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HERA at low-x

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A M Cooper-Sarkar, P R Newman

High Energy Physics - Phenomenology · Submitted 19 Oct 2026 · 27 references

Pith's verdict: An accurate, well-hedged encyclopedia reprint of HERA low-x physics whose Section 6 summary claims a bit more than the body's own evidence supports — worth refereeing after cleanup.

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arXiv ID2610.26393 (version 1)
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CitationA M Cooper-Sarkar, P R Newman. "HERA at low-x." arXiv:2610.26393 (2026). https://pith.science/paper/SYB2HVUI
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The paper

The paper claims

This review argues that HERA's low-$x$ data are only fully described once BFKL-type $\ln(1/x)$ resummation is added to standard QCD evolution, and that this dense-gluon state is the prerequisite, but not the proof, of saturation.

The reading

This review argues that HERA's low-$x$ legacy cannot be understood with the standard DGLAP evolution of quantum chromodynamics alone. The combined inclusive data are progressively worse described by NNLO DGLAP fits as data below $Q^2\approx 10$ GeV$^2$ are included, and the visible shortfall is the turn-over of the reduced cross section at small $x$, where the data want a larger longitudinal structure function $F_L$ than DGLAP predicts. Reading this as missing $\ln(1/x)$ resummation of the BFKL type, the authors show that adding it improves the fit by roughly 70 units of $\chi^2$ and restores a gluon density that rises at low $x$ instead of falling, reopening the dense-gluon picture. The review is equally explicit that saturation is not demonstrated at HERA, since non-saturating dipole fits describe the data as well as saturating ones and the extracted saturation scale is low. If the argument holds, HERA's measured legacy is a preference for BFKL-type dynamics that keeps a high-density gluon system, and hence saturation, as live physics for the next generation of machines.

What carries the argument

The argument is carried by two evolution equations and one observable. DGLAP evolution sums powers of $\alpha_s\ln(Q^2)$ under strong ordering of gluon transverse momenta; BFKL evolution sums powers of $\alpha_s\ln(1/x)$ without that ordering, and the data are read as preferring the latter as a supplement to the former. The discriminating observable is the reduced cross section $\sigma_r = F_2 - (y^2/Y_+)F_L$: because HERA reaches low $x$ only at low $Q^2$ and hence high $y$, the $F_L$ term, which is driven by the gluon through $\gamma^* g\to q\bar q$ boson-gluon fusion, produces a turn-over at small $x$ whose predicted size separates the schemes. The quantitative instruments are the $\chi^2$-versus-$Q^2_{\rm min}$ scans of HERAPDF-style fits and their resummed (NNLO+NLLx) and higher-twist (HHT) variants. In diffraction the machinery is the effective pomeron intercept extracted from the $x_{I\!P}$ dependence and the flat ratio of diffractive to inclusive cross sections, whose behaviour, along with geometric scaling in $\tau = Q^2 R_0^2(x)$ of $\sigma(\gamma^*p)$, is the paper's sharpest saturation hint.

Extended reading notes

Core claim

The paper's central case is built from the fit-quality scans of the combined H1 and ZEUS inclusive data: as the $Q^2_{\rm min}$ of the fitted data falls below about 10 GeV$^2$, the NNLO DGLAP $\chi^2$/d.o.f. rises, and the symptom is the low-$x$ turn-over of $\sigma_r = F_2 - (y^2/Y_+)F_L$, which wants a larger $F_L$, and hence a larger low-$x$ gluon, than DGLAP generates. The authors take this as evidence that the $\ln(Q^2)$ resummation of DGLAP must be supplemented by the $\ln(1/x)$ resummation of BFKL: NNLO+NLLx fits improve $\chi^2$ by about 70, describe the turn-over, and produce a gluon that is unsuppressed at low $x$ and low $Q^2$, in contrast to the valence-like falling gluon of the pure DGLAP fit. A higher-twist term in $F_L$ alone repairs the fit equally well, a degeneracy the paper leaves open. Diffractive data supply the 'more detailed hints' of saturation: a soft pomeron intercept $\alpha_{I\!P}(0)\approx 1.11$ in inclusive diffraction, a flat diffractive-to-inclusive ratio as energy grows, geometric scaling of $\sigma(\gamma^*p)$, and BFKL contributions required for gaps-between-jets and high-$|t|$ $J/\psi$ production. The stated bottom line is that saturation at HERA is controversial, but the $\ln(1/x)$ resummations are needed to preserve the high-density gluon system that saturation would require.

Load-bearing premise

The load-bearing premise is attribution: the degraded $\chi^2$ of NNLO DGLAP fits when data below $Q^2\approx 10$ GeV$^2$ are admitted is read as physics missing from the evolution equations (BFKL-type $\ln(1/x)$ resummation or higher twist) rather than as a sign that perturbative QCD itself has broken down at those scales, a worry the authors themselves raise when they concede that $\alpha_s(Q_0^2)$ may be too high for perturbation theory at such low $Q^2$.

Editorial extensions

If this is right

  • Global PDF sets built from HERA data without $\ln(1/x)$ resummation mis-describe the low-$x$, low-$Q^2$ corner, so fits used to predict hadron-collider observables must treat the low-$Q^2$ input boundary as part of the physics rather than a nuisance.
  • With BFKL-type resummation the low-$x$ gluon rises at low $Q^2$ instead of turning valence-like, restoring $g\to q\bar q$ as the driver of the sea and reviving the dense-gluon system that saturation would require.
  • Because either $\ln(1/x)$ resummation or a higher-twist term in $F_L$ can repair the inclusive fit, the inclusive data alone cannot decide between BFKL dynamics and gluon recombination; discriminating power must come from observables with different gluon dependence, such as charm or $F_L$.
  • The flat diffractive-to-inclusive ratio as $W$ grows makes diffraction the place where saturation, if real, should first become unambiguous, since diffractive scattering involves at least two exchanged gluons.
  • If the paper is right, the next lepton-hadron collider inherits a concrete objective: measure $F_L$ and forward-jet rates at low $x$ with $Q^2$ high enough to separate the two evolution schemes cleanly.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: if the BFKL preference is genuine, the same missing $\ln(1/x)$ physics should surface as a systematic low-$x$ deficit in other hard processes built on these PDFs, most testably in forward-jet and heavy-flavour rates at hadron colliders across a wide rapidity range.
  • The paper leaves the BFKL-versus-higher-twist degeneracy open; a way to break it is precision data on $F_2^{c\bar c}$ at low $x$, since the two repairs yield different gluon densities and charm is produced directly from gluons via boson-gluon fusion.
  • Because the extracted saturation scale is low ($Q_s^2 \lesssim 2$ GeV$^2$), this review's own evidence suggests saturation should be hunted where the effective scale is larger, in nuclei or in exclusive diffraction, rather than in inclusive $F_2$; the authors point toward this but do not state it as a program.
  • The equally good fit of the non-saturating IPnonsat model, reported in the paper, implies that geometric scaling and dipole-model success certify the shape of the dipole cross-section more than its non-linearity; an editorial reading is that only diffractive and exclusive data, which count exchanged gluons differently, can separate the two.
Original abstract

The field of Deep Inelastic Scattering is introduced with particular reference to the High Energy Ring Accelerator HERA. The formalism of the Quark Parton Model and its extension in perturbative Quantum Chromodynamics is outlined, defining the concept of a Parton Distribution Function. The HERA data on inclusive and semi-inclusive processes are reviewed with particular emphasis on results at low x and low Q 2 . Their theoretical imterpretation in terms of conventional QCD DGLAP evolution and its extension to BFKL evolution in the low-x regime is explored together with the possibility of the onset of a new non-linear regime which may lead to gluon saturation. The focus then moves to diffractive processes both quasi-elastic and diffractive dissociation, where more detailed hints of saturation are found.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-20: 1 request, 4,313 input tokens.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.06 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.17 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

The field of Deep Inelastic Scattering is introduced with particular reference to the High Energy Ring Accelerator HERA. The formalism of the Quark Parton Model and its extension in perturbative Quantum Chromodynamics is outlined, defining the concept of a Parton Distribution Function. The HERA data on inclusive and semi-inclusive processes are reviewed with particular emphasis on results at low x and low Q 2 . Their theoretical imterpretation in terms of conventional QCD DGLAP evolution and its extension to BFKL evolution in the low-x regime is explored together with the possibility of the onset of a new non-linear regime which may lead to gluon saturation. The focus then moves to diffractive processes both quasi-elastic and diffractive dissociation, where more detailed hints of saturation are found.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The field of Deep Inelastic Scattering is introduced with particular reference to the High Energy Ring Accelerator HERA. background 1.00 0.07 0.02 asserted 0.69
s1 The formalism of the Quark Parton Model and its extension in perturbative Quantum Chromodynamics is outlined, defining the concept of a Parton Distribution Function. background 1.00 0.09 0.02 asserted 0.85
s2 The HERA data on inclusive and semi-inclusive processes are reviewed with particular emphasis on results at low x and low Q 2 . contribution 0.71 0.12 0.05 asserted 0.76
s3 Their theoretical imterpretation in terms of conventional QCD DGLAP evolution and its extension to BFKL evolution in the low-x regime is explored together with the possibility of the onset of a new non-linear regime which may lead to gluon saturation. interpretation 0.96 0.34 0.29 hedged possibility 0.97
s4 The focus then moves to diffractive processes both quasi-elastic and diffractive dissociation, where more detailed hints of saturation are found. result 0.52 0.37 0.29 asserted 0.66

What kind of paper the classifier reports this is: review 1.00 · evidence from analysis of existing data 0.93.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 2. Regions in the (x, Q 2 ) plane covered by DIS experiments. Both HERA ad earlier fixed target DIS experiants are shown. From De￾venish and Cooper-Sarkar (2004). 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 (γ ∗ or Z ∗ ) is d 2σ(l ±N) dxdQ2 = 2πα2 Q4 x h Y+ F lN 2 (x, Q 2 ) − y 2 F lN L (x, Q 2 ) ∓ Y− xFlN 3 (x, Q 2 ) i , (1) where Y± = 1 ± (1 − y) 2 , F2, xF3 and FL are the structure functions of the hadron and the notation lN indicates lepton scattering…
Figure 4. This can be framed as a redefiniton of the parton density such that it includes the parton splitings and thus gains a scale dependence
Figure 5. Diagrams for gluon radiation from quarks in DIS via QCDC, together with the parton CM frame. From Devenish and Cooper￾Sarkar (2004)
Figure 9. The combined HERA data for the inclusive NC e + p and e − p reduced cross sections together with fixed-target data, Benvenuti et al. (1989); Arneodo et al. (1997), and the predictions of HERAPDF2.0 NNLO. The bands represent the total uncertainties on the predictions. Dashed lines indicate extrapolation into kinematic regions not included in the fit. From Abramowicz et al. (2015). 3 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…
Figure 10. F2 data showing the strong rise at small x. Left: the HERA data are from the 1994 runs and the data for Q 2 = 1.5 GeV2 and x > 10−3 (triangles) are from fixed target experiments. Right: The structure function F˜ 2, as extracted from the measured reduced cross sections of the final combined HERA data, for four values of Q 2 together with the predictions of HERAPDF2.0 NLO. The bands represent the total uncertainty on the predictions. From Abramowicz et al. (2015). final HERA data and for early data, and the quantity plotted is F2 rather than σr . The figures show that F2 rises…
Figure 11. The combined high-Q 2 HERA inclusive NC e + p reduced cross sections at √ s = 318 GeV with overlaid predictions from HERA￾PDF2.0 NNLO. The bands represent the total uncertainties on the predictions. From Abramowicz et al. (2015)
Figure 12. The combined low-Q 2 HERA inclusive NC e + p reduced cross sections at √ s = 318 GeV with overlaid predictions from HERA￾PDF2.0 NNLO. The bands represent the total uncertainties on the predictions. Dotted lines indicate extrapolation into kinematic regions not included in the fit. From Abramowicz et al. (2015)
Figure 13. The dependence of χ 2 /d.o.f. on Q 2 min of the LO, NLO and NNLO fits to the HERA combined inclusive data. From Abramowicz et al. (2015). Also shown are values for an NLO fit to the combined HERA I data Aaron et al. (2010c)
Figure 14. The reduced cross section σ˜ NC(x, Q 2 ) + 0.3i measured at three proton beam energies Ep = 460 GeV (diamonds, i=0), 575 GeV (squares, i=1) and 920 GeV (circles, i=2). The inner error bars represent the statistical errors, the full error bars include the statistical and systematic uncertainties added in quadrature, excluding the normalisation uncertainty. The curves represent the prediction from an NLO QCD fit. From Andreev et al. (2014)
Figure 15. The reduced cross section σ˜ NC(x, Q 2 ) as a function of y 2 /(1 + (1 − y) 2 ) for six values of x at Q 2 = 60 GeV2 , measured for proton beam energies of Ep = 920, 575 and 460 GeV. The inner error bars denote the statistical error, the outer error bars show statistical and systematic uncertainties added in quadrature. The luminosity uncertainty is not included in the error bars. From Andreev et al. (2014)
Figure 16. The proton structure functions FL(x, Q 2 ) (solid symbols) and F2(x, Q 2 ) (open symbols) measured by H1 (circles) and ZEUS (diamonds) in the region: left 2 ≤ Q 2 ≤ 25 GeV2 ; right 35 ≤ Q 2 ≤ 800 GeV2 The inner error bars represent the statistical uncertainties, the full error bars include the statistical and systematic uncertainties added in quadrature, including all correlated and uncorrelated uncertainties. The curves represent the prediction from an NLO QCD fit. From Andreev et al. (2014)
Figure 17. Left: combined reduced charm cross sections (full circles) as a function of x for given values of Q 2 , compared to the NLO QCD FFNS predictions based on the HERAPDF2.0 FF3A (solid lines), and various other FFN scheme calculations The shaded bands on the HERAPDF2.0 FF3A predictions show the theory uncertainties obtained by adding PDF, scale and charm-quark mass uncertainties in quadrature. Right: Ratio of reduced charm cross sections as a function of x for given values of Q 2 for the combined data (full circles) and the NLO (dashed-dotted lines) and NNLO (dashed lines) VFNS…
Figure 18. The combined HERA data for the inclusive NC e + p reduced cross sections at √ s = 318 GeV at very low Q 2 . From Abramowicz et al. (2015). Error bars represent the total uncertainties. 3.3 F p 2 at very low Q 2 A remarkable feature of the proton structure function F2 is its rapid rise at low x – as already shown in
Figure 19. The structure-function F2 as a function of xBj for low Q 2 as extracted from the HERA NC e + p cross sections, together with predictions from the Regge framework see Sec 4.4 for details. From Abt et al. (2017). kinematics, the slope ∂F2 ∂ ln Q2 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 F2 as a function of Q 2 is starting to abate as x < 10−3 and Q 2 < 2 GeV2 . This could be because we are entering the non-perturbative region, rather than evidence for satur
Figure 20. Left: the effective slope of F2 at low x, F2 ∼ x −λ(Q 2 ) . From Gayler (2002). Right: The slope ∂F2 ∂ ln Q2 as a function of x from early ZEUS data. From Breitweg et al. (1999d)
Figure 23. Gluon and sea quark momentum distributions compared. From a NLO QCD fit by ZEUS using the full HERA-I data sample. From Chekanov et al. (2003a). F cc¯ 2 in Sec. 4.2. The role of FL enters into the fitting of the reduced cross section of the inclusive data and is discussed in that context in the next section. 4.1 alternative fits to σr 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 25. Left: the χ 2 /d.o.f. as a function of Q 2 min for NNLO vs NNLO+NLLx fits to to HERA data. Right: The predictions for the reduced cross section for the fits to HERA data (NNLO+NLLx) and without (NNLO) ln(1/x) resummation. From Abdolmaleki et al. (2018). 10 1 10 2 10 3 Q2 [GeV 2 ] 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 F L ( x, Q 2 ) x = 8.77 10 5 x = 1.29 10 4 x = 1.69 10 4 x = 2.24 10 4 x = 3.19 10 4 x = 4.02 10 4 x = 5.40 10 4 x = 6.87 10 4 x = 9.58 10 4 x = 1.21 10 3 x = 1.57 10 3 x = 2.43 10 3 x = 3.03 10 3 x = 4.02 10 3 x = 5.41 10 3 x = 7.36 10 3 x = 9.86 10 3 x = 1.84 10 2 H1…
Figure 26. The predictions for FL for the fits to HERA data (NNLO+NLLx) and without (NNLO) ln(1/x) resummation. From Abdolmaleki et al. (2018). recombining gluons. The leading-twist perturbative QCD forms of the structure functions F2 and FL are augmented by simple twist-4 terms F HT 2 = F DGLAP 2 (1 + A HT 2 /Q 2 ) , (29) F HT L = F DGLAP L (1 + A HT L /Q 2 ) , (30) where A HT 2 and A HT L are free parameters in the fits. It is found that only the A HT L term is needed to produce fits with a better χ 2 , see
Figure 27. The up valence PDF xuv, the gluon PDF xg and the total singlet PDF xΣ for the final fits with (NNLO+NLLx) and without (NNLO) ln(1/x) resummation. From Abdolmaleki et al. (2018). 2 / GeV min 2 Q 2 4 6 8 10 12 14 16 18 20 / ndof 2 χ 1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26 HERAPDF2.0, RTOPT NLO HERAPDF2.0, RTOPT NNLO HHT, RTOPT NLO HHT, RTOPT NNLO
Figure 29. The predictions of HHT NNLO (top) and HERAPDF2.0 NNLO (bottom), both with Q 2 min = 3.5 GeV2 , compared to the HERA measurements of σr . The bands represent the experimental, i.e. fit, uncertainties. Extrapolations are indicated as dotted lines. From Abt et al. (2016)
Figure 30. The predictions for FL from HHT and HERAPDF2.0, both with Q 2 min = 3.5 GeV2 , compared to the separate direct measurements published by the H1 and ZEUS collaborations. The bands represent the experimental, i.e. fit, uncertainties on the predictions. Hatched bands represent extrapolations. From Abt et al. (2016)
Figure 31. Ratio of reduced charm cross sections as a function of x for given values of Q 2 for the combined data (full circles) and the NNLO VFNS predictions of the NNPDF group with respect to the FFNS predictions, σ cc nom red , obtained using HERAPDF2.0 FF3A (solid lines with shaded uncertainty bands). Results from two different GMVFN calculations are shown: without (FONLL-C, dotted lines with uncertainty bands) and with low-x resummation (FONLL-C+NLLsx, dashed lines). For the calculations the NNPDF3.1sx PDF set is used. For better clarity of the presentation the uncertainties of the…
Figure 32. Forward jet data of H1, Adloff et al. (1999), as a function of Bjorken-x for two ET jet cuts of 3.5 GeV and 5.0 GeV. (a) and (c) contain Monte-Carlo model predictions, whereas (b) and (d) show the results of a LO BFKL (full) and a fixed NLO (dashed) calculation. measurements. 4.3 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 33. Forward jet data of H1, Adloff et al. (1999), and ZEUS, Breitweg et al. (1999b), compared to prediction based on the BFKL formalism including sub-leading corrections, Kwiecinski et al. (1999). The three curves correspond to variation of model choices, such as the choice of scale and of an infra-red cut off (perturbative calculations cannot be done at low scale when the string coupling parameter becomes large). as a ’point-like’ component. The resolved comonent arises from the idea that the virtual photon can split into a q − q¯ pair and thence into more partons, i.e. that it has a…
Figure 34. The virtual-photon proton cross section, as extracted from the HERA e + p NC data at √ s = 318 and 300 GeV, plotted against its centre of mass energy W2 . From Abt et al. (2017)
Figure 35. σ γ ∗ p (Q 2 ) in the low-Q 2 regime as extracted from the HERA e + p NC data at √ s = 318 and 300 GeV for selected values of W, together with the predictions of the ALLM parametrisations and the Regge predictions. The values of σ γ ∗ p (Q 2 ) are multiplied by 2 i for different values of W to make the curves individually visible. From Abt et al. (2017)
Figure 37. The GBW dipole model fit to F2 data through the transition region to photoproduction. The data are plotted versus Q 2 in bins of y. The dashed curves show the original model, the full curve the improved model with DGLAP evolution. From Bartels et al. (2002a). through the transition region towards Q 2 = 0. The idea is that the virtual photon splits into a qq¯ pair3 (a colour dipole) with a transverse size r ∼ 1/Q, the splitting occurring a ‘long’ time ∼ 1/(mx) before the dipole interacts with the proton. The dipole then scatters coherently from the proton in a time which is short in…
Figure 39. Left: The gluon structure function as a function of x for various fixed values of virtuality Q 2 extracted in the dipole saturation model (IP-Sat), or the PDFs CT10 (NNLO), Lai et al. (2010), and MSTW 2008 (NNLO), Martin et al. (2009). The corresponding theoretical uncertainties are represented with bands between solid, dashed and dotted lines for IP-Sat, CT10 and MSTW, respectively. Right: The saturation scale extracted from the IP-Sat model and from an older parameterization set from Watt and Kowalski (2008), labeled by IP-Sat (old), as a function of 1/x at various…
Figure 40. IPsat and IPnonsat (linearized IPsat) independent fits to inclusive reduced cross-section HERA data. Figure from Mantysaari ¨ and Zurita (2018). in contrast to the predictions of DGLAP evolution, see
Figure 41. Schematic illustration of the dipole model concept, in which a virtual photon fluctuates to a qq¯ pair with relative momentum fractions z and 1 − z, forming a colour dipole of transverse size r ∼ 1/ √ 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 γ ∗ p cross section. However, only small modifications are required to adapt this to the cases of exclusive or inclusive diffraction. 5.2 Exclusive Production of Vector Mesons and Photons 5.2.1 Kinematics and…
Figure 42. (a) Generic representation of exclusive vector meson production via the quasi-elastic 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. hadronic scattering. Detailed models of these effects can be found for example in Kaidalov et al. (2003, 2010); Gotsman et al. (2007). Further characteristics of this soft regime are, Aid et al. (1996); Breitweg et al.…
Figure 43. Compilation of photoproduction cross section measurements as a function of the γp centre-of-mass energy, W. The total cross section and various vector meson production cross sections are included, with the approximate power law dependences σ ∝ Wδ indicated for each process. From Levy (2009)
Figure 44. Development of the effective pomeron intercept (see Eq. 37 and surrounding text) with a characteristic scale, µ 2 = (Q 2 + M2 V )/4, derived from fits to the W dependences of various vector meson production data, as well as a DVCS measurement (shown at µ 2 = Q 2 ). The dashed lines indicate the range of values which are typical in soft diffraction. From Aaron et al. (2010b). The transition from soft hadronic to perturbative behaviour is neatly mapped out in a single process in ρ 0 electroproduction data. As Q 2 increases, the t slope parameter B decreases, the W dependence becomes…
Figure 45. Exponential t slopes for vector meson electroproduction and photoproduction as a function of the characteristic scale Q 2 + M2 V . From Abramowicz et al. (2012). tudinal 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 46. J/ψ photoproduction cross section as a function of W, compared with a QCD calculation, MRT Martin et al. (2000), based on four different parametrisations of the gluon density of the proton and a dipole model, FMS Frankfurt et al. (2001b). The predictions are normalised to the data. From Aktas et al. (2006c). 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…
Figure 47. Dependence of HERA DVCS cross section measurements on Q 2 . The data are compared with a GPD-based model, Kumericki and Mueller (2010), and also with a dipole-based model, Marquet et al. (2007). See Section 5.2.3 for an explanation of the latter. From Aaron et al. (2009b). By the end of HERA operation, Q 2 , W and t distributions for DVCS had all been measured with good precision, Chekanov et al. (2009a); Aaron et al. (2009b, 2008). As shown in Figs. 44 and 45, the exponential t slope is B ≃ 5 GeV−2 and parametrisations of the cross section in the usual Wδ form yield δ ≈ 0.5 − 0.6.…
Figure 48. 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 zIP from the DPDFs enters the hard scattering. (c) A leading order process in resolved photoproduction involving a parton of momentum fraction xγ relative to the photon. fixed target configuration by HERMES, Airapetian et al. (2012b,a). Nonetheless, the H1 and ZEUS data provide important low x…
Figure 49. Example data on the dynamics of the slope parameter, B, describing the t dependence of DDIS, as measured using the H1 FPS. From Aaron et al. (2011). The semi-inclusive DDIS cross section is usually presented in the form of a diffractive reduced cross section σ D(3) r , which is related to the experimentally measured differential cross section analogously to the inclusive case (Sec. 2), such that d 3σ ep→eX p dxIP dx dQ2 = 2πα2 xQ4 · Y+ · σ D(3) r (xIP , x, Q 2 ) , (40) where Y+ = 1 + (1 − y) 2 and y is the usual Bjorken variable. The reduced cross section depends at moder
Figure 50. Comparisons between H1 and ZEUS DDIS data measured using the LRG method at example values of xIP = 0.001 and xIP = 0.01. Data for these and other xIP values are presented in Aaron et al. (2012b). For y not too close to unity, σ D(3) r = F D(3) 2 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. (2009b); Aaron et al. (2011). However, since the t dependence factorises to good approximation, this degree of freedom is usually integrated out. A more detailed exposition of the…
Figure 51. Summary of ZEUS data on tests of proton vertex factorisation in the form of possible Q 2 variations of the pomeron intercept in fits to LRG and LPS data, as well as ‘FPC’ data obtained by the MX decomposition method. From Chekanov et al. (2009b). there is no evidence for any variation of the effective pomeron intercept with Q 2 or β when considering data taken using either the LRG or LPS/FPS methods. Similar searches for a Q 2 or β dependence of α ′ IP or B have also yielded null results, Aaron et al. (2011), albeit with lesser precision. It is informative to compare the…
Figure 52. The ratio of diffractive to inclusive cross sections as a function of W, integrated over different MX regions at fixed Q 2 . Obtained using ZEUS 1994 data. From Breitweg et al. (1999c). DPDFs have been extracted using standard NLO QCD procedures, similar to those employed to extract inclusive proton PDFs (Sec. 4), 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 zIP (it is equal to β when used to describe the quark coupling to the exchange…
Figure 53. 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 Aktas et al. (2006d). z 0 0.2 0.4 0.6 0.8 1 q zf 0 0.01 0.02 0.03 0.04 2 = 6 GeV 2 Q ZEUS DPDF SJ exp. uncertainty ZEUS DPDF C z 0 0.2 0.4 0.6 0.8 1 q zf 0 0.01 0.02 0.03 0.04 ZEUS DPDF SJ exp. uncertainty ZEUS DPDF C z 0 0.2 0.4 0.6 0.8 1 g zf 0 0.2 0.4 0.6 2 = 6 GeV 2 Q ZEUS DPDF SJ exp. uncertainty ZEUS DPDF C z 0 0.2 0.4 0.6 0.8 1 g zf 0 0.2…
Figure 54. 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 Chekanov et al. (2010a). 5.3.6 Applications of Diffractive Parton Densities The DPDFs described in Section 5.3.5 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. (2006d)) in Monte Carlo generators, Jung (1995); Cox and Forshaw (2002); Navin…
Figure 55. Comparison of diffractive DIS dijet cross sections, measured double differentially in zIP 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 β sensitivity. From Chekanov et al. (2010a). Diffractive open charm production in DIS, Aktas et al. (2007a); Chekanov et al. (2003c), provides a further exclusive final state with high sensitivity to the diffractive gluon density via the lowest order contributing process, γ ∗g → cc¯. The charm quarks are usually tagged…
Figure 56. Comparisons between diffractive open charm (D ∗ ) production measurements and NLO QCD predictions based on DPDFs. From Aktas et al. (2007a)
Figure 57. Summary of H1 measurements of the diffractive longitudinal structure function F D L , exploiting data taken with variations in the proton beam energy. To allow comparisons between measurements at different xIP values, the data are normalised by a diffractive flux factor fIP/p (see Section 5.3.4). The data are compared with an NLO QCD prediction based on DPDFs. From Aaron et al. (2012d). 5.3.7 Limitations of Diffractive Parton Densities Whilst models based on proton vertex factorisation and DPDFs work well to describe all diffractive processes beyond the lowest MX resonance region…
Figure 58. Fraction of 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). Left: H1 data, from Aaron et al. (2010a) Right: ZEUS data, from Chekanov et al. (2010a). a two-gluon exchange model of exclusive qq¯, Bartels et al. (1996b); Lotter (1997), and qqg¯ , Bartels et al. (1999b, 2002b), production. However, progress has been limited by significant theoretical uncertainties associated with the low β qqg¯ contribution and a lack of direct evidence for the high β higher…
Figure 59. An example decomposition of the β dependence of ZEUS DDIS data, Breitweg et al. (1999c), into different dipole terms, according to the parametrisation in Golec-Biernat and Wusthoff (1999b). The dashed, dotted and dot-dashed curves correspond to leading twist transverse qq¯, leading twist transverse qqg¯ and higher twist longitudinal qq¯ contributions, respectively
Figure 60. Example data at very large |t|. Left: Fraction of dijet photoproduction events in which the two jets with the highest transverse energy are separated by a pseudorapidity ’gap’ of size ∆η in which the total transvere energy is less than 0.5 GeV, Chekanov et al. (2007c). ZEUS and H1 data are compared with a prediction that includes a BFKL pomeron contribution. From Chekanov et al. (2007c). Right: Dependence on |t| of proton dissociative J/ψ production integrated over proton dissociation masses satisfying M2 Y /W2 < 0.05. The data are compared with a prediction (‘GLMN LL’, Gotsman et…

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