The hadronic final state at HERA

The hadronic final state in electron-proton collisions at HERA has provided a rich testing ground for development of the theory of the strong force, QCD. In this review, over 200 publications from the H1 and ZEUS Collaborations are summarized. Short distance physics, the measurement of processes at high-energy scales, has provided rigorous tests of perturbative QCD and constrained the structure of the proton as well as allowing precise determinations of the strong coupling constant to be made. Nonperturbative or low-energy processes have also been investigated and results on hadronization interpreted together with those from other experiments. Searches for exotic QCD objects, such as pentaquarks, glueballs, and instantons, have been performed. The subject of diffraction has been reinvigorated through its precise measurement, such that it can now be described by perturbative QCD. After discussion of HERA, the H1 and ZEUS detectors, and the techniques used to reconstruct differing hadronic final states, the above subject areas are elaborated on. The major achievements are then condensed further in a final section summarizing what has been learned.

Figure 1

Schematic illustration of a generic electron-proton DIS process, in which an exchanged photon of virtuality $Q^2$ couples to a quark carrying a fraction $x$ of the proton’s longitudinal momentum. The electron-proton center-of-mass energy is denoted as $\sqrt{s}$ and the photon-proton center-of-mass energy (equivalently the invariant mass of the hadronic final state) is denoted as $W$.

Figure 2

Event displays of the (top) ZEUS and (bottom) H1 detectors showing a neutral current and a charged current DIS event, respectively.

Figure 3

The distribution of the mass difference $m(D^{*})-m(D^0)$ for $D^{*}$ candidates and a background estimate. From [368].

Figure 4

Diagrams of a track in the $X\text{−}Y$ plane. The direction of the struck quark, experimentally approximated as a jet axis, is used to define the impact parameter $\delta$.

Figure 5

Distribution of the significance at which the transverse distance between the primary and secondary vertices is nonzero. Included is a decomposition of the data by fitting Monte Carlo templates for light, $c$, and $b$ quarks. From [30].

Figure 6

ZEUS jet energy scale uncertainty as a function of (a) ${\eta }^{\text{jet}}$ and (b) $E_T^{\text{jet}}$, showing that the scale is known to 1%. In (a), the relative difference between the hadronic jet transverse energy $E_T^{\text{jet}}$ and the transverse momentum of the scattered electron, calculated using the double-angle method $p_T^{\mathrm{DA}}$ is shown as a function of the jet pseudorapidity ${\eta }^{\text{jet}}$. In (b), the relative difference between the data and Monte Carlo simulation is shown for the quantity $⟨r_{e\text{jet}}⟩$ as a function of the jet transverse energy, where $r_{e\text{jet}}$ is the ratio of the jet to electron transverse energies.

Figure 7

Schematic illustration of the beam line instrumentation for the example of the H1 experiment. The detector components shown are all described in the text with the exception of the 103 m photon tagger which was used for luminosity determination via the Bethe-Heitler $ep\to ep\gamma$ process and the proton dissociation taggers which were used to identify forward rapidity gaps (see Sec. 5).

Figure 8

(a) Boson-gluon-fusion and (b) $u$-channel QCD Compton processes. Along with $s$-channel QCD Compton scattering, these are the LO QCD processes in DIS and direct photoproduction, i.e., the lowest order process involving at least one power (or vertex) of ${\alpha }_s$.

Figure 9

An example of a LO resolved jet photoproduction process, containing a hard scattering between a quark from the photon and a gluon from the proton.

Figure 10

(a) Measurement of $d\sigma /d{E}_{T,B}^{\text{jet}}$ for inclusive-jet production in the Breit frame in DIS for different jet algorithms. The lower part shows the hadronization corrections applied to the NLO calculations. (b) Measurement of $d^2\sigma /d{Q}^{2}d{P}_T$ in different regions of $Q^2$ for inclusive-jet production in the Breit frame in DIS using the $k_T$ jet algorithm. Note that $P_T$ is equivalent to $E_{T,B}^{\text{jet}}$. The data in (a) and (b) are compared with NLO QCD predictions (corrected for hadronization and $Z^0$ effects). (a) From [68] and (b) from [26].

Figure 11

Raw distribution in $x_{\gamma }^{\text{meas}}$ for photoproduction events with two or more jets. The direct- and resolved-photon Monte Carlo predictions are fitted to the data with free normalization. From [400].

Figure 12

Measured cross section $d\sigma /d{\overline{E}}_T$ for (a) $x_{\gamma }^{\text{obs}}>0.75$ and (b) $x_{\gamma }^{\text{obs}}<0.75$, compared with NLO QCD predictions (corrected for hadronization) using photon PDFs, AFG04 (solid line) and CJK (dashed line). From [345].

Figure 13

Measured jet shapes in $e^{+}{e}^{-}$ scattering ([143]), from ZEUS in DIS ([265]) and photoproduction ([262]), and in $p\overline{p}$ collisions ([52]; [44]). From [287].

Figure 14

Measurement of the dijet cross section differential in the cosine of the dijet scattering angle $d\sigma /d|\cos {\theta }^{*}|$ for events with two broad jets or two narrow jets. The data are compared with Pythia Monte Carlo predictions. From [331].

Figure 15

Differential cross section for prompt photon production in DIS with the data compared with various theoretical predictions. An $\mathcal{O}({\alpha }^3)$ calculation with radiation from the quark line (lower solid line) and additionally including radiation from the lepton line and interference between the two (shaded band around upper solid line) is shown. The photon is also treated by Martin et al. as a partonic constituent of the proton (hatched area surrounding dashed line) and including photon radiation, i.e., adding the prediction corresponding to the thick line, from the quark line (dot-dashed line). See text for further details of the models. From [375].

Figure 16

Differential cross sections for prompt photons in photoproduction. The data are compared with two QCD calculations, one based on collinear factorization in NLO (hatched histogram) and the other based on $k_T$ factorization. The lower parts show the ratio of the NLO QCD prediction to the data. From [31].

Figure 17

Measurements of cross sections $d\sigma /d{p}_T^{D^{*}}$ and $d\sigma /d{\eta }^{D^{*}}$ for $D^{*}$ production in DIS compared with NLO QCD predictions using two different proton PDFs. The lower plot for each variable shows the ratio of theory to data where each is first normalized to its corresponding total cross section. From [34].

Figure 18

Beauty (left) and charm (right) jet photoproduction cross sections $d\sigma /d{p}_T^{\text{jet}}$ (upper) and $d\sigma /d{\eta }^{\text{jet}}$ (lower) compared to Pythia Monte Carlo and NLO QCD predictions. From [73].

Figure 19

Summary of beauty photoproduction measurements at HERA as a function of the transverse momentum of the $b$ quark compared with predictions from NLO QCD. The measurements, made using different final states and in different kinematic regions, were corrected for branching ratios and extrapolated to the specified kinematic region at the quark level for this comparison.

Figure 20

Measurement of the reduced charm cross section at fixed $Q^2$ as a function of $x$, having combined all available published H1 and ZEUS data. The data are compared with a parametrization of the proton PDF from the ABM ([178]) group at NLO and NNLO in QCD. From [77].

Figure 21

(a) Measurement of the ratio of the dijet to the total cross section in DIS, compared with NLO QCD predictions (corrected for hadronization). (b) Values of ${\alpha }_s$ vs the scale of the process extracted from distributions such as that in (a). The solid line shows the two-loop solution of the renormalization group equation. (a) From [28], and (b) from [29].

Figure 22

Values of ${\alpha }_s$ from individual H1 and ZEUS measurements and the world average. Note the extraction in the Herapdf1.6 fit is a preliminary result.

Figure 23

Measurement of the difference in azimuth between the two highest transverse energy jets for (a) $x_{\gamma }^{\text{obs}}>0.75$ and (b) $x_{\gamma }^{\text{obs}}\le 0.75$ compared with NLO QCD [$\mathcal{O}(\alpha {\alpha }_s^2)$, corrected for hadronization] and Monte Carlo predictions. From [345].

Figure 24

Schematic illustration of $ep$ scattering with a forward jet taking a fraction $x_{\text{jet}}$ of the proton momentum. The evolution in the longitudinal momentum fraction $x$ from large $x_{\text{jet}}$ to small $x_{\mathrm{Bj}}$ is indicated. From [159].

Figure 25

Measurement of forward jet production at low $x_{\mathrm{Bj}}$ in DIS compared with (a) NLO DGLAP QCD (corrected for hadronization) and (b), (c) Monte Carlo models. From [159].

Figure 26

Measurement of the cross section as a function of the Bjorken scaling variable $x$ for events in which two jets are reconstructed in the forward direction and one central. The data are compared with second- and third-order QCD calculations. From [15].

Figure 27

Measurement of the dijet cross section in the ${\gamma }^{*}p$ frame in DIS as a function of $x_{\gamma }^{\text{jets}}$ in different regions of $Q^2$ and $E_T^{*}$. The data ([150]) are compared with three NLO QCD calculations in DIS for $2\to 2$ processes, Disent, Jetvip dir, and Nlojet for 2 jets, with an NLO QCD calculation for $2\to 3$ processes, Nlojet for 3 jets, and with an NLO QCD calculation in DIS for $2\to 2$ processes, including a component of resolved virtual photon structure, Jetvip full. All of the theory predictions are corrected for hadronization effects. From [383].

Figure 28

Measurements of inclusive jet photoproduction at the smallest jet transverse energies reached at HERA (extending to $E_T^{\text{jet}}=5\mathrm{GeV}$). The data are compared with NLO QCD calculations before and after correcting for underlying event and hadronization effects, the combination of which is labeled ${\delta }_{\text{hadr}}$. From [124].

Figure 29

A schematic representation of an event with multiparton interactions.

Figure 30

Measurement of the four-jet photoproduction cross section as a function of $x_{\gamma }^{\text{obs}}$ for low invariant masses of the four-jet system $25<M_{4\mathrm{j}}<50\mathrm{GeV}$. The predictions from the Herwig and Pythia Monte Carlo models are shown both with ([286]; [605]) and without multiparton interactions, as is the direct-photon component of the Herwig prediction. Each prediction is normalized to the measured high-mass ($M_{4\mathrm{j}}>50\mathrm{GeV}$) cross section by the factors given in the legend. From [359].

Figure 31

Distributions of mean event shapes: the thrust variable $\tau$, which measures the longitudinal momentum components projected onto the boson axis; the thrust variable ${\tau }_C$, which maximizes the sum of the longitudinal momenta in the current hemisphere; the broadening variable $B$, which measures the scalar sum of transverse momenta with respect to the boson axis; the squared jet mass ${\rho }_0$, normalized to 4 times the squared scalar momentum sum in the current hemisphere; and the $C$ parameter. The data, as a function of the scale $Q$, are compared with the result of a fit based on NLO QCD with power corrections (solid lines) as well as the contribution from NLO QCD alone (dashed lines). From [162].

Figure 32

Extracted values from (a) H1 and (b) ZEUS for ${\alpha }_0$ and ${\alpha }_s$ for fits of the $\mathrm{NLO}+\mathrm{NLL}+\mathrm{PC}$ predictions to all event-shape variables. The solid lines represent the $1\sigma$ contours and for (b) the dashed line represents the 95% C.L. contour. The world-average value of ${\alpha }_s$ and its uncertainty are shown as the bands, taken from [235]. (a) From [162]. (b) From [343].

Figure 33

Mean charged multiplicity $⟨n_{\text{ch}}⟩$ as a function of the energy scale for ZEUS DIS measurements in the Breit and hadronic center-of-mass frames, compared with results from $e^{+}{e}^{-}$ and fixed-target DIS experiments. From [357].

Figure 34

The number of charged particles per event per unit $x_p$ as a function of $Q$ in bins of $x_p$. Data are shown for H1 and ZEUS and are compared with those from $e^{+}{e}^{-}$ experiments ([576]; [253]; [537]; [82]). From [71].

Figure 35

The number of charged particles per event per unit $x_p$ as a function of $Q$ in bins of $x_p$. The shaded bands represent NLO QCD calculations by Kretzer ([531]) with their renormalization uncertainties. Additional calculations are shown : Kniehl, Kramer, and Pötter (KKP) ([524]); Albino, Kniehl, and Kramer (AKK) ([173], [174]); and de Florian, Sassot, and Stratmann (DSS) ([394], [395]). From [71].

Figure 36

Measurements of fragmentation fractions for various charm hadrons in DIS and photoproduction at HERA and also in $e^{+}{e}^{-}$ annihilation at LEP. From [78].

Figure 37

$D^{*}$ fragmentation function $f(z)$ for ZEUS data compared to measurements in $e^{+}{e}^{-}$ collisions. To compare the shapes, the data sets were normalized to $1/(\text{bin}\text{width})$ for $z>0.3$. From [366].

Figure 38

Normalized $D^{*}$ cross sections as a function of an estimator $z_{\text{hem}}$ of the fragmentation function $z$, showing the best fits of an NLO QCD prediction using the Kartvelishvili fragmentation function with the uncertainty range on the $\alpha$ parameter shown. The scales of the processes are (a) $8.4<\sqrt{\widehat{s}}<18\mathrm{GeV}$ and (b) $\sqrt{\widehat{s}}<8.4\mathrm{GeV}$. From [21].

Figure 39

Photoproduction cross section, differential in $P_T^2$ of the $J/\psi$ meson, for H1 data, compared with LO and NLO QCD predictions from the CS model and the CS and CO models combined. The uncertainty on the theory (shaded band) arises from the difference between results using an LO or a higher-order color-octet contribution ([280]). From [25].

Figure 40

Helicty parameter $\nu$ for the decay azimuthal angle in the frame where the polarization axis in the $J/\psi$ meson rest frame is defined by the flight direction of the $J/\psi$ meson in the $\gamma p$ rest frame. The value can vary between $-1$ and $+1$ and is shown here for different inelasticity values. From [365].

Figure 41

Raw distributions in sphericity, $Q^2$, charged-particle multiplicity, total transverse energy, isotropy, and transverse jet energy, compared with predictions for DIS events and for instanton-induced events (QCDINS), after applying a cut on a multivariate discriminant designed to enhance instantonlike signatures. From [123].

Figure 42

(a) The measured $K_S^0{K}_S^0$ invariant-mass spectrum compared with a fit including signal and background contributions. (b) The background-subtracted $K_S^0{K}_S^0$ invariant-mass spectrum with the fit for the signals also shown. From [358].

Figure 43

(a) $d/p$ and $\overline{d}/\overline{p}$ production rates in DIS (ZEUS) and in photoproduction (H1) as a function of $p_T/M$. (b) $\overline{d}/d$ and $\overline{p}/p$ production rates in DIS as a function of $p_T/M$. From [349].

Figure 44

Schematic illustration of the dipole model concept in which a virtual photon fluctuates to a $q\overline{q}$ pair with relative momentum fractions $z$ and $1-z$, forming a color dipole of transverse size $r\sim 1/\sqrt{Q^2}$. The dipole then scatters elastically from the proton according to a dipole cross section. The example shown corresponds to the amplitude for the total inclusive ${\gamma }^{*}p$ cross section. However, only small modifications are required to adapt this to the cases of exclusive or inclusive diffraction.

Figure 45

(a) Generic representation of exclusive vector meson production via the quasielastic scattering of a real or virtual photon. The commonly used kinematic variables discussed in the text are indicated. (b) Lowest-order perturbative diagram for hard vector meson production, involving the exchange of a pair of gluons between the proton and the real or virtual photon.

Figure 46

Compilation of photoproduction cross-section measurements as a function of the $\gamma p$ center-of-mass energy $W$. The total cross section and various vector meson production cross sections are included, with the approximate power law dependences $\sigma \propto W^{\delta }$ indicated for each process.

Figure 47

Development of the effective Pomeron intercept [see Eq. (9) and surrounding text] with a characteristic scale ${\mu }^2=(Q^2+M_V^2)/4$, derived from fits to the $W$ dependences of various vector meson production data, as well as a DVCS measurement (shown at ${\mu }^2=Q^2$). The dashed lines indicate the range of values which are typical in soft diffraction. From [24].

Figure 48

Exponential $t$ slopes for vector meson electroproduction and photoproduction as a function of the characteristic scale $Q^2+M_V^2$. From [76].

Figure 49

$J/\psi$ photoproduction cross section as a function of $W$, compared with a QCD calculation, MRT ([554]), based on four different parametrizations of the gluon density of the proton and a dipole model, FMS ([452]). The predictions are normalized to the data. From [158].

Figure 50

Dependence of HERA DVCS cross-section measurements on $Q^2$. The data are compared with a GPD-based model ([533]) and also with a dipole-based model ([549]). See Sec. 5b3 for an explanation of the latter. From [18].

Figure 51

Exclusive $J/\psi$ photoproduction data compared with versions of the b-Sat dipole model ([527]) both with (eikonalized) and without (1-Pomeron) saturation effects included. An extrapolation into the kinematic regime of a possible future $ep$ collider is also included. From [64].

Figure 52

Sketches of diffractive $ep$ processes. (a) Inclusive DDIS at the level of the quark-parton model, illustrating the kinematic variables discussed in the text. (b) Dominant leading-order QCD diagram for hard scattering in DDIS or direct photoproduction, in which a parton of momentum fraction $z_{\mathrm{IP}}$ from the DPDFs enters the hard scattering. (c) A leading-order process in resolved photoproduction involving a parton of momentum fraction $x_{\gamma }$ relative to the photon.

Figure 53

Example data on the dynamics of the slope parameter $B$ describing the $t$ dependence of DDIS, as measured using the H1 FPS. From [35].

Figure 54

Comparisons between H1 and ZEUS DDIS data measured using the LRG method at $x_{\mathrm{IP}}=0.001$, $x_{\mathrm{IP}}=0.003$, and $x_{\mathrm{IP}}=0.01$. From [36].

Figure 55

Summary of ZEUS data on tests of proton vertex factorization in the form of possible $Q^2$ variations of the Pomeron intercept in fits to LRG and LPS data, as well as FPC (forward plug calorimeter) data obtained by the $M_X$ decomposition method. From [361].

Figure 56

The ratio of diffractive to inclusive cross sections as a function of $W$, integrated over different $M_X$ regions at fixed $Q^2$. Obtained using ZEUS 1994 data. From [267].

Figure 57

DPDFs extracted from an H1 fit to inclusive LRG data only. The two different sets of DPDFs shown (Fit A and Fit B) have very similar fit qualities, illustrating the lack of sensitivity to the gluon density at large momentum fractions. From [160].

Figure 58

Diffractive quark (left) and gluon (right) distributions extracted from a ZEUS fit to inclusive LRG and LPS data, with diffractive dijet data also included. From [371].

Figure 59

Comparison of diffractive DIS dijet cross sections, measured double differentially in $z_{\mathrm{IP}}$ and $Q^2$, with NLO QCD predictions based on DPDFs. In this case, the data shown were used together with inclusive diffraction measurements in the fit, to improve the high-$\beta$ sensitivity. From [371].

Figure 60

Comparisons between diffractive open charm ($D^{*}$) production measurements and NLO QCD predictions based on DPDFs. From [165].

Figure 61

Summary of H1 measurements of the diffractive longitudinal structure function $F_L^D$, exploiting data taken with variations in the proton beam energy. To allow comparisons between measurements at different $x_{\mathrm{IP}}$ values, the data are normalized by a diffractive flux factor $f_{\mathrm{IP}/p}$ (see Sec. 5c4). The data are compared with an NLO QCD prediction based on DPDFs. From [41].

Figure 62

Diffractive dijet photoproduction data shown as ratios of measured cross sections to DPDF-based NLO predictions (i.e., as measurements of the rapidity gap survival probability). (a)–(c) H1 data. From [23]. Bottom panel: ZEUS data. From [371].

Figure 63

An example decomposition of the $\beta$ dependence of ZEUS DDIS data ([267]) into different dipole terms, according to the parametrization in [476]. The dashed, dotted and dot-dashed curves correspond to leading twist transverse $q\overline{q}$, leading twist transverse $q\overline{q}g$, and higher twist longitudinal $q\overline{q}$ contributions, respectively. The solid curves represent the sum of the three.

Figure 64

Leading neutron structure function data (integrated over neutron transverse momenta up to 200 MeV), divided by a parametrization ([501]) of the pion flux ${\mathrm{\Gamma }}_{\pi }$ at $x_L\equiv 1-x_{\mathrm{IP}}=0.73$. The estimated contribution from standard fragmentation processes is shown as a shaded band. After accounting for this, the data are compared with two parametrizations of the pion structure function ([207]; [470]) and with a parametrization of the proton structure function ([17]) multiplied by $2/3$. From [27].

Figure 65

Dependence on $|t|$ of proton dissociative $J/\psi$ production integrated over proton dissociation masses satisfying $M_Y^2/W^2<0.05$. The data are compared with a prediction [GLMN LL ([481])] which is based on DGLAP evolution and is expected to be valid for $|t|<m_{\psi }^2$. They are also compared with a BFKL-based prediction [EMP LL ([433])] and a more phenomenological approach [FSZ ([453])]. From [376].