancies, usually at the edges of the accessible phase space. An investigation of the residual differences and crude combination may be found in Newman and Ruspa [203]. The comparison of final data in the most precise region is illustrated in Figure 50. Whilst there is good agreement in the bulk of the phase space, residual disagreements are apparent at small $\beta$ and small $x_{\mathbb{P}}$. These discrepancies have only a small influence on the interpretation of the data discussed in the following.

Various models have been used to interpret DDIS data. Since it has become standard, Sections 5.3.3–5.3.7 focus mainly on the ‘proton vertex factorisation’ approach, in which the diffractive exchange (‘pomeron’) has a resolved structure. The relevance of DDIS to dipole models is discussed separately in Section 5.3.8. It is worth noting that a third approach, in which rapidity gaps are randomly produced from final state colour interactions in standard DIS events, has been at least partially successful. This model is attractive in its simplicity. Indeed an early incarnation, Buchmuller and Hebecker [88], in which diffractive final states simply emerge randomly due to colour rotations in one ninth of all low $x$ DIS events works remarkably well for inclusive data. Closely related ideas are encoded in the ‘Semi-classical model’, Buchmuller et al. [89, 90]. The ‘Soft Colour Interaction’ model, Edin et al. [134], offers a rather different, but again related, approach, which later developed into the ‘Generalised Area Law’, Rathsman (1999) model. All of these approaches have enjoyed success, particularly in comparisons with $F_{2}^{D}$ and $\sigma_{r}^{D}$ data, but none provides a comprehensive description, breaking down for example when confronted with more exclusive processes such as diffractive dijet production, Adloff et al. [33].

Factorisation Studies

Cleanly interpreting a three (or even four)-fold differential cross section is far from easy. It is simplified considerably if the dependences on the relevant kinematic variables can be factorised. QCD hard scattering collinear factorisation, when applied to diffractive DIS, Collins (1998), implies that the cross section for the process $e p \to e X Y$ can be written in terms of convolutions of partonic cross sections $\hat{\sigma}^{ei}(\hat{x},Q^{2})$ with diffractive parton density functions (DPDFs) $f_{i}^{D}$ as

with additional implicit dependences on $\mu_{F}$, $\mu_{R}$ and $\alpha_{S}$ as in the analogous inclusive case. The partonic cross sections are the same as those for inclusive DIS. This factorisation formula is valid for sufficiently large $Q^{2}$ and fixed $x_{\mathbb{P}}$ and $t$.

The experimental data over most of the accessible phase space are compatible with ‘proton vertex’ factorisation, Ingelman and Schlein [165], in which the DPDFs may be further factorised into a term containing only variables associated with the virtual photon vertex ($\beta$, $Q^2$) and a term containing variables associated with the proton vertex ($x_{\mathbb{P}}$, $t$):

This is equivalent to considering the diffractive exchange as a ‘pomeron’ with a partonic structure given by the parton distributions $f_i(\beta,Q^2)$, the variable $\beta$ corresponding to the fraction of the pomeron longitudinal momentum carried by the struck quark. The term $f_{\mathbb{P}/p}(x_{\mathbb{P}},t)$ represents the probability that a pomeron with particular values of $x_{\mathbb{P}}$ and $t$ couples to the proton and is called the ‘pomeron flux factor’. Whilst this approach cannot be valid to ultimate precision (e.g. it is clearly violated in the high $\beta$ region where higher twist processes such as exclusive vector meson production play an important role), it is justified theoretically for the leading twist case, Blumlein and Robaschik [75].

As first shown in Adloff et al. [27], a good description of the data at relatively large $x_{\mathbb{P}}$ cannot be obtained by considering pomeron exchange alone. As in the case of total hadronic cross sections, Donnachie and Landshoff [130], sub-leading exchanges become important away from the asymptotic limit ($x_{\mathbb{P}}\to0$ in this case). An additional sub-leading exchange (IR) is therefore usually considered, contributing at low $x_{\mathbb{P}}$ and $\beta$ and exhibiting a similar factorisation to the pomeron term, such that Eq. (43) is modified to

where $n_{\mathrm{IR}}$ sets the relative normalisation of the sub-leading term. In practice, the sub-leading exchange is not well constrained in the HERA kinematic regime.

With the above ansatz, a rather complete description is obtained of all HERA inclusive diffractive data. The energy dependence of DDIS is encoded in the pomeron flux factor, which is parametrised based on Regge phenomenology, as discussed further in § 7.7. The DPDFs are treated in a similar way to the case of inclusive DIS, as described in § 7.8. Full details of the standard fitting scheme adopted for the most recent data by both collaborations can be found in Aktas et al. [49]. An improved heavy flavour treatment is described in Chekanov et al. [108].

Energy Dependences and Soft Phenomenology

In the fits to DDIS data described in § 7.6.1, the $x_{\mathbb{P}}$ and $t$ dependences are parametrised using a flux factor motivated by Regge theory,

The parameters $B_{\mathbb{P}}$ and $\alpha_{\mathbb{P}}'$ and their uncertainties are obtained from fits to FPS or LPS data, Aktas et al. [46], which also take sub-leading contributions into account. The $x_{\mathbb{P}}$ dependence of the data principally determines the effective pomeron intercept $\alpha_{\mathbb{P}}(0)$, appropriate to the DDIS data. The most precise determinations are obtained from LRG data, with remarkable consistency between statistically independent data sets:

the model dependence uncertainties being highly correlated between the different data sets.

These $\alpha_{\mathbb{P}}(0)$ values are only slightly larger than their counterpart obtained from fits to total, Donnachie and Landshoff [130]; Cudell et al. [116], and diffractive, Goulianos [160]; Aad et al. [1], cross sections in $pp$ and $p\bar{p}$ scattering and are compatible with results from soft photoproduction at HERA, Adloff et al. [26]; Breitweg et al. [77]. They are significantly smaller than the results obtained from the energy dependences of hard exclusive processes at HERA (§ 7.2) and are much smaller than values predicted based on leading-logarithmic BFKL, Balitsky and Lipatov [61]; Kuraev et al. [179, 180], or other ‘hard pomeron’, Donnachie and Landshoff [132], approaches. This supports the picture of the dominating process in DDIS being the application of a hard virtual photon probe to an approximately factorisable exchange object, closely related to the ’soft pomeron’ of soft hadronic scattering.

The slope of the effective DDIS pomeron trajectory can be determined from the $x_{\mathbb{P}}$ dependence of the slope parameter $B$, as obtained using the LPS and FPS detectors. In this case, the H1 and ZEUS results are:

Whilst the precision on these results is limited, the trajectory slope is clearly smaller than the canonical value from soft $pp$ and $p\bar{p}$ scattering of $\alpha_{\mathbb{P}}'\sim0.25$, Donnachie and Landshoff [128]; Abe et al. [16]. As discussed in § 7.2, soft vector meson photoproduction data also show lower $\alpha_{\mathbb{P}}'$ values than those usually taken from $pp$ and $p\bar{p}$ scattering, Breitweg et al. [84], suggesting that this is a heavily process dependent parameter, for example due to different absorptive corrections.

The validity of the factorisation assumption implicit in the fits to inclusive diffractive data is tested by adding further free parameters, which allow the pomeron intercept and the slope parameter $B$ to vary freely between different $Q^2$ or $\beta$ bins. A summary of ZEUS results on possible variations of $\alpha_{\mathbb{P}}(0)$ with $Q^2$ is shown in Figure 51. Both here and in corresponding H1 studies (see e.g. Aaron et al. [12]), there is no evidence for any variation of the effective pomeron intercept with $Q^{2}$ or $\beta$ when considering data taken using either the LRG or LPS/FPS methods. Similar searches for a $Q^{2}$ or $\beta$ dependence of $\alpha_{\mathbb{P}}^{\prime}$ or $B$ have also yielded null results, Aaron et al. [10], albeit with lesser precision.

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It is informative to compare the photon–proton centre-of-mass energy ($W$ or equivalently $1/x$ at fixed $Q^{2}$) dependences of the diffractive and the inclusive cross sections. Basic Regge pole phenomenology predicts that the growth with centre-of-mass energy of the diffractive cross section ($\sim(W^{2})^{\alpha_{\mathbb{P}}(t)-2}$ at fixed $Q^{2}$ and $\beta$, with $\langle t\rangle\sim1/B \sim0.2\ \mathrm{GeV}^{2}$) should be faster than that of the total cross section ($\sim(W^{2})^{\alpha_{\mathbb{P}}(0)-1}$). However, in numerous studies of HERA data, the ratio of diffractive to inclusive cross sections has been found to be relatively flat as a function of $W$. The first example, Breitweg et al. [82], is shown in Figure 52. See also [49, 12]. This flatness represents a clear breakdown of the simple Regge approach, showing conclusively that there is no universal pomeron in virtual photon–proton scattering, as already discussed in § 7.2. The flatness of the ratio has been interpreted, [154, 156], as evidence for saturation of low $x$ parton densities, which is expected to be visible in diffraction at higher $x$ than in the inclusive case, due to the exchange of two, rather than one, gluon in the simplest interpretation. Saturation effects would tend to reduce the diffractive cross section relative to the inclusive cross section in a manner which becomes more important as $x$ falls or $W$ grows. Whilst the flatness of the ratio of these cross sections is thus a highly suggestive observation, it has also been explained without invoking saturation, for example in models which explain rapidity gap formation as a random process, due to soft colour rearrangements, [90, 134].

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QCD Phenomenology and Diffractive Parton Densities

Figure 50 immediately leads to the conclusion that the scaling violations in diffraction remain of positive sign ($\partial\sigma^{D}/\partial Q^{2} > 0$) up to large values of $\beta\sim0.5$, which may be compared with $x \sim0.1$ for the inclusive cross section.⁴ This is indicative of a large role of gluons in the diffractive exchange, Ahmed et al. [40]. This qualitative conclusion is formalised through the extraction of DPDFs, as described in more detail in this section.

The DPDFs in Eq. (42) represent probability distributions for partons $i$ in the proton under the constraint that the proton is scattered to a system $Y$ with a specified four-momentum. They are essentially equivalent to the fracture functions developed in an earlier approach, Trentadue and Veneziano [219]. They are not known from first principles, but can be determined from fits to the data using the DGLAP evolution equations, Blümlein and Robaschik [75]. Due to kinematic constraints, it is not possible to access the full range of $\beta$ and $Q^{2}$ using data from only one value of $x_{\mathbb{P}}$ at HERA. The parametrisations of the $x_{\mathbb{P}}$ dependence of the DPDFs described in § 7.7, together with the proton vertex factorisation assumption introduced in § 7.6.1, is therefore adopted, such that data from multiple $x_{\mathbb{P}}$ values can be used simultaneously in extracting DPDFs which then depend on $\beta$ and $Q^{2}$ only.

DPDFs have been extracted using standard NLO QCD procedures, similar to those employed to extract inclusive proton PDFs (§ 5), but with smaller numbers of free parameters for the momentum fraction dependence of the parton densities at the starting scale for evolution. This momentum fraction is usually denoted $z$ or $z_{\mathrm{IP}}$ (it is equal to $\beta$ when used to describe the quark coupling to the exchange boson). Only two distinct parton densities are generally considered: a gluon density and a singlet quark density, which is assumed to be flavour symmetric between up, down and strange quarks and their antiquarks. This latter assumption, particularly in regard to the strange quark density, is rather ad hoc and remains to be tested in detail.

Example results obtained when fitting inclusive $\sigma_r^D$ data alone can be found in Aktas et al. [49]; Chekanov et al. [108], see Figure 53. Since $\sigma_r^D$ measures essentially the charge-squared-weighted sum of quarks, these fits result in tight constraints (to around 5%) on the singlet quark density. From the scaling violations, the gluon distribution is also constrained (to around 10–15%) at moderate $z$ values. However, the sensitivity to the gluon density with $z \sim0.5$ is poor, due to the inevitable dominance of the evolution by the $q \to qg$ splitting as $z \to1$. An additional constraint can come from diffractive jet production (see § 7.9) such that the high $z$ gluon precision is improved considerably. However, this comes at the expense of additional theoretical assumptions on the large $\beta$ dynamics. Results of fits of this type can be found in Aktas et al. [52]; Chekanov et al. [108], see Figure 54.

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In all fits performed, the DPDFs are dominated by the gluon density, which extends to large values of $z$ and accounts for typically 60% (ZEUS Chekanov et al. [108]) to 70% (H1 Aktas et al. [49]) of the total longitudinal momentum of the diffractive exchange. It is interesting to note that the DPDFs, particularly the quark distribution, resemble the parton densities of the photon, Nisius [208]. The hadronic structure of the photon is generated from an initial $\gamma\to q\bar{q}$ splitting, with lower $x$ structure emerging from further splittings and evolution. This superficial similarity is as expected if diffraction emerges from a single gluon exchange, dressed in a manner which neutralises the colour. The structure then develops from $g \to q\bar{q}$ similarly to the photon case, but with an admixture of $g \to gg$ initial splittings.

More sophisticated approaches to DPDFs exist, notably those in which a direct, hard pomeron contribution is also included, Martin et al. [194], or those which include a higher twist contribution at large $\beta$ (see § 7.11), Royon et al. [215]; Golc-Biernart and Luszczak (2007). Whilst these contributions arise naturally in QCD and must be present at some level, the evidence both from these fits and from hadronic final state comparisons (§ 7.9) is that they are numerically small, compared with the standard resolved structure contribution. The DPDF approach does, however, appear to undergo an infrared breakdown at larger $Q^2$ scales than is the case in inclusive QCD fits, Aktas et al. [49]; Chekanov et al. [108]. Whilst this may provide further evidence for saturation effects in DDIS, Frankfurt et al. [147], it may also be a consequence of enhanced higher twist contributions, or of a less quickly convergent QCD order expansion in the diffractive than the inclusive case.

Applications of Diffractive Parton Densities

The DPDFs described in § 7.8 have been used to predict a wide variety of observables in DIS and photoproduction at HERA, as well as at the Tevatron and the LHC. At one level, this is done through implementations (usually of the H1 Fit B DPDFs, Aktas et al. [49]) in Monte Carlo generators, Jung [168]; Cox and Forshaw [115]; Navin [201]. In particular, the Rapgap model has been used extensively as an experimental tool for modelling both inclusive and hadronic final state diffractive data at HERA. The DPDFs have also been interfaced to various NLO QCD calculations in order to compare their predictions with measurements.

Early measurements of inclusive final state observables in DDIS were sufficient to rule out a diffractive exchange with a quark-dominated structure, by showing that the basic event topology is consistent with the boson–gluon fusion process, $\gamma^{*}g \to q\bar{q}$ (Figure 48(b)), yielding copious high $p_{T}$ particle production. The full list of inclusive final state observables measured in DDIS comprises charged particle spectra, multiplicities and their rapidity correlations, [28 b]; [95], energy flows, [28]; [95], and event shapes (thrust and sphericity), [79]; [30]; [95]. These measurements are universally in agreement with predictions based on DPDFs extracted from $\sigma_{r}^{D}$ data, reinforcing the picture of a diffractive exchange dominated by gluons extending to large momentum fractions.

The most precise tests of DPDFs in diffractive DIS have been obtained from exclusive final state observables formed from relatively high transverse momentum jet production, the rate for which is closely related to the diffractive gluon density. Dijet measurements have been made as a function of many different variables, [52 c]; [101]; [13], and have been extensively used in comparisons with DPDF-based NLO predictions, [52]; [108]. An example is shown in Figure 55. This sort of comparison has again been universally successful. The strong experimental evidence for proton vertex factorisation and the applicability of DPDFs to diffractive dijets in DIS, together with the relatively high precision with which the momentum fraction $z$ can be reconstructed from the kinematics of the diffractive final state and the jet pair, led to the jet data being used as an input to DPDF extractions, [52, 108], as discussed in § 7.8.

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Diffractive open charm production in DIS, [51, 98], provides a further exclusive final state with high sensitivity to the diffractive gluon density via the lowest order contributing process, $\gamma^{*}g \to c\bar{c}$. The charm quarks are usually tagged through the reconstruction of a $D^{*}$ meson using the $D^{*}-D^{0}$ mass difference technique. However, due to larger backgrounds and smaller statistics, charm observables have not yielded the precision tests of diffractive factorisation and DPDFs that have been achieved with jets. In principle, due to the weaker kinematic constraints on $M_{X}$, it should be possible to probe smaller $z$ values by using charm as a tag. However, the problems of forward-going track reconstruction experienced by both collaborations has limited progress in this direction. Nonetheless, as can be seen from the example comparisons in Figure 56, open charm production is well described by DPDF-based models, providing complementary support for this approach and a constraint on the heavy flavour treatment in diffractive QCD fits.

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A final class tests of the consistency of the DPDF model in DIS arises from the inclusive diffractive cross sections. An H1 measurement of the diffractive charged current cross section is shown to be consistent with DPDF-based predictions in [49]. The rather ad hoc flavour-democratic assumptions made in the standard QCD fitting procedures (§ 7.8) are consistent with the data within the large statistical uncertainties, but the diffractive charged current cross section is the only measurement made to date which is sensitive to the light quark flavour decomposition of the diffractive quark density. The diffractive longitudinal structure function, $F_{L}^{D}$, Newman [202],

is a further independent observable with sensitivity to both the quark and the gluon densities in novel ways. In particular, it provides a unique test of the low-$x$ gluon. Significantly non-zero results have been obtained, Aaron et al. [14], using the Rosenbluth technique, comparing diffractive reduced cross section data at fixed $x_{\mathbb{P}}$, $\beta$ and $Q^2$, but different $ep$ centre-of-mass energy, exploiting the data taken at $\sqrt{s}=460$ GeV and $\sqrt{s}=525$ GeV at the very end of the HERA running. The sensitivity to $F_L^{D}$ requires highly challenging measurements at large $y$ (small scattered electron energy). Nonetheless, this structure function has been measured over a fairly large kinematic region. As summarised in Figure 57, the data are again supportive of the DPDF approach.

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Limitations of Diffractive Parton Densities

Whilst models based on proton vertex factorisation and DPDFs work well to describe all diffractive processes beyond the lowest $M_X$ resonance region in DIS, they fail spectacularly when DPDFs extracted from HERA $\sigma_r^{D}$ data are applied to diffractive $p\bar{p}$ scattering at the Tevatron or LHC. For example, predictions for diffractive dijet production at the Tevatron exceed the data by a factor of around 10, Affolder et al. [38]; Klasen and Kramer [173]. This limitation is expected as part of the QCD hard scattering factorisation theorem for diffraction, Collins [113], and is usually interpreted in terms of multiple scattering, or ‘absorptive’ effects, which occur in the presence of beam remnants. These effects can be parametrised in terms of a ‘rapidity gap survival probability’, Dokshitzer et al. [127]; Bjorken [73]. Similar effects are emerging at the LHC, Chatrchyan et al. (2013).

Measurements of diffractive dijet photoproduction in DIS have been pursued as a control experiment for gap destruction models. In a lowest order interpretation, direct photon processes should be unaffected by such effects, whereas they are expected in resolved photon processes, where the photon interacts through its hadronic structure (Figure 48(c)). For example, a theoretical model which successfully describes the Tevatron result predicts a suppression of the resolved contribution by a factor of 0.34, Kaidalov et al. [169]. Multiple measurements of diffractive dijet photoproduction have been made by H1 and ZEUS, and have been compared with NLO calculations based on DPDFs extracted in DIS. Whilst the experimental data are just about compatible between the two collaborations, the conclusions from H1 and ZEUS differ. Notably, neither collaboration sees any evidence for $x_\gamma$ dependence of the survival probability. H1 data, Aktas et al. (2007c); Aaron et al. [7], suggest a suppression of the data by a factor of around 0.6 relative to NLO QCD predictions, independently of $x_\gamma$; ZEUS results, which correspond to larger jet transverse energies, Chekanov et al. [105, 108], suggest a smaller level of suppression and are, in fact, consistent with no suppression whatsoever. Selected data from both collaborations are shown in Figure 58. These apparent inconsistencies are at least partially resolved by a model in which a more careful treatment of point-like, as distinct from fully resolved, photon structure is introduced, Kaidalov et al. [170], leading to a much larger, and $E_T$ dependent, survival probability for $0.1 \lesssim x_\gamma< 1$.

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Diffractive DIS in Dipole Models

The application of the dipole picture to inclusive DDIS is challenging due to the need for higher multiplicity fluctuations of the virtual photon ($q\bar{q}$ and perhaps others) in order to describe the large $M_X$, small $\beta$ region. This need was first shown in a quantified manner through the “BEKW” fits to $F_2^{D}$ data in Bartels et al. [67]. In this parametrisation, the data at low and moderate $\beta$ are described in terms of $q\bar{q}$ and $q\bar{q}g$ dipole fluctuations of transversely polarised photons, whilst the high $\beta$ region contains a $Q^2$-suppressed higher twist contribution from $q\bar{q}$ fluctuations of longitudinally polarised photons. This approach was further developed for comparisons with inclusive DDIS data, for example in the saturation model, Golec-Biernat and Wüsthoff [156]; Golec-Biernat and Luszczak [153]. An example decomposition of the $\beta$ dependence of DDIS data is shown in Figure 59. Predictions have also been made for hadronic final state observables, based on a two-gluon exchange model of exclusive $q\bar{q}$, Bartels et al. [66]; Lotter [186], and $q\bar{q}g$, Bartels et al. [68, 71], production. However, progress has been limited by significant theoretical uncertainties associated with the low $\beta$ $q\bar{q}g$ contribution and a lack of direct evidence for the high $\beta$ higher twist $q\bar{q}$ term.

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Very Hard Diffraction and the BFKL Pomeron

For the diffractive processes discussed in the previous sections, $|t|$ is generally smaller than typical hadronic mass scales. However, diffractive processes have also been studied at HERA at $|t|$ values that are large enough to provide a hard scale in their own right. Under such circumstances, the $t$ channel pomeron exchange ought to be genuinely hard in the sense that it couples as a whole to individual partons. Cases where the hard sub-process satisfies $\hat{s} \gg-\hat{t} \gg\Lambda_{\mathrm{QCD}}$ correspond to a perturbatively calculable limit of Regge theory. There is no particular ordering in transverse momentum within the colour singlet exchange, such that the dynamics may be driven primarily by BFKL evolution. Diffractive processes at large $|t|$ therefore represent a promising area in which to search for low $x$ parton dynamics driven by the BFKL pomeron.

One striking signature for perturbative colour singlet exchange is the photoproduction of pairs of jets separated by a large rapidity gap. For such ‘gap between jets’ configurations, $|t| \approx E_{\mathrm{T,jet}}^2$ is very large, typically $|t| > 25\ \mathrm{GeV}^2$ in HERA studies. Calculations using the leading-logarithmic BFKL approximation, Mueller and Tang [200] (with appropriate modifications, Forshaw and Sutton [142]; Enberg et al. [135]), can be applied to the scattering between a parton from the proton and a parton from the resolved structure of the photon. However, the situation is complicated by the possibility of secondary scattering and low rapidity gap survival. The HERA data, Derrick et al. [123]; Adloff et al. [36]; Chekanov et al. [103], have shown clear evidence for events with little energy flow between jets, which occur more frequently than expected from hadronisation fluctuations in standard Monte Carlo models of jet photoproduction. The order of magnitude of the signal is also in agreement with that expected from the BFKL calculations. Example data from both ZEUS and H1 are shown in Figure 60 (left). The fraction of dijet events $f(\Delta\eta)$ ($E_{\mathrm{T,jet}} > 6\ \mathrm{GeV}$, $E_{\mathrm{T,jet}}^2 > 5\ \mathrm{GeV}$) for which the pseudorapidity region between the jets contains transverse energy of less than $0.5\ \mathrm{GeV}$ is shown as a function of the pseudorapidity separation between the jets $\Delta\eta$. The data are successfully described by a HERWIG Monte Carlo prediction that incorporates a BFKL pomeron contribution, whereas models without such a contribution are unable to describe the data. Whilst this is highly suggestive, and is in fact perhaps the best evidence for BFKL dynamics at HERA, more quantitative conclusions have been precluded by the uncertainties in the perturbative calculations, the gap survival probability and the residual non-diffractive contributions.

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Another candidate process in which to observe perturbative colour singlet exchange is quasi-elastic vector meson production at large $|t|$. Although $|t|$ values are smaller here than in the gaps-between-jets case and the proton essentially always dissociates, and the exclusive production mechanism at the photon vertex (Figure 42(b)) implies significantly reduced complications from gap destruction effects, even in photoproduction. The possibility of perturbative colour singlet exchange has long been considered for this process, Ginzburg et al. [152], and detailed calculations have been performed in the leading-logarithmic BFKL framework, Forshaw and Ryskin [141]; Bartels et al. [65]; Forshaw and Poludniowski [140]; Enberg et al. [137]; Poludniowski et al. [210].

BFKL evolution is expected to be relevant for $|t| > m_V^2$, where $m_V$ is the vector meson mass. For the case of heavy vector meson production, there is an interim region where $|t| < m_V^2$, where $|t|$ remains large enough to apply perturbative techniques. In this case, the transverse momenta along the gluon ladder remain ordered and DGLAP dynamics are expected to apply, [158, 74]. Studies have been made at HERA for light vector mesons, [47, 97], and $J/\psi$ mesons, [45, 109], as well as exclusively produced photons, [5]. The $J/\psi$ channel offers adequate statistics as well as a relatively clean theoretical interpretation, and thus it has yielded the most precise tests. An example analysis is shown in Figure 60 (right). As expected in a BFKL based prediction, the $t$ dependence generally follows the expected approximate power law behaviour at large $|t|$ and the effective trajectory slope $\alpha^\prime_{\mathrm{P}}$ is much smaller than that describing soft diffraction. However, the specific BFKL model predictions compared with the data leave room for improvement.

Summary

In summary, the low $x$ data from HERA have produced many surprising and interesting results that can be understood in terms of the perturbative dynamics of a gluon-rich system. Inclusive DIS at low $x$ reveals a progressively growing gluon distribution driven by the $g \to gg$ (and ultimately $g \to q\bar{q}$) splittings. The dominant feature of diffractive DIS data is scattering from a composite net-colour-singlet exchange with a structure dominated by gluons at high fractional momenta, while exclusive processes in the perturbative domain are produced by the exchange of gluon pairs.

To fully understand the inclusive data, it seems to be necessary to go beyond the conventional DGLAP $\ln(Q^2)$ resummation to the include $\ln(1/x)$ resummation of BFKL. There are similar hints in charm production data and forward-jet production, while in diffraction, describing data on rapidity gaps between jets and very high $|t|$ $J/\psi$ production also requires BFKL contributions.

Whether saturation effects have been demonstrated at HERA is controversial, but one needs the $\ln(1/x)$ resummations in order to preserve the high density gluon system that is a pre-requisite of saturation. Particularly significant are the attempts to describe high density gluon dynamics in terms of semi-classical gluon fields – the so-called ‘colour glass condensate’, and the dipole models which have been developed to confront these ideas with data.

See Also: other contributions to the encyclopedia