<table><caption>**Table 18.1:** The main processes relevant to global PDF analyses, ordered in three groups: fixed-target experiments, HERA and the $p\bar{p}$ Tevatron / $pp$ LHC. For each process we give an indication of their dominant partonic subprocesses, the primary partons which are probed and the approximate range of $x$ constrained by the data. This list expands as more processes are measured and calculated with sufficient precision.</caption><thead><tr><th>Process</th><th>Subprocess</th><th>Partons</th><th>x range</th></tr></thead><tbody><tr><td>ℓ<sup>±</sup> {p, n} → ℓ<sup>±</sup> X</td><td>γ<sup>*</sup>q → q</td><td>q, q̄, g</td><td>x ≥ 0.01</td></tr><tr><td>ℓ<sup>±</sup> n/p → ℓ<sup>±</sup> X</td><td>γ<sup>*</sup> d/u → d/u</td><td>d/u</td><td>x ≥ 0.01</td></tr><tr><td>pp → μ<sup>+</sup> μ<sup>−</sup> X</td><td>uū, d<span style="text-decoration: overline">d</span> → γ<sup>*</sup></td><td>q̄</td><td>0.015 ≤ x ≤ 0.35</td></tr><tr><td>pn/pp → μ<sup>+</sup> μ<sup>−</sup> X</td><td>(u<span style="text-decoration: overline">d</span>)/(u<span style="text-decoration: overline">ū</span>) → γ<sup>*</sup></td><td>d/ū</td><td>0.015 ≤ x ≤ 0.35</td></tr><tr><td>ν(ν̄) N → μ<sup>−</sup>(μ<sup>+</sup>) X</td><td>W<sup>*</sup>q → q'</td><td>q, q̄</td><td>0.01 ≤ x ≤ 0.5</td></tr><tr><td>νN → μ<sup>−</sup> μ<sup>+</sup> X</td><td>W<sup>*</sup>s → c</td><td>s</td><td>0.01 ≤ x ≤ 0.2</td></tr><tr><td>ν̄N → μ<sup>+</sup> μ<sup>−</sup> X</td><td>W<sup>*</sup>s̄ → c̄</td><td>s̄</td><td>0.01 ≤ x ≤ 0.2</td></tr><tr><td>e<sup>±</sup> p → e<sup>±</sup> X</td><td>γ<sup>*</sup>q → q</td><td>g, q, q̄</td><td>10<sup>−4</sup> ≤ x ≤ 0.1</td></tr><tr><td>e<sup>+</sup> p → ν̄X</td><td>W<sup>+</sup> {d, s} → {u, c}</td><td>d, s</td><td>x ≥ 0.01</td></tr><tr><td>e<sup>±</sup> p → e<sup>±</sup> c<span style="text-decoration: overline">c</span>X, e<sup>±</sup> b<span style="text-decoration: overline">b</span>X</td><td>γ<sup>*</sup>c → c, γ<sup>*</sup>g → c<span style="text-decoration: overline">c</span></td><td>c, b, g</td><td>10<sup>−4</sup> ≤ x ≤ 0.01</td></tr><tr><td>e<sup>±</sup> p → jet+X</td><td>γ<sup>*</sup>g → q<span style="text-decoration: overline">q</span></td><td>g</td><td>0.01 ≤ x ≤ 0.1</td></tr><tr><td>p<span style="text-decoration: overline">p</span>, pp → jet(dijet)+X</td><td>gg, gg, qq → 2j</td><td>g, q</td><td>0.00005 ≤ x ≤ 0.5</td></tr><tr><td>p<span style="text-decoration: overline">p</span> → (W<sup>±</sup> → ℓ<sup>±</sup>ν) X</td><td>u<span style="text-decoration: overline">d</span> → W<sup>+</sup>, d<span style="text-decoration: overline">d</span> → W<sup>−</sup></td><td>u, d, s, u<span style="text-decoration: overline">ū</span>, d<span style="text-decoration: overline">d</span>, s̄</td><td>x ≥ 0.05</td></tr><tr><td>pp → (W<sup>±</sup> → ℓ<sup>±</sup>ν) X</td><td>u<span style="text-decoration: overline">d</span> → W<sup>+</sup>, d<span style="text-decoration: overline">d</span> → W<sup>−</sup></td><td>u, d, s, u<span style="text-decoration: overline">ū</span>, d<span style="text-decoration: overline">d</span>, s̄, g</td><td>x ≥ 0.001</td></tr><tr><td>p<span style="text-decoration: overline">p</span>(pp) → (Z → ℓ<sup>±</sup>ℓ<sup>−</sup>)X</td><td>uu, dd, .. (u<span style="text-decoration: overline">ū</span>, ..) → Z</td><td>u, d, s, ..(g)</td><td>x ≥ 0.001</td></tr><tr><td>pp → W<sup>−</sup>c, W<sup>+</sup>c̄</td><td>gs → W<sup>−</sup>c</td><td>s, s̄</td><td>x ∼ 0.01</td></tr><tr><td>pp → (γ<sup>*</sup> → ℓ<sup>±</sup>ℓ<sup>−</sup>)X</td><td>u<span style="text-decoration: overline">ū</span>, d<span style="text-decoration: overline">d</span>, .. → γ<sup>*</sup></td><td>q̄, g</td><td>x ≥ 10<sup>−5</sup></td></tr><tr><td>pp → (γ<sup>*</sup> → ℓ<sup>±</sup>ℓ<sup>−</sup>)X</td><td>uγ, dγ, .. → γ<sup>*</sup></td><td>γ</td><td>x ≥ 10<sup>−2</sup></td></tr><tr><td>pp → b<span style="text-decoration: overline">b</span>X, t<span style="text-decoration: overline">t</span>X</td><td>gg → b<span style="text-decoration: overline">b</span>, t<span style="text-decoration: overline">t</span></td><td>g</td><td>x ≥ 10<sup>−5</sup>, 10<sup>−2</sup></td></tr><tr><td>pp → t<span style="text-decoration: overline">t</span>(t̄)X</td><td>bu(b<span style="text-decoration: overline">d</span>) → td(t<span style="text-decoration: overline">u</span>)</td><td>b, d/u</td><td>x ≥ 10<sup>−2</sup></td></tr><tr><td>pp → exclusive J/ψ, γ</td><td>γ<sup>*</sup>(gg) → J/ψ, γ</td><td>g</td><td>x ≥ 10<sup>−5</sup>, 10<sup>−4</sup></td></tr><tr><td>pp → γX</td><td>gg → γq, g<span style="text-decoration: overline">g</span> → γq̄</td><td>g</td><td>x ≥ 0.005</td></tr></tbody></table>

It is often convenient to write the evolution equations in terms of the gluon, non-singlet ($q^{NS}$) and singlet ($q^S$) quark distributions, such that

$$q^{NS} = q_i - \bar{q}_i \quad (\text{or } q_i - q_j), \qquad q^S = \sum_i (q_i + \bar{q}_i) \cdot (18.24)$$

The non-singlet distributions have non-zero values of flavor quantum numbers, such as isospin and baryon number. The DGLAP evolution equations then take the form

$$\frac{\partial q^{NS}}{\partial \ln \mu^2} = \frac{\alpha_s(\mu^2)}{2\pi} P_{qq} \otimes q^{NS}, \\ \frac{\partial}{\partial \ln \mu^2} \begin{pmatrix} q^S \\ g \end{pmatrix} = \frac{\alpha_s(\mu^2)}{2\pi} \begin{pmatrix} P_{qq} & 2n_f P_{qg} \\ P_{qg} & P_{gg} \end{pmatrix} \otimes \begin{pmatrix} q^S \\ g \end{pmatrix}, \quad (18.25)$$

where $P$ are splitting functions that describe the probability of a given parton splitting into two others, and $n_f$ is the number of (active) quark flavors. The leading-order Altarelli-Parisi [22] splitting functions are

$$P_{qq} = \frac{4}{3} \left[ \frac{1+x^2}{(1-x)} \right]_+ = \frac{4}{3} \left[ \frac{1+x^2}{(1-x)_+} \right] + 2\delta(1-x), \quad (18.26)$$

$$P_{qg} = \frac{1}{2} \left[ x^2 + (1-x)^2 \right], \quad P_{qg} = \frac{4}{3} \left[ \frac{1+(1-x)^2}{x} \right], \quad (18.27)$$

$$P_{gg} = 6 \left[ \frac{1-x}{x} + x(1-x) + \frac{x}{(1-x)_+} \right] + \left[ \frac{11}{2} - \frac{n_f}{3} \right] \delta(1-x), \quad (18.28)$$

<figure><img src="image_82.png" /><figcaption>**Figure 18.4:** The bands are $x$ times the unpolarized parton distributions $f(x)$ (where $f = u_v, d_v, \bar{u}, \bar{d}, s \simeq s̄, c = c̄, b = b̄, g$) obtained in the NNLO MSHT20 global analysis [26] (top) at scales $\mu^2 = 10 \text{ GeV}^2$ (left) and $\mu^2 = 10^4 \text{ GeV}^2$ (right), with $\alpha_s(M_Z^2) = 0.118$. The polarized parton distributions $f(x)$ obtained in the NLO NNPDFpol1.1 fit [27] (bottom).</figcaption></figure>

where the notation $[F(x)]_+$ defines a distribution such that for any sufficiently regular test function, $f(x)$,

$$\int_0^1 dx f(x) [F(x)]_+ = \int_0^1 dx (f(x) - f(1)) F(x) \cdot (18.29)$$

In general, the splitting functions can be expressed as a power series in $\alpha_s$. The series contains both terms proportional to $\ln \mu^2$ and to $\ln(1/x)$ and $\ln(1-x)$. The leading-order DGLAP evolution sums up the $(\alpha_s \ln \mu^2)^n$ contributions, while at next-to-leading order (NLO) the sum over the $\alpha_s(\alpha_s \ln \mu^2)^{n-1}$ terms is included [28, 29]. The NNLO contributions to the splitting functions and the DIS coefficient functions are also all known [30–32], as are the N$^3$LO corrections to the coefficient functions [33]. Some Mellin moments of the N$^3$LO splitting functions are also known [34], together with some other pieces of specific information.

In the kinematic region of very small $x$, one may also sum leading terms in $\ln(1/x)$, independent of the value of $\ln \mu^2$. At leading order, LLx, this is done by the BFKL equation for the unintegrated distributions (see Refs. [35, 36]) The leading-order $(\alpha_s \ln(1/x))^n$ terms result in a power-like growth, $x^{-\omega}$ with $\omega = (12\alpha_s \ln 2)/\pi$, at asymptotic values of $\ln 1/x$. The next-to-leading $\ln 1/x$ (NLLx) contributions are also available [37, 38]. They are so large (and negative) that the results initially appeared to be perturbatively unstable. Methods, based on a combination of collinear and small-$x$ resummations, have been developed which reorganize the perturbative series into a more stable hierarchy [39–42], and this has been used as the basis for a framework for including the corrections in phenomenological studies [43, 44]. There are some limited indications that small-$x$ resummations become necessary for sufficient precision for $x \lesssim 10^{-3}$ at low scales [15, 16]. At sufficiently small $x$ it is expected that evolution should also be affected by non-linear/saturation effects due to a high density of partons, i.e. the gluon density would be so high that gluon-gluon recombination effects would become significant, see [45–47] for reviews. However, there is not yet any very convincing indication for a non-linear regime for $Q^2 \gtrsim 2 \text{ GeV}^2$ in hadron DIS. Better evidence for saturation may be found in more exclu-
sive final states such as dihadron correlations in proton-nucleus collisions, e.g. [48, 49].

The precision of the experimental data demands that at least NLO, and preferably NNLO, DGLAP evolution be used in comparisons between QCD theory and experiment. Beyond the leading order, it is necessary to specify, and to use consistently, both a renormalization and a factorization scheme. The renormalization scheme used almost universally is the modified minimal subtraction (MS) scheme [50, 51]. The most popular choices for the factorization scheme is also MS [52]. Historically, sometimes the DIS [53] scheme was adopted, in which there are no higher-order corrections to the F2 structure function, and this was extended to a specific definition appropriate to small x [54]. assimilated in the parton distribution functions.

The discussion above relates to the $Q^2$ behavior of leading-twist (twist-2) contributions to the structure functions. Higher-twist terms, which involve their own non-perturbative input, exist. These die off as powers of $Q$; specifically twist-$n$ terms are damped by $1/Q^{n-2}$. Provided a cut, say $W^2 > 15$ GeV$^2$ is imposed, the higher-twist terms appear to be numerically unimportant for $Q^2$ above a few GeV$^2$, except possibly for very small $x$ and more definitely for $x$ close to 1 [55–57], though it is important to note that they are likely to be larger in $xF_3(x, Q^2)$ than in $F_2(x, Q^2)$ (see e.g. [58]) due to a lack of a constraining sum rule for $xF_3(x, Q^2)$.

## 18.3 Determination of parton distributions

The parton distribution functions (PDFs) can be determined from an analysis of data for deep inelastic lepton-nucleon scattering and for related hard-scattering processes initiated by nucleons; see Refs. [59–65] for reviews. Table 18.1 highlights some of the processes and their primary sensitivity to PDFs. LHC data are playing an increasing role [66], and new processes are continually being added to the list. Fixed-target and collider experiments have complementary kinematic reach (as is shown in Fig. 18.3), which enables the determination of PDFs over a wide range in $x$ and $\mu^2$. As more precise LHC data for $W^{\pm}$, $Z$, $\gamma$, jet, $bb$ and $t\bar{t}$ production become available, tighter constraints on the PDFs are expected in a wider kinematic range. At present about half the constraint on PDFs comes from LHC data [26], but much of the constraint still comes from structure functions.

Recent determinations and releases of the unpolarized PDFs up to NNLO have been made by five groups: MSHT [26], NNPDF [67], CT [68], ATLAS [69] and ABMP [70]. The CJ group produce PDFs at NLO and concentrate on the high $x$, low $Q^2$ regime, including data with lower kinematic cuts [71]. All groups start evolution at $Q_0^2 = 1-4$ GeV$^2$. Most groups use input PDFs of the form $xf = x^a(...)(1-x)^b$ with 14-32 free parameters in total. In these cases the PDF uncertainties are made available using the “Hessian” formulation. The free parameters are expanded around their best fit values, and orthogonal eigenvector sets of PDFs depending on linear combinations of the parameter variations are obtained. The uncertainty is then the quadratic sum of the uncertainties arising from each eigenvector. The NNPDF group combines a Monte Carlo representation of the probability measure in the space of PDFs with the use of neural networks. Fits are performed to a very large number of “replica” data sets obtained by allowing individual data points to fluctuate randomly by amounts determined by the size of the data uncertainties. This results in a set of replicas of unbiased PDF sets. In this case the best prediction is the average obtained using all PDF replicas and the uncertainty is the standard deviation over all replicas. It is possible to convert the eigenvectors of Hessian-based PDFs to Monte Carlo replicas [72] and vice versa [73]. Recently accumulated information on N$^3$LO calculation has been used by MSHT in order to produce a first set of approximate N$^3$LO global PDFs [74] together with theoretical uncertainties, and an alternative approach to theoretical uncertainty has been made in [75].

In these analyses, the u, d and s quarks are taken to be massless, but the treatment of the heavy c and b quark masses, $m_Q$, differs, and has a long history, which may be traced from Refs. [76–87]. The MSHT, CT, NNPDF and HERAPDF analyses use different variants of the General-Mass Variable-Flavour-Number Scheme (GM-VFNS). This combines fixed-order contributions to the co-

efficient functions (or partonic cross sections) calculated with the full $m_Q$ dependence, with the all-order resummation of contributions via DGLAP evolution in which the heavy quarks are treated as massless after starting evolution at some transition point. Transition matrix elements are computed, following [79], which provide the boundary conditions between $n_f$ and $n_f + 1$ PDFs. The ABMP analysis uses for structure function calculations a FFNS where only the three light (massless) quarks enter the evolution, while the heavy quarks enter the partonic cross sections with their full $m_Q$ dependence. The GM-VFNS and FFNS approaches yield different results: in particular $\alpha_s(M_Z^2)$ and the large-$x$ gluon PDF at large $Q^2$ are both significantly smaller in the FFNS. It has been argued [56, 57, 86] that the difference is due to the slow convergence of the $\ln^n(Q^2/m_Q^2)$ terms in certain regions in a FFNS. The final HERA combination of heavy flavour structure function data has been published [88], and the evolution of these measurements and their interpretation may be traced in [89].

The recent determinations of the groups fitting a variety of data and using a GM-VFNS (MSHT, NNPDF and CT) have converged, so that now a good agreement has been achieved between the resulting PDFs. Indeed, the recent global fit PDF sets, CT18 [68], MSHT2018 [26], and NNPDF3.1 [90], have been combined [91] using the Monte Carlo approach [72] mentioned above. The single combined set of PDFs is discussed in detail in Ref. [92].

For illustration, we show in Fig. 18.4 the PDFs obtained in the NNLO MSHT analysis [26] at scales $\mu^2 = 10$ and $10^4$ GeV$^2$ (and an equivalent plot for polarized PDFs from NNPDF [27]). The values of $\alpha_s$ found by MSHT [93] may be taken as representative of those resulting from the GM-VFNS analyses

$$ \begin{aligned} \text{NLO: } \alpha_s(M_Z^2) &= 0.1203 \pm 0.0015, \\ \text{NNLO: } \alpha_s(M_Z^2) &= 0.1174 \pm 0.0013, \end{aligned} $$

where the error (at 68% C.L.) corresponds to the uncertainties resulting from the data fitted (the uncertainty that might be expected from the neglect of higher orders is at least as large). A similar results is found by the NNPDF group [94], who find $\alpha_s(M_Z^2) = 0.1185 \pm 0.0012$ at NNLO and the CT group [68], who obtain $\alpha_s(M_Z^2) = 0.1164 \pm 0.0026$. The ABMP analysis [70], which uses a FFNS, finds $\alpha_s(M_Z^2) = 0.1147 \pm 0.0011$ at NNLO.

As a first step towards the inclusion of higher order electroweak corrections a recent development has been a vastly increased understanding of the photon content of the proton. Sets of PDFs with a photon contribution were considered in Refs. [95–97]. However, due to weak data constraints, the uncertainty was extremely large. Subsequently, there has been a much improved understanding of the separation into elastic and inelastic contributions [98–100]. This gives much more theoretical precision, since the elastic contribution, arising from coherent emission of a photon from the proton, can be directly related to the well-known proton electric and magnetic form factors; the model dependence of the inelastic (incoherent) contribution, related to the quark PDFs, is at the level of tens of percent. A final and decisive development directly relating the entire photon contribution to the proton structure function [101] resulted in a determination of the photon content of the proton as precise as that of the light quarks. The framework has been applied within global fits to PDFs via an iterative procedure in [102] and to provide the low-scale input photon PDF in [103, 104]. A further development is the calculation of photon-initiated collider processes directly from structure functions [105].

There are also some calculations of PDFs using lattice QCD, see [106, 107] for a recent review. There has been significant recent progress in the development of efficient algorithms for generating ensembles of gauge field configurations and in tools for extracting the required information from correlation functions. However, there remain a number of sources of systematic uncertainty; discretization effects, pion mass dependence, finite volume effects, excited state contamination and renormalization, which need to be determined and ideally minimised. Moreover, PDFs cannot be determined directly in Euclidean lattice QCD because they depend on quantum fields at light-like separations. Traditionally
the lattice has been used to determine matrix elements of twist two operators which can be related to integer moments of PDFs. Until recently these did not always agree well with determinations using global fit PDFs [106]. However, there have been recent improvements [108–110], though uncertainties are still relatively large. An alternative approach [111] suggested the computation of the matrix elements of frame-dependent, equal-time correlators in the large momentum limit. This leads to so-called Quasi-PDFs, which in the infinite momentum frame can be related to light-cone PDFs via a matching procedure, and there is the closely related psuedo PDF appraoch [112]. Recent results using physical values of the light quark masses are in [113–117].

Comprehensive sets of PDFs are available from the LHAPDF library [118], which can be linked directly into a user's programme to provide access to recent PDFs in a standard format. This also includes many nuclear and polarized PDFs.

**Nuclear PDFs:** The study of the parton distributions for nucleons within nuclei, so-called nuclear parton distribution functions (nPDFs), is now reaching a level of maturity and sophistication similar to nucleon PDFs (and some nuclear target data is often included in determinations of nucleon PDFs), though they are still typically performed at NLO in perturbative QCD. The PDFs are also a function of the nucleon number of the nucleus, A. The nPDFs are obtained via fits to deep inelastic scattering data and dilepton (Drell-Yan) and pion production from proton-nucleus, and most recently also from jet and heavy flavour production data. There are a number of recent examples of NLO analyses, TUJU21 [119], nCTEQ15HQ(HiX) [120, 121], EPPS21 [122], nNNPDF3.0 [123], while NNLO analyses with a smaller selection of data types now also exist [119, 124, 125]. A comparison of the nuclear modification factors for nCTEQ15HQ [120], EPPS21 [122] and nNNPDF3.0 [123] is shown in Fig. 18.6 where, for example, for the up quark $R_u^{\text{Pb}} = (Z_{Pb}u_{Pb} + (A_{Pb} - Z_{Pb})d_{Pb})/(Z_{Pb}u_p + (A_{Pb} - Z_{Pb})d_p)$ i.e. it involves “physical” nuclear PDFs and is normalised such that $R_u^{\text{Pb}} = 1$ corresponds to no nuclear effects.

Much of the heavy-nucleus data included are in the form of ratios to proton or deuteron measurements. Initially most nuclear PDFs were related to a particular proton PDF via a nuclear modification factor, i.e.

$$f_i^{p/A}(x, Q^2) = R_i^A(x, Q^2) f_i^p(x, Q^2). \quad (18.30)$$

However, it is now common to parameterise the nuclear PDFs directly but so that they become equal to proton PDFs in the limit $A = 1$, and this approach has been adopted in [119–121, 123]. There is some variation in whether charged current neutrino DIS data is used as well as neutral current DIS data since there is no clear compatibility in the modification factors obtained [126, 127]. Recently, LHC data from vector boson production [128–131] in proton-lead collisions has been used along with DIS data in most fits, though the most recent data [132] shows some tendency to require NNLO corrections. LHC jet data [133] has been included in some fits [122, 123] giving extra constraint on the gluon within nuclei. Further information at smaller $x$ values is also obtained from heavy meson production at LHCb [134]. Single inclusive hadron production e.g. [135, 136], is already sometimes used. All the PDF extractions above are based on the Hessian formulation, except for the NNPDF studies which use Monte Carlo replicas. Agreement between the different nuclear PDFs is generally good, though uncertainty is still large compared to proton PDFs. As well as improved constraints from further LHC data, nPDFs would be very significantly improved by data from a potential high-energy Electron-Ion Collider [137–139].

**Double Parton Distributions** Double parton scattering (DPS) occurs when two pairs of partons initiate two distinct hard scattering processes in a single collision. It becomes more prevalent at higher collider energies as the parton flux increases. Hence, it is becoming relevant at the LHC, and even more so for planning of higher energy colliders. It has a long theoretical history [140–144], and in recent years experimental studies have

started. DPS is often suppressed, but this is less the case in certain kinematic regions, e.g. when the total transverse momentum of the two hard-scattering processes is small [145–148], when they have a large rapidity separation [149, 150], or when single particle production is suppressed by coupling constants, e.g. like-sign W production [151]. There have been results with some evidence for DPS found from both the Tevatron [152, 153] involving final states with multijets and photons, and at the LHC [154–156] in states with multiple vector bosons or mesons. Recent developments have been involved in putting the theoretical foundations of DPS on a much more rigorous footing and describing it in terms of well-defined double partons distributions (DPDs) which extend the concept of parton distribution functions (PDFs) to the case of two partons [145–148, 157–159]. The current knowledge of DPDs is still limited, though there are a number of attempts to impose theoretical constraints as a limit on the forms [160–162]. The DPDs can be calculated quite accurately when the transverse separation of the two partons is small, since in this limit the observed partons are produced by a splitting from a single parton. This limit is complicated by potential double counting [148], and an approach to deal with this has been devised [159]. The full splitting functions for DPS have been calculated at LO [148], and there is progress at NLO, e.g. [163].

**Polarized PDFs:** For spin-dependent structure functions, data exists for a more restricted range of $Q^2$ and has lower precision, so that the scaling violations are not seen so clearly. However, spin-dependent (or polarized) parton distributions have been extracted by comparison to data using NLO global analyses which include measurements of the $g_1$ structure function in inclusive polarized DIS, ‘flavour-tagged’ semi-inclusive DIS data, open-charm production in DIS and results from polarized pp scattering at RHIC. There are recent results on DIS from JLAB [164] (for $g_1^n/F_1^n$), COMPASS [165, 166] and CLAS [167]. NLO analyses are given in Refs. [168–171] and more recent extractions [172, 173]. Improved parton-to-hadron fragmentation functions, needed to describe the semi-inclusive DIS (SIDIS) data, can be found in Refs. [174–177].

The DSSV collaboration includes in their NLO analysis to extract polarized PDFs all the world data, inclusive and semi-inclusive DIS, double spin asymmetries in jet [178], dijet [179] and inclusive $\pi^0$-production [180], but not yet the single spin asymmetries in $W^\pm$, $Z^0$ production [181, 182]. A determination [183], using the NNPDF methodology, concentrates just on the inclusive polarized DIS data, and finds the uncertainties on the polarized gluon PDF have been underestimated in the earlier analyses. An update to this [27], where jet and $W^\pm$ data from pp collisions

<figure><img src="image_83.png" /><figcaption>**Figure 18.5:** Ensemble of replicas (dotted blue lines) for the NLO gluon helicity density $\Delta g(x, Q^2)$ at $Q^2 = 10 \text{ GeV}^2$ shown along with its statistical average (solid blue line) and variance (dot-dashed blue lines). The corresponding results from the DSSV14 fit (black lines) [172] and the NNPDF1.1 analysis (green lines) [27] are shown for comparison. Figure taken from Ref. [184].</figcaption></figure>
<figure><img src="image_84.png" /><figcaption>**Figure 18.6:** Comparison of the nNNPDF3.0, nCTEQ15HQ and EPPS21 nuclear PDFs. The curves shown are ratios to the result in the limit of no nuclear corrections. Plot from NNPDF collaboration (Juan Rojo - private communication).</figcaption></figure>

and open-charm DIS data have been included via reweighting, reduces the uncertainty and suggests a positive polarized gluon PDF. The evidence for a positive gluon polarization is strengthened by the most recent jet [185–187] and inclusive $\pi^0$ [188] data. The DSSV group has recently implemented a Monte Carlo sampling strategy to extract helicity parton densities and their uncertainties from a reference set of longitudinally polarized scattering data [184]. A simultaneous extraction of spin-dependent parton distributions and fragmentation functions has recently also been performed [189]. Calculations of polarised PDFs on the lattice are frequently produced by the same studies as for unpolarised PDFs. These are usually in better agreement with the corresponding moments obtained from global fits, than the unpolarised PDFs, and also tend to have much more competitive uncertainties, see [106, 107] for a review and e.g. [109, 190] for more recent results.

A comparison of the polarized gluon PDFs obtained in the NLO analyses of NNPDF [27] and DSSV [184] is shown in Fig. 18.5 at scale $\mu^2 = 10 \text{ GeV}^2$. The world data of the inclusive structure function $g_1$ for proton and deuterium included in these analysis are shown in Fig. 18.16 and Fig. 18.17.

## 18.4 The hadronic structure of the photon

Besides the direct interactions of the photon, it is possible for it to fluctuate into a hadronic state via the process $\gamma \to q\bar{q}$. While in this state, the partonic content of the photon may be resolved, for example, through the process $e^+e^- \to e^+e^-\gamma^*\gamma \to e^+e^-X$, where the virtual photon emitted by the DIS lepton probes the hadronic structure of the quasi-real photon emitted by the other lepton. The perturbative LO QED contributions to this process with $\gamma \to q\bar{q}$ in conjunction with $\gamma^*q(\bar{q}) \to q(\bar{q})$, are subject to QCD corrections due to the radiation of gluons from these quarks.

Often the equivalent-photon approximation is used to express the differential cross section for deep inelastic electron-photon scattering in terms of the structure functions of the transverse quasi-real photon times a flux factor $N_\gamma^T$ (for these incoming quasi-real photons of transverse polarization)

$$ \frac{d^2\sigma}{dxdQ^2} = N_{\gamma}^{T} \frac{2\pi\alpha^2}{xQ^4} \left[ (1+(1-y)^2) F_2^{\gamma}(x,Q^2) - y^2 F_L^{\gamma}(x,Q^2) \right], \quad (18.31) $$

where we have used $F_2^\gamma = 2xF_T^\gamma + F_L^\gamma$ (where $F_T$ is the transverse structure function), not to be confused with $F_2^\gamma$ of Sec. 18.2. Complete formulae are given, for example, in the comprehensive review of [191].

The hadronic photon structure function, $F_2^\gamma$, evolves with in-

creasing $Q^2$ from the ‘hadron-like’ behavior, calculable via the vector-meson-dominance model, to the dominating ‘point-like’ behaviour, calculable in perturbative QCD. Due to the point-like coupling, the logarithmic evolution of $F_2^\gamma$ with $Q^2$ has a positive slope for all values of $x$, see Fig. 18.18. The ‘loss’ of quarks at large $x$ due to gluon radiation is over-compensated by the ‘creation’ of quarks via the point-like $\gamma \to q\bar{q}$ coupling. The logarithmic evolution was first predicted in the quark-parton model ($\gamma^*\gamma \to q\bar{q}$) [192, 193], and then an improved expression was obtained using QCD corrections in the limit of large $Q^2$ [194]. The evolution is now known to NLO [195–197]. The NLO data analyses to determine the parton densities of the photon can be found in Refs. [198–200].

## 18.5 Diffractive DIS (DDIS)

Some 10% of DIS events are diffractive, $\gamma^*p \to X+p$, in which the slightly deflected proton and the cluster $X$ of outgoing hadrons are well-separated in rapidity [201]. Besides $x$ and $Q^2$, two extra

<figure><img src="image_85.png" /><figcaption>**Figure 18.7:** Diffractive parton distributions, $x_P z f_{a/p}^D$, obtained from fitting to the ZEUS data with $Q^2 > 5 \text{ GeV}^2$ [202], H1 data with $Q^2 > 8.5 \text{ GeV}^2$ assuming Regge factorization [203], and from MRW2006 [204] using a more perturbative QCD approach [204]. Only the Pomeron contributions are shown and not the secondary Reggeon contributions, which are negligible at the value of $x_P = 0.003$ chosen here. The H1 2007 Jets distribution [205] is similar to H1 2006 Fit B.</figcaption></figure>

variables are needed to describe a DDIS event: the fraction $x_P$ of
the proton's momentum transferred across the rapidity gap and $t$, the square of the 4-momentum transfer of the proton. The DDIS data [206, 207] are usually analysed using two levels of factorization. First, the diffractive structure function $F_2^D$ satisfies collinear factorization, and can be expressed as the convolution [208]

$$F_2^D = \sum_{a=q,g} C_2^a \otimes f_{a/p}^D, \qquad (18.32)$$

with the same coefficient functions as in DIS (see Eq. (18.21)), and where the diffractive parton distributions $f_{a/p}^D$ ($a = q, g$) satisfy DGLAP evolution. Second, Regge factorization is assumed [209],

$$f_{a/p}^D(x_{\text{IP}}, t, z, \mu^2) = f_{\text{IP}/p}(x_{\text{IP}}, t) f_{a/\text{IP}}(z, \mu^2), \qquad (18.33)$$

where $f_{a/\text{IP}}$ are the parton densities of the Pomeron, which itself is treated like a hadron, and $z \in [x/x_{\text{IP}}, 1]$ is the fraction of the Pomeron's momentum carried by the parton entering the hard subprocess. The Pomeron flux factor $f_{\text{IP}/p}(x_{\text{IP}}, t)$ is taken from Regge phenomenology. There are also secondary Reggeon contributions to Eq. (18.33). A sample of the $t$-integrated diffractive parton densities, obtained in this way, is shown in Fig. 18.7. A more recent extraction of the parton densities may be found in [210].

Although collinear factorization holds as $\mu^2 \to \infty$, there are non-negligible corrections for finite $\mu^2$ and small $x_{\text{IP}}$. Besides the resolved interactions of the Pomeron, the perturbative QCD Pomeron may also interact directly with the hard subprocess, giving rise to an inhomogeneous evolution equation for the diffractive parton densities analogous to the photon case. The results of the MRW analysis [204], which includes these contributions, are also shown in Fig. 18.7.

Unlike the inclusive case, the diffractive parton densities cannot be directly used to calculate diffractive hadron-hadron cross sections, since account must first be taken of “soft” rescattering effects.

## 18.6 Three-dimensional structure of hadrons

Generalized parton distributions (GPDs) and their complement, transverse momentum dependent distributions (TMDs) describe the three-dimensional structure of hadrons. While GPDs encode the transverse position of a parton in a nucleon, TMDs, encompassing both the parton distributions (TMD PDF) and fragmentation functions (TMD FF), encode their transverse momenta and lead to observable transverse momenta in the final state. Both TMDs and GPDs derive, via integration over the appropriate variable, from Wigner distributions [211–213] that depend on the average transverse momentum and position of partons.

### 18.6.1 Generalized parton distributions

The parton distributions of the proton of Sec. 18.3 are given by the diagonal matrix elements $\langle P, \lambda | \hat{O} | P, \lambda \rangle$, where $P$ and $\lambda$ are the 4-momentum and helicity of the proton, and $\hat{O}$ is a twist-2 quark or gluon operator. However, there is new information in the so-called generalised parton distributions (GPDs) defined in terms of the off-diagonal matrix elements $\langle P', \lambda' | \hat{O} | P, \lambda \rangle$; see Refs. [214–219] for reviews. Unlike the diagonal PDFs, the GPDs cannot be regarded as parton densities, but are to be interpreted as probability amplitudes.

<figure><img src="image_86.png" /><figcaption>Figure 18.8: Schematic diagrams of the three distinct kinematic regions of the imaginary part of $H_q$. The proton and quark momentum fractions refer to $P^+$, and $x$ covers the interval (-1,1). In the ERBL domain the GPDs are generalisations of distribution amplitudes which occur in processes such as $p\bar{p} \to J/\psi$.</figcaption></figure>

The physical significance of GPDs is best seen using light-cone coordinates, $z^\pm = (z^0 \pm z^3)/\sqrt{2}$, and in the light-cone gauge, $A^+ = 0$. It is conventional to define the generalised quark distributions in terms of quark operators at light-like separation

$$ \begin{aligned} F_q(x, \xi, t) &= \frac{1}{2} \int \frac{dz^-}{2\pi} e^{ix\bar{P}^+ z^-} \langle P' | \bar{\psi}(-z/2) \gamma^+ \psi(z/2) | P \rangle \Big|_{z^+=z^1=z^2=0} \\ &= \frac{1}{2\bar{P}^+} \\ & \quad \times \left( H_q(x, \xi, t) \bar{u}(P') \gamma^+ u(P) + E_q(x, \xi, t) \bar{u}(P') \frac{i\sigma^{+\alpha} \Delta_\alpha}{2m} u(P) \right) \end{aligned} \qquad (18.34) $$

with $\bar{P} = (P + P')/2$ and $\Delta = P' - P$, and where we have suppressed the helicity labels of the protons and spinors. We now have two extra kinematic variables:

$$t = \Delta^2, \quad \xi = -\Delta^+/((P + P')^+). \qquad (18.35)$$

We see that $-1 \le \xi \le 1$. Similarly, we may define GPDs $\tilde{H}_q$ and $\tilde{E}_q$ with an additional $\gamma_5$ between the quark operators in Eq. (18.34); and also an analogous set of gluon GPDs, $H_q, E_q, \tilde{H}_q$ and $\tilde{E}_q$. After a Fourier transform with respect to the transverse components of $\Delta$, we are able to describe the spatial distribution of partons in the impact parameter plane in terms of GPDs [220, 221].

For $P' = P$, $\lambda' = \lambda$ the matrix elements reduce to the ordinary PDFs of Sec. 18.2.1

$$H_q(x, 0, 0) = q(x), \quad H_q(-x, 0, 0) = -\bar{q}(x), \quad H_g(x, 0, 0) = xg(x), \qquad (18.36)$$

$$\tilde{H}_q(x, 0, 0) = \Delta q(x), \quad \tilde{H}_q(-x, 0, 0) = \Delta \bar{q}(x), \quad \tilde{H}_g(x, 0, 0) = x \Delta g(x), \qquad (18.37)$$

where $\Delta q = q \uparrow -q \downarrow$ as in Eq. (18.18). No corresponding relations exist for $E, \tilde{E}$ as they decouple in the forward limit, $\Delta = 0$.

The functions $H_q, E_q$ are even in $x$, and $\tilde{H}_q, \tilde{E}_q$ are odd functions of $x$. We can introduce valence and ‘singlet’ quark distributions which are even and odd functions of $x$ respectively. For example

$$H_q^V(x, \xi, t) \equiv H_q(x, \xi, t) + H_q(-x, \xi, t) = H_q^V(-x, \xi, t), \qquad (18.38)$$

$$H_q^S(x, \xi, t) \equiv H_q(x, \xi, t) - H_q(-x, \xi, t) = -H_q^S(-x, \xi, t). \qquad (18.39)$$

All the GPDs satisfy relations of the form

$$H(x, -\xi, t) = H(x, \xi, t) \quad \text{and} \quad H(x, -\xi, t)^* = H(x, \xi, t), \qquad (18.40)$$

and so are real-valued functions. Moreover, the moments of GPDs, that is the $x$ integrals of $x^n H_q$ etc., are polynomials in $\xi$ of order $n+1$. Another important property of GPDs are Ji's sum rule [214]

$$\frac{1}{2} \int_{-1}^{1} dx \, x (H_q(x, \xi, t) + E_q(x, \xi, t)) = J_q(t), \qquad (18.41)$$

where $J_q(0)$ is the total angular momentum carried by quarks and antiquarks of flavour $q$, with a similar relation for gluons.

To visualize the physical content of $H_q$, we Fourier expand $\psi$ and $\bar{\psi}$ in terms of quark, antiquark creation ($b, d$) and annihilation ($b^\dagger, d^\dagger$) operators, and sketch the result in Fig. 18.8. There are two types of domain: (i) the time-like or ‘annihilation’ domain, with $|x| < |\xi|$, where the GPDs describe the wave functions of a $t$-channel $q\bar{q}$ (or gluon) pair and evolve according to modified ERBL equations [222, 223]; (ii) the space-like or ‘scattering’ domain, with $|x| > |\xi|$, where the GPDs generalise the familiar $\bar{q}, q$ (and gluon) PDFs and describe processes such as ‘deeply virtual Compton scattering’ ($\gamma^* p \to \gamma p$), $\gamma p \to J/\psi p$, etc., and evolve according to modified DGLAP equations. The splitting functions for the evolution of GPDs are known to NLO [224–226].
<figure><img src="image_87.png" /></figure>

**Figure 18.9:** (Left) Sivers function in the momentum space for u quark [227] at $x = 0.1$ as a function of transverse momentum $k_T$ for different values of $Q$. The bands are 68%CI. (Right) Momentum space density function as defined in [227] for the unpolarized u quark in a proton totally polarized in the $y$ direction. The white cross indicates the origin with respect to which a shift of the distribution along the $x$-direction due to the Sivers function can be seen. The width of the patterns in the plot roughly indicates the uncertainty of the extraction.

GPDs describe new aspects of proton structure and must be determined from experiment. We can parametrise them in terms of ‘double distributions’ [228,229], which reduce to diagonal PDFs as $\xi \to 0$. Alternatively, flexible SO(3)-based parametrisations have been used to determine GPDs from DVCS data [230,231]; a more recent summary may be found in Ref. [232,233].

### 18.6.2 Transverse momentum dependent distributions

For a proton, there are eight independent transverse dependent distribution (TMD) PDFs, at leading twist, three of which correspond to the usual unpolarized, longitudinally polarized and transversely polarized quark parton distributions [234,235]. The novel TMD PDFs have physical interpretations. For example, the Sivers function [236] represents the distribution of unpolarized partons inside a transversely polarized hadron. For (pseudo)scalar particles, such as kaon and pions, there are two independent leading-twist TMD FFs, one being the ordinary unpolarized fragmentation function and the other the Collins FF [237] which is related to the probability of a polarized quark fragmenting into an unpolarized hadron.

Factorization of TMDs has been shown for semi-inclusive DIS, for the Drell-Yan process as well as for electron-position annihilation into dihadrons [238–243]. Recently first TMD global fits have become available [244–252], although potential problems with consistent descriptions are highlighted in remain [253–255]. The results of a recent extraction of Sivers function from a global fit to polarized Semi-Inclusive DIS, polarized pion-induced Drell-Yan and W+-/Z boson production data is shown in Fig. 18.9 [227].

Because TMD PDFs encode nonperturbative information about transverse momentum and polarization degrees of freedom, they are important for descriptions of multi-scale, non-inclusive collider observables, for example, production of electroweak gauge bosons at LHC [256] and can have an effect on determination of the W boson mass [257]. The combination of TMD PDFs and FFs can give consistent global description of spin and azimuthal asymmetries and provide predictions. Some recent reviews of this rapidly developing field are given here [256,258–261].

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<figure><img src="image_88.png" /><figcaption>**Figure 18.10:** The proton structure function $F_2^p$ measured in electromagnetic scattering of electrons and positrons on protons, and for electrons/positrons (SLAC, HERMES, JLAB) and muons (BCDMS, E665, NMC) on a fixed target. Statistical and systematic errors added in quadrature are shown. The H1+ZEUS combined values are obtained from the measured reduced cross section and converted to $F_2^p$ with a HERAPDF NLO fit, for all measured points where the predicted ratio of $F_2^p$ to reduced cross-section was within 10% of unity. The data are plotted as a function of $Q^2$ in bins of fixed $x$. Some points have been slightly offset in $Q^2$ for clarity. The H1+ZEUS combined binning in $x$ is used in this plot; all other data are rebinned to the $x$ values of these data. For the purpose of plotting, $F_2^p$ has been multiplied by $2^{1-x}$, where $i_x$ is the number of the $x$ bin, ranging from $i_x = 1$ ($x = 0.85$) to $i_x = 26$ ($x = 0.0000085$). Only data with $W^2 > 3.5$ GeV$^2$ is included. Plot from CJ collaboration (Shujie Li - private communication).</figcaption></figure>

References: **H1 and ZEUS**—H. Abramowicz et al., Eur. Phys. J. C75, 580 (2015) (for both data and HERAPDF parameterization); **BCDMS**—A.C. Benvenuti et al., Phys. Lett. B223, 485 (1989) (as given in [260]) E665—M.R. Adams et al., Phys. Rev. D54, 3006 (1996); **NMC**—M. Arneodo et al., Nucl. Phys. B483, 3 (1997); **SLAC**—L.W. Whitlow et al., Phys. Lett. B282, 475 (1992); **HERMES**—A. Airapetian et al., JHEP 1105, 126 (2011); **JLAB**—Y. Liang et al., Jefferson Lab Hall C E94-110 collaboration, nucl-ex/0410027, M.E. Christy et al., Jefferson Lab Hall C E94-110 Collaboration, Phys. Rev. C70, 015206 (2004), S. Malace et al., Jefferson Lab Hall C E00-116 Collaboration, Phys. Rev. C80, 035207 (2009), V. Tvaskis et al., Jefferson Lab Hall C E99-118 Collaboration, Phys. Rev. C81, 055207 (2010), M. Osipenko et al., Jefferson Lab Hall B CLAS6 Collaboration, Phys. Rev. D67, 092001 (2003).
<figure><img src="image_89.png" /><figcaption>**Figure 18.11:** The deuteron structure function $F_2^d$ measured in electromagnetic scattering of electrons/positrons (SLAC, HERMES, JLAB) and muons (BCDMS, E665, NMC) on a fixed target, shown as a function of $Q^2$ for bins of fixed $x$. Statistical and systematic errors added in quadrature are shown. For the purpose of plotting, $F_2^d$ has been multiplied by $2^{1-x}$, where $i_x$ is the number of the $x$ bin, ranging from 1 ($x=0.85$) to 29 ($x=0.00076$). Only data with $W^2 > 3.5$ GeV$^2$ is included. Plot from CJ collaboration (Shujie Li – private communication). References: BCDMS—A.C. Benvenuti et al., Phys. Lett. **B237**, 592 (1990). E665, NMC, SLAC, HERMES—same references as Fig. 18.10; JLAB—S. Malace et al., Jefferson Lab Hall C E00-116 Collaboration, Phys. Rev. **C80**, 035207 (2009), V. Tvaskis et al., Jefferson Lab Hall C E99-118 Collaboration, Phys. Rev. **C81**, 055207 (2010), J. Seely (MIT, LNS) et al., Jefferson Lab Hall C E03-103 Collaboration, Phys. Rev. Lett. **103**, 202301 (2009), M. Osipenko et al., Jefferson Lab Hall B CLAS6 Collaboration, Phys. Rev. **C73**, 045205 (2006).</figcaption></figure>
<figure><img src="image_90.png" /></figure>

**Figure 18.12:** a) The deuteron structure function $F_2$ measured in deep inelastic scattering of muons on a fixed target (NMC) is compared to the structure function $F_2$ from neutrino-iron scattering (CCFR and NuTeV) using $F_2^\mu = (5/18)F_2^\nu - x(s+\bar{s})/6$, where heavy-target effects have been taken into account. The data are shown versus $Q^2$, for bins of fixed $x$. The NMC data have been rebinned to CCFR and NuTeV $x$ values. For the purpose of plotting, a constant $c(x) = 0.05i_x$ is added to $F_2$, where $i_x$ is the number of the $x$ bin, ranging from 0 ($x = 0.75$) to 7 ($x = 0.175$). For $i_x = 8$ ($x = 0.125$) to 11 ($x = 0.015$), $2c(x)$ has been added. References: **NMC**—M. Arneodo et al., Nucl. Phys. **B483**, 3 (1997); **CCFR/NuTeV**—U.K. Yang et al., Phys. Rev. Lett. **86**, 2741 (2001); **NuTeV**—M. Tzanov et al., Phys. Rev. **D74**, 012008 (2006). b) The proton structure function $F_2^p$ mostly at small $x$ and $Q^2$, measured in electromagnetic scattering of electrons and positrons (H1, ZEUS), electrons (SLAC), and muons (BCDMS, NMC) on protons. Lines are ZEUS Regge and HERAPDF parameterizations for lower and higher $Q^2$, respectively. The width of the bins can be up to 10% of the stated $Q^2$. Some points have been slightly offset in $x$ for clarity. The H1+ZEUS combined values for $Q^2 \ge 3.5 \text{ GeV}^2$ are obtained from the measured reduced cross section and converted to $F_2^p$ with a HERAPDF NLO fit, for all measured points where the predicted ratio of $F_2^p$ to reduced cross-section was within 10% of unity. A turn-over is visible in the low-$x$ points at medium $Q^2$ (3.5 $	ext{GeV}^2$ and 6 $	ext{GeV}^2$) for the H1+ZEUS combined values. In order to obtain $F_2^p$ from the measured reduced cross-section, $F_L$ must be estimated; for the points shown, this estimate is obtained from HERAPDF2.0. No $F_L$ value consistent with the HERA data can eliminate the turn-over. This may indicate that at low $x$ and $Q^2$ there are contributions to the structure functions that cannot be described in standard DGLAP evolution.

References: **H1 and ZEUS**—F.D. Aaron et al., JHEP **1001**, 109 (2010) (data for $Q^2 < 3.5 \text{ GeV}^2$), H. Abramowicz et al., Eur. Phys. J. **C75**, 580 (2015) (data for $Q^2 \ge 3.5 \text{ GeV}^2$ and HERAPDF parameterization); **ZEUS**—J. Breitweg et al., Phys. Lett. **B487**, 53 (2000) (ZEUS Regge parameterization); **BCDMS, NMC, SLAC**—same references as Fig. 18.10.

Statistical and systematic errors added in quadrature are shown for both plots.
<figure><img src="image_91.png" /><figcaption>Figure 18.13: a) The charm-quark structure function $F_2^{cc\bar{c}}(x)$, i.e. that part of the inclusive structure function $F_2^p$ arising from the production of charm quarks, measured in electromagnetic scattering of positrons on protons (H1, ZEUS) (the values are obtained from the measured reduced cross section and converted to $F_2^{c\bar{c}}$ using the PDFs from the MMHT NNLO fit) and muons on iron (EMC). For the purpose of plotting, a constant $c(Q) = 0.07 i_Q^{1.7}$ is added to $F_2^{c\bar{c}}$ where $i_Q$ is the number of the $Q^2$ bin, ranging from 1 ($Q^2 = 2.5 \text{ GeV}^2$) to 12 ($Q^2 = 2000 \text{ GeV}^2$). References: **H1 and ZEUS run I +II combination**—H. Abramowicz et al., Eur. Phys. J. C78, 473 (2018); **EMC**—J.J. Aubert et al., Nucl. Phys. B213, 31 (1983).</figcaption></figure>

b) The bottom-quark structure function $F_2^{b\bar{b}}(x)$. For the purpose of plotting, a constant $c(Q) = 0.01 i_Q^{1.6}$ is added to $F_2^{b\bar{b}}$ where $i_Q$ is the number of the $Q^2$ bin, ranging from 1 ($Q^2 = 2.5 \text{ GeV}^2$) to 12 ($Q^2 = 2000 \text{ GeV}^2$). References: **H1 and ZEUS run I combination**—H. Abramowicz et al., Eur. Phys. J. C78, 473 (2018).

For both plots, statistical and systematic errors added in quadrature are shown. The data are given as a function of $x$ in bins of $Q^2$. Points may have been slightly offset in $x$ for clarity. Some data have been rebinned to common $Q^2$ values. Also shown is the MMHT2014 parameterization given at several $Q^2$ values (L. A. Harland-Lang et al., Eur. Phys. J. C75, 204 (2015)).
<figure><img src="image_92.png" /><figcaption>**Figure 18.14:** The structure function $xF_3^{\gamma Z}$ measured in electroweak scattering of a) electrons on protons (H1 and ZEUS) and b) muons on carbon (BCDMS). The line in a) is the HERAPDF parameterization. References: **H1 and ZEUS**—H. Abramowicz *et al.*, Eur. Phys. J. C75, 580 (2015) (for both data and HERAPDF parameterization); **BCDMS**—A. Argento *et al.*, Phys. Lett. B140, 142 (1984).</figcaption></figure>

c) The structure function $xF_3$ of the nucleon measured in ν-Fe scattering. The data are plotted as a function of $Q^2$ in bins of fixed $x$. For the purpose of plotting, a constant $c(x) = 0.5(i_x - 1)$ is added to $xF_3$, where $i_x$ is the number of the $x$ bin as shown in the plot. The NuTeV and CHORUS points have been shifted to the nearest corresponding $x$ bin as given in the plot and slightly offset in $Q^2$ for clarity. References: **CCFR**—W.G. Seligman *et al.*, Phys. Rev. Lett. 79, 1213 (1997); **NuTeV**—M. Tzanov *et al.*, Phys. Rev. D74, 012008 (2006); **CHORUS**—G. Önengüt *et al.*, Phys. Lett. B632, 65 (2006).

Statistical and systematic errors added in quadrature are shown for all plots.
<figure><img src="image_93.png" /><figcaption>**Figure 18.15:** Top panels: The longitudinal structure function $F_L$ as a function of $x$ in bins of fixed $Q^2$ measured on the proton (except for the SLAC data which also contain deuterium data). BCDMS, NMC, and SLAC results are from measurements of $R$ (the ratio of longitudinal to transverse photon absorption cross sections) which are converted to $F_L$ by using the BDCMS parameterization of $F_2$ (A.C. Benvenuti et al., Phys. Lett. B223, 485 (1989)). It is assumed that the $Q^2$ dependence of the fixed-target data is small within a given $Q^2$ bin. Some of the other data may have been rebinned to common $Q^2$ values. Some points have been slightly offset in $x$ for clarity. Also shown is the MSTW2008 parameterization given at three $Q^2$ values (A.D. Martin et al., Eur. Phys. J. C63, 189 (2009)). References: **H1**—V. Andreev et al., Eur. Phys. J. C74, 2814 (2014); **ZEUS**—S. Chekanov et al., Phys. Lett. B682, 8 (2009); H. Abramowicz et al., Phys. Rev. D90, 072002 (2014); BCDMS—A. Benvenuti et al., Phys. Lett. B223, 485 (1989); NMC—M. Arneodo et al., Nucl. Phys. B483, 3 (1997); SLAC—L.W. Whitlow et al., Phys. Lett. B250, 193 (1990) and numerical values from the thesis of L.W. Whitlow (SLAC-357).</figcaption></figure>

Bottom panel: The longitudinal structure function $F_L$ as a function of $Q^2$. Some points have been slightly offset in $Q^2$ for clarity. References: **H1**—V. Andreev et al., Eur. Phys. J. C74, 2814 (2014); **ZEUS**—H. Abramowicz et al., Phys. Rev. D90, 072002 (2014).

The results shown in the bottom plot require the assumption of the validity of the QCD form for the $F_2$ structure function in order to extract $F_L$. Statistical and systematic errors added in quadrature are shown for both plots.
<figure><img src="image_94.png" /><figcaption>**Figure 18.16:** World data on the spin-dependent structure function $g_1^p$ as a function of $Q^2$ for various values of $x$. The lines represent the $Q^2$ dependence for each value of $x$, as determined from a NLO QCD fit. The dashed ranges represent the region with $W^2 < 10$ (GeV/$c^2$)$^2$. References: EMC—J. Ashman *et al.*, Phys. Lett. **B**206, 363 (1988); Nucl. Phys. **B**328, 1 (1989); E143—K. Abe *et al.*, Phys. Rev. **D**58, 112003 (1998); SMC—B. Adeva *et al.*, Phys. Rev. **D**58, 112001 (1998); HERMES—A. Airapetian *et al.*, Phys. Rev. **D**75, 012007 (2007); E155—P.L. Anthony *et al.*, Phys. Lett. **B**493, 19 (2000); COMPASS—M.G. Alekseev *et al.*, Phys. Lett. **B**690, 466 (2010), C. Adolph, *et al.*, Phys. Lett. **B**753, 18 (2016); CLAS—K.V. Dharmawardane *et al.*, Phys. Lett. **B**641, 11 (2006) (which also includes resonance region data not shown on this plot — there is also low $W^2$ CLAS data in Y. Prok *et al.*, Phys. Rev. **C**90, 025212 (2014) and N. Guler *et al.*, Phys. Rev. **C**92, 055201 (2015)).</figcaption></figure>
<figure><img src="image_95.png" /><figcaption>**Figure 18.17:** World data on the spin-dependent structure function $g_1^d$ as a function of $Q^2$ for various values of $x$. The lines represent the $Q^2$ dependence for each value of $x$, as determined from a NLO QCD fit. The dashed ranges represent the region with $W^2 < 10$ (GeV/c$^2$)$^2$. **CLAS**—K.V. Dharmawardane *et al.*, Phys. Lett. **B641**, 11 (2006) **HERMES**—A. Airapetian *et al.*, Phys. Rev. **D75**, 012007 (2007); **SMC**—B. Adeva *et al.*, Phys. Rev. **D58**, 112001 (1998); **E155**—P.L. Anthony *et al.*, Phys. Lett. **B463**, 339 (1999); **E143**—K. Abe *et al.*, Phys. Rev. **D58**, 112003 (1998); **COMPASS**—C. Adolph, *et al.*, Phys. Lett. **B769**, 34 (2017);</figcaption></figure>
<figure><img src="image_96.png" /><figcaption>**Figure 18.18:** The hadronic structure function of the photon $F_2^\gamma$ divided by the fine structure constant $\alpha$ measured in $e^+e^-$ scattering, shown as a function of $Q^2$ for bins of $x$. Data points have been shifted to the nearest corresponding $x$ bin as given in the plot. Some points have been offset in $Q^2$ for clarity. Statistical and systematic errors added in quadrature are shown. For the purpose of plotting, a constant $c(x) = 1.5i_x$ is added to $F_2^\gamma/\alpha$ where $i_x$ is the number of the $x$ bin, ranging from 1 ($x = 0.0055$) to 8 ($x = 0.9$). References: **ALEPH**–R. Barate *et al.*, Phys. Lett. **B458**, 152 (1999); A. Heister *et al.*, Eur. Phys. J. **C30**, 145 (2003); **DELPHI**–P. Abreu *et al.*, Z. Phys. **C69**, 223 (1995); **L3**–M. Acciarri *et al.*, Phys. Lett. **B436**, 403 (1998); M. Acciarri *et al.*, Phys. Lett. **B447**, 147 (1999); M. Acciarri *et al.*, Phys. Lett. **B483**, 373 (2000); **OPAL**–A. Ackerstaff *et al.*, Phys. Lett. **B411**, 387 (1997); A. Ackerstaff *et al.*, Z. Phys. **C74**, 33 (1997); G. Abbiendi *et al.*, Eur. Phys. J. **C18**, 15 (2000); G. Abbiendi *et al.*, Phys. Lett. **B533**, 207 (2002) (note that there is overlap of the data samples in these last two papers); **AMY**–S.K. Sahu *et al.*, Phys. Lett. **B346**, 208 (1995); T. Kojima *et al.*, Phys. Lett. **B400**, 395 (1997); **JADE**–W. Bartel *et al.*, Z. Phys. **C24**, 231 (1984); **PLUTO**–C. Berger *et al.*, Phys. Lett. **142B**, 111 (1984); C. Berger *et al.*, Nucl. Phys. **B281**, 365 (1987); **TASSO**–M. Althoff *et al.*, Z. Phys. **C31**, 527 (1986); **TOPAZ**–K. Muramatsu *et al.*, Phys. Lett. **B332**, 477 (1994); **TPC/Two Gamma**–H. Aihara *et al.*, Z. Phys. **C34**, 1 (1987).</figcaption></figure>
# 19. Fragmentation Functions in $e^{+}e^{-}$, ep, and pp Collisions

Revised August 2023 by O. Biebel (Ludwig-Maximilians U.), D. de Florian (ICAS and ICIFI, UNSAM), D. Milstead (Stockholm U.) and W. Vogelsang (Tübingen U.).

## 19.1 Introduction to fragmentation

Quarks and gluons produced in hard-scattering reactions will ultimately give rise to the colorless hadronic bound states that may be observed in the detector. The associated hadronization process is described by fragmentation functions $D_i^h(x, \mu^2)$ ($i = q, \bar{q}, g$) which are universal functions representing, in the simplest picture, a measure of the probability density that an outgoing parton produces a hadron $h$. Here, $x$ is the fraction of the parton's momentum transferred to the hadron, and $\mu$ is a 'resolution' scale known as factorization scale. The $D_i^h(x, \mu^2)$ may be viewed as the final-state analogs of the initial-state parton distribution functions (PDFs) addressed in Section 18 of this Review. They are also sometimes referred to as timelike distributions since they are primarily accessed in $e^{+}e^{-}$ annihilation via a timelike intermediate boson. (See Refs. [1,2] for introductory reviews, and Refs. [3-5] for summaries of experimental and theoretical research in this field).

The cleanest laboratory for the study of fragmentation functions is provided by semi-inclusive electron-positron annihilation, $e^{+}e^{-} \to \gamma/Z \to h+X$. The cross section for this reaction may be expressed in terms of 'fragmentation structure functions' $F_{T,L,A}$ that are directly related to the fragmentation functions. At center-of-mass (CM) energy $\sqrt{s} = q^2$ we have

$$ \frac{1}{\sigma_0} \frac{d^2 \sigma^h}{dx d \cos \theta} = \frac{3}{8} (1 + \cos^2 \theta) F_T^h(x, q^2) + \frac{3}{4} \sin^2 \theta F_L^h(x, q^2) + \frac{3}{4} \cos \theta F_A^h(x, q^2). \quad (19.1) $$

Here, $q$ is the four-momentum of the intermediate photon or Z-boson, with $q^2 > 0$, and $x = 2P_h \cdot q/q^2$ with the hadron's four-momentum $P_h$ is the fragmentation counterpart of the familiar DIS Bjorken variable. (Note that $x = 2E_h/\sqrt{s} \le 1$ in terms of the energy $E_h$ of the produced hadron in the CM frame of the electron positron pair.) Furthermore, in the same frame, $\theta$ is the hadron's angle relative to the electron beam direction. Eq. (19.1) is the most general form for unpolarized inclusive single-particle production via vector bosons [6]. The fragmentation structure functions $F_T$ and $F_L$ represent the contributions from $\gamma/Z$ polarizations transverse or longitudinal with respect to the direction of motion of the hadron. The parity-violating term with the asymmetric fragmentation function $F_A$ arises from the interference between vector and axial-vector contributions. Various normalization factors $\sigma_0$ are used in the literature, ranging from the total cross section $\sigma_{tot}$ for $e^{+}e^{-} \to$ hadrons, including all weak and QCD contributions, to $\sigma_0 = 4\pi\alpha^2 N_c/3s$ with $N_c = 3$, the lowest-order QED cross section for $e^{+}e^{-} \to \mu^{+}\mu^{-}$ times the number of colors $N_c$. LEP1 measurements of the three fragmentation structure functions are shown in Fig. 19.1.

Integration of Eq. (19.1) over all $\theta$ yields the total fragmentation structure function $F^h = F_T^h + F_L^h$:

$$ \begin{aligned} \frac{1}{\sigma_0} \frac{d\sigma^h}{dx} &= F^h(x, q^2) \\ &= \sum_i \int_x^1 \frac{dz}{z} C_i \left(z, \alpha_s(\mu), \frac{q^2}{\mu^2}\right) D_i^h\left(\frac{x}{z}, \mu^2\right). \end{aligned} \quad (19.2) $$

On the right we have written the factorized expression for the structure function in terms of a sum over convolutions of the fragmentation functions $D_i^h$ for partons $i = u, \bar{u}, d, \bar{d}, \dots, g$ with perturbative coefficient functions $C_i$. Since photons and Z bosons do not distinguish between quarks and antiquarks, $e^{+}e^{-}$ annihilation primarily constrains the combinations $D_i^h + D_i^h$. Gluon fragmentation contributes only at higher order in perturbation theory or by scaling violations. Corrections to the factorized expression in Eq. (19.2) are suppressed by inverse powers of $q^2$. They arise from quark and hadron mass terms and from non-perturbative effects.

<figure><img src="image_97.png" /><figcaption>Figure 19.1: LEP1 measurements of total transverse ($F_T$), longitudinal ($F_L$), and asymmetric ($F_A$) fragmentation structure functions [7]. Data points with relative errors greater than 100% are omitted.</figcaption></figure>

Analogous factorized expressions as in Eq. (19.2) may be written for each of the structure functions $F_{T,L,A}$ individually.

The fragmentation functions obey the momentum sum rule constraint

$$ \sum_h \int_0^1 dx \, x D_i^h(x, \mu^2) = 1, \quad (19.3) $$

separately for each flavor $i$. Note that the sum rule involves a sum over all possible produced hadrons. The dependence of the functions $D_i^h$ on the factorization scale $\mu^2$ will be discussed in the next section.

Measurements of hadron production in deeply-inelastic lepton-proton scattering and hadron-hadron scattering are complementary to those in $e^{+}e^{-}$ annihilation. The former process, $\ell p \to \ell' + h + X$, is known as semi-inclusive deep-inelastic scattering (SIDIS). Here, in analogy with Eq. (19.2), the high virtuality of the photon in DIS also permits factorization of the cross section in terms of fragmentation functions, PDFs for the incoming proton, and perturbative hard-scattering cross sections. Likewise, factorization also occurs for $pp \to h + X$ at large transverse momentum of the produced hadron, and for $pp \to \text{jet}(h) + X$, where the hadron is part of a fully reconstructed jet. The fragmentation functions contributing to $e^{+}e^{-} \to h + X$, $\ell p \to \ell' + h + X$, and $pp \to h + X$, $pp \to \text{jet}(h) + X$ are universal in the sense that the same functions appear in the factorized expressions for the three reactions. Modern QCD analyses of fragmentation functions "globally" take into account experimental data sets for all three types of processes in order to obtain optimal sets of fragmentation functions.

Electron-positron annihilation has the advantage that there is no hadronic initial state and hence no beam remnant. This is in contrast to $\ell p \to \ell' + h + X$ or $pp \to h + X$, which are affected by hadron remnant contributions associated with the partons of the initial-state hadron(s) which are collaterally involved in the hard lepton-parton or parton-parton collision. On the other hand,
<figure><img src="image_98.png" /><figcaption>Figure 19.2: Cross section for $e^{+}e^{-} \to h + X$ for all charged hadrons [8–26], (a) for different CM energies $\sqrt{s}$ versus $x$, and (b) for various ranges of $x$ versus $\sqrt{s}$. For (a) the distributions have been scaled by $c(\sqrt{s}) = 10^i$ with $i$ ranging from $i=0$ ($\sqrt{s}=12$ GeV) to $i=13$ ($\sqrt{s}=202$ GeV).</figcaption></figure>

$e^{+}e^{-} \to h + X$ has little sensitivity to $D_q^h$ and is insensitive to the charge asymmetries $D_q^h - D_{\bar{q}}^h$. These quantities are best constrained in proton–(anti-)proton and electron-proton scattering, respectively. Especially the latter provides an environment that allows the study of the influence of initial-state QCD radiation on the fragmentation process, of the partonic and spin structure of the hadron target, and of the target remnant system. (See Ref. [27] for a comprehensive review of the measurements and models of fragmentation in lepton-hadron scattering).

Moreover, unlike $e^{+}e^{-}$ annihilation where $q^2 = s$ is fixed by the collider energy, lepton-hadron scattering has two independent scales, $Q^2 = -q^2$ and the invariant mass squared, $W^2 \approx Q^2(1-x)/x$, of the hadronic final state, which both can vary by several orders of magnitudes for a given CM energy, thus allowing the study of fragmentation in different environments by a single experiment. For example, in photoproduction the exchanged photon is quasi-real ($Q^2 \approx 0$), leading to processes akin to hadron-hadron scattering. In DIS ($Q^2 \gg 1$ GeV$^2$), using factorization, the hadronic fragments of the struck quark can be directly compared with quark fragmentation in $e^{+}e^{-}$ in a suitable frame. Results from lepton-hadron experiments quoted in this report primarily concern fragmentation in the DIS regime. Studies performed by lepton-hadron experiments of fragmentation with photoproduction data containing high transverse momentum jets or particles are also reported, when these are directly comparable to DIS and $e^{+}e^{-}$ results.

Fragmentation studies in lepton-hadron collisions are usually performed in one of two frames in which the target hadron and the exchanged boson are collinear. The hadronic center-of-mass frame (HCMS) is defined as the rest system of the exchanged boson and incoming hadron, with the $z^*$-axis defined along the direction of the exchanged boson. The positive $z^*$ direction de-

fines the so-called current fragmentation region. Fragmentation measurements performed in the HCMS often use the Feynman-$x$ variable $x_F = 2p_z^*/W$, where $p_z^*$ is the longitudinal momentum of the particle in this frame. As $W$ is the invariant mass of the hadronic final state, $x_F$ ranges between -1 and 1.

The Breit system [28, 29] is related to the HCMS by a longitudinal boost such that the time component of $q$ vanishes, i.e, $q = (0, 0, 0, -Q)$. In the parton model, the struck parton then has the longitudinal momentum $Q/2$ which becomes $-Q/2$ after the collision. As compared with the HCMS, the current fragmentation region of the Breit frame is more closely matched to the partonic scattering process, and is thus appropriate for direct comparisons of fragmentation functions in DIS with those from $e^{+}e^{-}$ annihilation. The variable $x_p = 2p^*/Q$, where $p^*$ is the particle's momentum in the current region of the Breit frame, is used at HERA for measurements in the Breit frame, enabling rather direct comparisons of DIS and $e^{+}e^{-}$ results.

## 19.2 Scaling violations and QCD corrections

As mentioned, the coefficient functions for the fragmentation structure functions in $e^{+}e^{-} \to h + X$ are amenable to QCD perturbation theory. For each of the structure functions $F_{T,L,A}(x, q^2)$ in Eq. (19.1) (and hence for the total structure function $F^h$ in Eq. (19.2)) the coefficient function has an expansion of the form

$$C_{a,i} \left( z, \alpha_s(\mu), \frac{q^2}{\mu^2} \right) = (1 - \delta_{aL}) \delta_{iq} \delta(1 - z) + \frac{\alpha_s(\mu)}{2\pi} c_{a,i}^{(1)} \left( z, \frac{q^2}{\mu^2} \right) + \left( \frac{\alpha_s(\mu)}{2\pi} \right)^2 c_{a,i}^{(2)} \left( z, \frac{q^2}{\mu^2} \right) + .(19,4)$$
<figure><img src="image_99.png" /><figcaption>Figure 19.3: (a) The distribution $1/N \cdot dN/dx_F$ for all charged particles in DIS lepton-hadron experiments at different values of W, measured in the HCMS [30–33]. (b) Scaling violations of the fragmentation structure function for all charged particles in the current region of the Breit frame of DIS [34, 35] and in $e^{+}e^{-}$ interactions [19, 36]. The data are shown as a function of $\sqrt{s}$ for $e^{+}e^{-}$ results, and as a function of Q for the DIS results, each within the same indicated intervals of the scaled momentum $x_p$. The data for the four lowest intervals of $x_p$ are multiplied by factors 50, 10, 5, and 3, respectively for better visibility.</figcaption></figure>

where $a = T, L, A$. At the zeroth order in the strong coupling $\alpha_s$ the coefficient functions $C_g$ for gluons vanish, while for (anti-) quarks $C_i = g_i(s)\delta(1-z)$ (except for $F_L$ for which the leading contribution is of order $\alpha_s$, as indicated in Eq. (19.4)). Here $g_i(s)$ is the appropriate electroweak coupling. In particular, $g_i(s)$ is proportional to the squared charge of the quark $i$ at $s \ll M_Z^2$, when weak effects can be neglected. The full electroweak prefactors $g_i(s)$ can be found in Ref. [6]. The first-order QCD corrections to the coefficient functions have been calculated in Refs. [37,38], and the second-order terms in [39–41]. Thus, the coefficient functions are known to NNLO, except for $F_L$. We note that beyond the leading order the coefficient functions, and hence the fragmentation functions, start to depend on the choice of factorization scheme. The standard choice in the literature is the $\overline{\text{MS}}$ scheme.

The simplest parton-model approach would predict scale-independent (‘scaling’) $x$-distributions for both the structure function $F^h$ and the parton fragmentation functions $D_i^h$. Perturbative QCD corrections lead to logarithmic scaling violations via the evolution equations [42]

$$ \frac{\partial}{\partial \ln \mu^2} D_i^h(x, \mu^2) = \sum_j \int_x^1 \frac{dz}{z} P_{ji}(z, \alpha_s(\mu^2)) D_j^h\left(\frac{x}{z}, \mu^2\right), \quad (19.5) $$

where the functions $P_{ij}(z, \alpha_s(\mu^2))$ describe the splitting process $i \to j + X$, where parton $j$ carries the longitudinal momentum fraction $z$ of parton $i$. Note that for fragmentation the relevant splitting functions are $P_{ji}$ (rather than $P_{ij}$ as for the PDFs) since $D_j^h$ represents the fragmentation of the final parton. Usually the system of evolution equations is decomposed into a $2 \times 2$ flavor-singlet sector comprising the gluon and the sum of all quark and antiquark fragmentation functions, and scalar (‘non-singlet’) equations for quark-antiquark and flavor differences.

The splitting functions in Eq. (19.5) have the perturbative expansion

$$ P_{ji}(z, \alpha_s) = \frac{\alpha_s}{2\pi} P_{ji}^{(0)}(z) + \left(\frac{\alpha_s}{2\pi}\right)^2 P_{ji}^{(1)}(z) + \left(\frac{\alpha_s}{2\pi}\right)^3 P_{ji}^{(2)}(z) + \dots, \quad (19.6) $$

where the leading-order (LO) functions $P^{(0)}(z)$ [42, 43] are the same as those for the initial-state parton distributions. The next-to-leading order (NLO) corrections $P^{(1)}(z)$ have been calculated in Refs. [44–48] (there are well-known misprints in the journal version of Ref. [45]). Ref. [48] also includes the spin-dependent case. The timelike functions are different from, but related to, their spacelike counterparts, see also Ref. [49]. The connections between the two sets of functions has facilitated recent calculations of the next-to-next-to-leading order (NNLO) quantities $P_{qq}^{(2)}(z)$ and $P_{gg}^{(2)}(z)$ in Eq. (19.6) [40,50]. In the same way, the corresponding off-diagonal quantities $P_{qq}^{(2)}$ and $P_{gg}^{(2)}$ were recently obtained in Ref. [51] with the help of constraints from the momentum sum rule Eq. (19.3) [50] and of the limit of $C_A = C_F = n_f$ for which QCD becomes supersymmetric. An uncertainty still remains for the $P_{qq}^{(2)}$ kernel, which however does not affect the logarithmic behavior at small and large momentum fractions. With the exception of Ref. [47], all these higher-order results refer to the standard $\overline{\text{MS}}$ scheme with a fixed number $n_f$ of light flavors. When the threshold for the production of a heavier quark flavor is crossed in the course of the scale evolution, fragmentation functions change. The NLO treatment of these flavor thresholds in the evolution has been addressed in Ref. [52].

The phenomenological effect of scale evolution is similar in the timelike and spacelike cases: As the scale increases, one observes a scaling violation in which the $x$-distribution is shifted towards lower values. This can be seen from Fig. 19.2 where a set of measurements of the total fragmentation structure function in $e^{+}e^{-}$ annihilation are shown. In particular, the figure on the right exhibits the dependence on $\sqrt{q^2} = \sqrt{s}$ at fixed values of $x$. QCD analyses of these data are discussed in Section 19.5 below.

The NLO coefficient functions for SIDIS, $ep \to e+h+X$, have been presented in Refs. [37, 38] Corresponding results have also been obtained for the case that a non-vanishing hadron transverse momentum is required in the HCMS frame [53, 54].

Scaling violations in DIS are shown in Fig. 19.3 for both the HCMS and the Breit frames. In Fig. 1.3(a) the distribution in terms of $x_F = 2p_z^*/W$ shows a steeper slope in ep data than for the lower-energy $\mu p$ data for $x_F > 0.15$, indicating the scaling
violations. At smaller values of $x_F$ in the current jet region, the multiplicity of particles substantially increases with $W$, owing to the increased phase space available for the fragmentation process. The EMC data access both the current region and the region of the fragmenting target remnant system. At higher values of $|x_F|$, due to the extended nature of the remnant, the multiplicity in the target region far exceeds that in the current region. For acceptance reasons the remnant hemisphere of the HCMS is only accessible by the lower-energy fixed-target experiments.

Using hadrons from the current hemisphere in the Breit frame, measurements of fragmentation functions and the production properties of particles in ep scattering have been reported in Refs. [34, 35, 55–58]. Fig. 19.3(b) compares results from ep scattering and $e^{+}e^{-}$ experiments; the latter results have been divided by two as they cover both event hemispheres. The agreement between the DIS and $e^{+}e^{-}$ results is fairly good. However, processes in DIS which are not present in $e^{+}e^{-}$ annihilation, such as boson-gluon fusion and initial-state QCD radiation, can depopulate the current region. These effects become most prominent at low values of $Q$ and $x_p$. Hence, when compared with $e^{+}e^{-}$ annihilation data at $\sqrt{s} = 5.2, 6.5$ GeV [59] not shown here, the DIS particle rates tend to lie below those observed in $e^{+}e^{-}$ annihilation. A ZEUS study [60] finds that the direct comparability of the ep data to $e^{+}e^{-}$ results at low scales is improved if twice the energy in the current hemisphere of the Breit frame, $2E_B^{cr}$, is used instead of $Q/2$ as the fragmentation scale. Choosing $2E_B^{cr}$ for the fragmentation scale approximates QCD radiation effects relevant at low scales, as detailed in Ref. [29].

## 19.3 Fragmentation functions for small particle momenta

The higher-order timelike splitting functions in Eq. (19.6) are singular at small values of $x$. They show a double-logarithmic enhancement, with leading terms of the form $\alpha_s^k(\ln^{2k-2}x)/x$ at the $k$th order of perturbation theory, corresponding to poles $\alpha_s^k(N-1)^{1-2k}$ for the Mellin moments

$$P^{(k)}(N) = \int_{0}^{1} dx \, x^{N-1} P^{(k)}(x). \qquad (19.7)$$

Despite large cancellations between leading and non-leading logarithms at non-asymptotic values of $x$, the resulting small-$x$ rise in the timelike splitting functions dwarfs that of their spacelike counterparts for the evolution of the parton distributions in Section 18 of this Review, see Fig. 1 of Ref. [50]. Consequently, in fragmentation the fixed-order approximation to the evolution breaks down orders of magnitude earlier in $x$ than in DIS.

The pattern of the known coefficients and other considerations suggest that the double-logarithmic terms sum to all-order expressions without any pole at $N=1$, such as [61, 62]

$$P_{gg}^{LL}(N) = -\frac{1}{4} \left( N - 1 - \sqrt{(N-1)^2 \cdot 24 \alpha_s / \pi} \right) \qquad (19.8)$$

for the gluon-to-gluon splitting function at leading logarithmic order. Keeping the first three terms in the resulting expansion of Eq. (19.5) around $N=1$ and taking the Mellin inverse yields a Gaussian in the variable $\xi = \ln(1/x)$ for the small-$x$ fragmentation functions,

$$xD(x, q^2 = s) \propto \exp \left[ -\frac{1}{2\sigma^2} (\xi - \xi_p)^2 \right], \qquad (19.9)$$

with the peak position and width varying with the energy as [63] (see also Ref. [2])

$$\xi_p \approx \frac{1}{4} \ln \left( \frac{s}{\Lambda^2} \right), \quad \sigma \propto \left[ \ln \left( \frac{s}{\Lambda^2} \right) \right]^{3/4} . \qquad (19.10)$$

Next-to-leading logarithmic corrections to the above predictions have been calculated [64]. In the method of Ref. [65], see also Refs. [66, 67], the corrections are included in an analytical form known as the ‘modified leading logarithmic approximation’ (MLLA). Alternatively they can be used to compute higher-moment corrections

to the shape in Eq. (19.9) [68]. The small-$x$ resummation of the coefficient functions for semi-inclusive $e^{+}e^{-}$ annihilation and of the timelike spitting functions in the standard $\overline{\text{MS}}$ scheme was extended in Refs. [69–73] and has reached full next-to-next-to-leading logarithmic accuracy. Applications of these results to gluon and quark jet multiplicities have been presented in Refs. [74].

Fig. 19.4 shows the $\xi$ distribution for charged particles produced in the current region of the Breit frame in DIS and in $e^{+}e^{-}$ annihilation. Consistently with Eq. (19.9) (the ‘hump backed plateau’) and Eq. (19.10) the distributions have a Gaussian shape, with the peak position and area increasing with CM energy ($e^{+}e^{-}$) and $Q^2$ (DIS).

<figure><img src="image_100.png" /><figcaption>Figure 19.4: Distribution of the normalized fragmentation cross sections in $\xi = \ln(1/x_p)$ at several CM energies ($e^{+}e^{-}$) [10, 11, 16–19, 57, 58, 75–78] and for intervals of $Q^2$ (DIS). At each energy only one representative measurement is displayed. For clarity some measurements at intermediate CM energies ($e^{+}e^{-}$) or $Q^2$ ranges (DIS) are not shown. The DIS measurements (*) have been scaled by a factor of 2 for direct comparability with the $e^{+}e^{-}$ results. Fits of simple Gaussian functions are overlaid for illustration.</figcaption></figure>

The predicted energy dependence of the peak in the $\xi$ distribution (see Eq. (19.10)) is explained by soft gluon coherence (angular ordering), i.e., the destructive interference of the color wavefunction of low energy gluon radiation, which correctly predicts the suppression of hadron production at small $x$. Of course, a decrease at very small $x$ is expected on purely kinematical grounds, but this would occur at particle energies proportional to their masses, i.e., at $x \propto m/\sqrt{s}$ and hence $\xi \sim \frac{1}{2} \ln s$. Thus, if the suppression were purely kinematic, the peak position $\xi_p$ would vary twice as rapidly with the energy, which is ruled out by the data in Fig. 19.5. The $e^{+}e^{-}$ and DIS data agree well with each other, demonstrating the universality of hadronization and the MLLA prediction. Measurements of the higher moments of the $\xi$ distribution in $e^{+}e^{-}$ [19, 78–80] and DIS [58] have also been performed and show consistency with each other.

The average charged-particle multiplicity is another observable sensitive to fragmentation functions for small particle momenta. Perturbative predictions using both NLO [89] and MLLA [90, 91] have been obtained by solving Eq. (19.5) yielding

$$\langle n_G(Q^2) \rangle \propto \alpha_s^b(Q^2) \cdot \exp \left[ \frac{c}{4\pi b_0 \sqrt{\alpha_s(Q^2)}} \cdot \left( 1 + 6a_2 \frac{\alpha_s(Q^2)}{\pi} \right) \right], \qquad (19.11)$$

where $b = \frac{1}{4} + \frac{10}{27} \frac{n_f}{4\pi b_0}$, $c = \sqrt{96\pi}$, with $b_0 = (33 - 2n_f)/(12\pi)$,
<figure><img src="image_101.png" /><figcaption>**Figure 19.5:** Evolution of the peak position, $\xi_p$, of the $\xi$ distribution with the CM energy $\sqrt{s}$. The MLLA QCD prediction using $\alpha_s(s = M_Z^2) = 0.118$ is superimposed to the data of Refs. [10, 12, 15, 19, 56, 57, 76, 77, 80–88].</figcaption></figure>

cf. Section 9 of this Review, for $n_f$ contributing quark flavors. Higher-order corrections to Eq. (19.11) are known up to next-to-next-to-next-to-leading order (N$^3$LO), for details and references see [92]. The term proportional to $a_2 \approx -0.502 + 0.0421 n_f - 0.00036 n_f^2$ in Eq. (19.11) is the contribution due to NNLO corrections [93]. The quantity $\langle n_G(Q^2) \rangle$ refers to the average number of gluons, while for $\langle n_q(Q^2) \rangle$ for quarks a correction factor $1/r$ is required due to the different color factors in quark and gluon couplings, so that $\langle n_q(Q^2) \rangle = \langle n_G(Q^2) \rangle/r$. The correction factor depends only weakly on $Q^2$; higher-order corrections up to N$^3$LO on the asymptotic value $r = C_A/C_F = 9/4$ [94] are quoted in [92].

Employing the hypothesis of ‘Local Parton-Hadron Duality’ (LPHD) [90], i.e., that the color charge of partons is balanced locally in phase space and, hence, their hadronization occurs locally such that (Mellin transformed) parton and hadron inclusive distributions directly correspond, Eq. (19.11) can be applied to describe average charged particle multiplicities obtained in $e^{+}e^{-}$ annihilation. The equation can also be applied to $e^{\pm}p$ scattering if the current fragmentation region of the Breit frame is considered for measuring the average charged-particle multiplicity. Fig. 19.6 shows corresponding data and fits of Eq. (19.11) where apart from an LPHD normalization factor a constant offset has been allowed for, so that $\langle n_{ch}(Q) \rangle = K_{LHPD} \cdot \langle n_G(Q) \rangle / r + n_0$.

In hadron-hadron collisions beam remnants, e.g. from single-diffractive (SD) scattering where one colliding proton is negligibly deflected while hadrons related with the other colliding proton are well-separated in rapidity from the former proton, contribute to the measurement of the hadron multiplicity from a hard parton-parton scattering, making interpretation of the data more model dependent. Experimental results are usually given for inelastic processes or for non-single diffractive processes (NSD). Due to the large beam particle momenta at Tevatron and LHC, not all final state particles can be detected within the limited detector acceptance. Therefore, experiments at Tevatron and LHC quote particle multiplicities for limited ranges of pseudo-rapidity $\eta = -\ln \tan(\vartheta/2)$ or at central rapidity, i.e. $\eta = 0$, as shown in Fig. 19.6.

A universality of the average particle multiplicities in $e^{+}e^{-}$ and $p(\bar{p})$ processes has been reported in Ref. [124] when considering an effective collision energy $Q_{\text{eff}} = \sqrt{s}/k$ in $p(\bar{p})$ reduced by a factor of $k \approx 3$, plus a constant offset of $n_0 \approx 2$. A more detailed review is available in Ref. [125]. According to the investigations presented in Ref. [126] the universality of the energy dependence of average particle multiplicities also applies to hadron-hadron and nucleus-nucleus collisions for both full and central rapidity multiplicities. Evidence for this universality is given by the good agreement for the energy dependence of Eq. (19.11) when fit to the $p(\bar{p})$ data as shown in Fig. 19.6.

<figure><img src="image_102.png" /><figcaption>**Figure 19.6:** Average charged-particle multiplicity $\langle n_{ch} \rangle$ as a function of $\sqrt{s}$ or Q for $e^{+}e^{-}$ and $p\bar{p}$ annihilations, and pp and ep collisions. The indicated errors are statistical and systematic uncertainties added in quadrature, except when no systematic uncertainties are given. All NNLO QCD curves are from Eq. (19.11) with fitted normalization, $K_{LHPD}$, and offset, $n_0$, using a fixed $\alpha_s(M_Z^2) = 0.1184$ [95] and for $e^{+}e^{-}$ annihilation data $n_f = 3, 4$, or 5 depending on $\sqrt{s}$, else $n_f = 3$. $e^{+}e^{-}$: Contributions from $K_S^0$ and $\Lambda$ decays included. Data compiled from Refs. [8, 10, 16, 16, 22, 77, 83, 96–106]; $e^{+}p$: Multiplicities have been measured in the current fragmentation region of the Breit frame. Data compiled from Refs. [35, 57, 58, 60, 107]; $p(\bar{p})$: Measured values above 20 GeV refer to non-single diffractive (NSD) processes. Central pseudorapidity multiplicities $(dn/d\eta)|_{|\eta|}|$... refer to either $|\eta| < 2.5$ (CMS: $|\eta| < 2.4$) or $|\eta| = 0$ (UA5, CMS, ALICE: $|\eta| < 0.5$). Data compiled from Refs. [108–123].</figcaption></figure>

## 19.4 Fragmentation models

Although the scaling violations can be calculated perturbatively, the actual form of the parton fragmentation functions is non-perturbative. Perturbative evolution gives rise to a shower of quarks and gluons (partons). Multi-parton final states from leading and higher order matrix element calculations are linked to these parton showers using factorization prescriptions, also called matching schemes, see Ref. [127] for an overview.

Phenomenological schemes are then used to model the carry-over of parton momenta and flavor to the hadrons. Implemented in Monte Carlo event generators (see Section 43 of this Review), these schemes have been tuned using $e^{+}e^{-}$ data and provide good description of hadron collisions as well, thus providing evidence of the universality of fragmentation. However, $e^{+}e^{-}$ mainly fix the quark jet fragmentation while it provides less constraints for modelling the gluon jet fragmentation.

## 19.5 Phenomenology of quark and gluon fragmentation functions

The fragmentation functions are solutions to the evolution equations Eq. (19.5), but need to be specified at some initial scale $\mu_0^2$ (usually around 1 GeV$^2$ for light quarks and gluons, and at $m_Q^2$ for heavy quarks). A typical parameterization for a given light hadron is [128, 131–139]

$$D_i^h(x, \mu_0^2) = N_i x^{\alpha_i} (1-x)^{\beta_i} (1+\gamma_i(1-x))^{\delta_i}, \quad (19.12)$$

where as indicated the normalization $N_i$, and the parameters $\alpha_i$, $\beta_i$, $\gamma_i$ and $\delta_i$ depend on the type i of the fragmenting parton. Heavy flavor fragmentation into heavy mesons is discussed in Sec. 19.8 below. The parameters of Eq. (19.12) are obtained by
<figure><img src="image_103.png" /><figcaption>**Figure 19.7:** Comparison of up, strange, charm and gluon NLO fragmentation functions for $\pi^{+} + \pi^{-}$ at the mass of the Z. The different lines correspond to the results of the analyses performed in Refs. [128–130].</figcaption></figure>

performing global fits to data on various hadron types for different combinations of partons and hadrons in $e^{+}e^{-}$, lepton-hadron and hadron-hadron collisions. We note that the choice of parameterization of the fragmentation functions at the initial scale necessarily introduces a bias since it imposes a certain form of the functions. This bias is largely avoided in neural network approaches which offer a wide flexibility of the initial functions and have recently been applied to fragmentation functions as well [130]. Sets of fragmentation functions are now available for pions, kaons, protons, neutrons, $\eta$ mesons, $\Lambda$ baryons, and charged hadrons [129, 130, 132–144]. They are all at NLO level, except for Refs. [130, 141, 144] which have been performed at NNLO level. The latter sets are restricted to the analysis of $e^{+}e^{-}$ annihilation data. Recently, data from hadron-hadron collisions have been added in the framework of the neural network approach at NLO accuracy for charged hadrons [145]. It is noteworthy that the NNLO effects lead to an improvement in the theoretical description of the data in $e^{+}e^{-}$ annihilation.

Data from $e^{+}e^{-}$ annihilation present the cleanest experimental source for the measurement of fragmentation functions, but cannot be used to disentangle quark from antiquark fragmentation. Since the bulk of the $e^{+}e^{-}$ annihilation data is obtained at the mass of the Z-boson, where the electroweak couplings are roughly the same for the different partons, it provides the most precise determination of the flavor-singlet combination of quark and antiquark fragmentation functions. Flavor-tagged results [146], distinguishing between the light quark, charm and bottom contributions are of particular value for flavor decomposition, even though those measurements cannot be unambiguously interpreted in perturbative QCD.

The most relevant source for quark-antiquark (and also flavor) separation is provided by SIDIS data. Semi-inclusive measurements are usually performed at much lower scales than for $e^{+}e^{-}$ annihilation. The inclusion of SIDIS data in global fits allows for a wider coverage in the evolution of the fragmentation functions, resulting at the same time in a stringent test of the universality of the distributions. Charged-hadron production data in hadronic collisions also have sensitivity to (anti-)quark fragmentation functions. A recent analysis [129] performs a simultaneous extraction of parton distributions and fragmentation functions based on SIDIS data.

The gluon fragmentation function $D_g^h(x)$ can be extracted, in

<figure><img src="image_104.png" /><figcaption>**Figure 19.8:** Scaled momentum spectra of (a) $\pi^{\pm}$, (b) $K^{\pm}$, and (c) $p, \bar{p}$ at $\sqrt{s} = 10, 29$, and 91 GeV [24, 26, 85, 147, 148].</figcaption></figure>
<figure><img src="image_105.png" /><figcaption>Figure 19.9: (a) $1/N \cdot dn/dx_F$ for identified strange particles in DIS at various values of W [149–151]. (b) $1/N \cdot dn/dz$ for measurements of pions in fixed-target DIS experiments [152–154].</figcaption></figure>

<figure><img src="image_106.png" /><figcaption>Figure 19.10: Selection of inclusive (a) $\pi^0$ and (b) charged-hadron production data from pp [118, 155–160] and $p\bar{p}$ [114, 161, 162] collisions.</figcaption></figure>

principle, from the longitudinal fragmentation structure function $F_L$ in Eq. (19.2), as the coefficient functions $C_{L,i}$ for quarks and gluons are comparable at order $\alpha_s$. However at NLO, i.e., including the $O(\alpha_s^2)$ coefficient functions $C_{L,i}^{(2)}$ [39], quark fragmentation is dominant in $F_L$ over a large part of the kinematic range, reducing the sensitivity to $D_g^h$. This distribution could be determined also by analyzing the scale evolution of the fragmentation func-

tions. This possibility is limited by the lack of sufficiently precise data at energy scales away from the Z-resonance and the dominance of the quark contributions at medium and large values of $x$. In $e^{+}e^{-}$ annihilation, $D_g^h$ can also be deduced from the study of three-jet events in which the gluon jet is identified, for example, by tagging the other two jets with heavy quark decays. To leading order, the measured distributions of $x = E_{had}/E_{jet}$ for
<table><caption>**Table 19.1:** Classification of spin- and transverse-momentum dependent quark fragmentation functions. For simplicity we have left out the ubiquitous label for flavor *i* of the fragmenting quark and for hadron species *h*. Each of the functions carries the argument $(x, x^2k_T^2)$ (plus dependence on a factorization scale), where $xk_T = p_T^h$ is the hadron's transverse momentum. $\lambda$ and $\Lambda$ are the quark's and hadron's helicities, respectively, and $\vec{s}_T$ and $\vec{s}_T$ are their transverse spin vectors. We have defined $[\vec{a} \times \vec{b}] = a^1b^2 - a^2b^1$. Finally, $m_h$ is the mass of the produced hadron.</caption><thead><tr><th rowspan="2">hadron pol.</th><th colspan="3">quark polarization</th></tr><tr><th>unpolarized</th><th>long. polarized</th><th>transv. polarized</th></tr></thead><tbody><tr><td>unpol.</td><td>D</td><td>-</td><td>[k<sub>T</sub> &sdot; s<sub>T</sub>]<sub>m<sub>h</sub></sub> H<sup>&perp;</sup></td></tr><tr><td>long. pol.</td><td>-</td><td>&lambda; &Lambda; G<sub>L</sub></td><td>[k<sub>T</sub> &sdot; s<sub>T</sub>]<sub>m<sub>h</sub></sub> &Lambda; H<sub>L</sub><sup>&perp;</sup></td></tr><tr><td>transv. pol.</td><td>[k<sub>T</sub> &sdot; s<sub>T</sub>]<sub>m<sub>h</sub></sub> D<sub>T</sub><sup>&perp;</sup></td><td>k<sub>T</sub> &sdot; s<sub>T</sub><sub>m<sub>h</sub></sub> &lambda; G<sub>T</sub><sup>&perp;</sup> (s<sub>T</sub> &sdot; s<sub>T</sub>) H<sub>T</sub> + [k<sub>T</sub> &sdot; s<sub>T</sub>]<sub>m<sub>h</sub></sub> k<sub>T</sub> &sdot; s<sub>T</sub><sub>m<sub>h</sub></sub> H<sub>T</sub><sup>&perp;</sup></td><td></td></tr></tbody></table>

particles in gluon jets can be identified directly with the gluon fragmentation function $D_g^h(x)$.

Data for $p(\bar{p}) \to h + X$ provide much more direct constraint on $D_g^h$. At variance with $e^+e^-$ annihilation and SIDIS, here gluon fragmentation contributes already at the lowest order in the coupling constant. At large $x \gtrsim 0.5$, where information from $e^+e^-$ is sparse, data from hadronic colliders significantly improve ex- tractions of $D_g^h$ [131, 132, 138, 140]. Recent LHC data has been included in the NLO analyses [128, 138, 139] of pion-fragmentation functions; see Sec. (17.7) for more details. Note that these anal- yses are currently the only ones that ‘globally’ incorporate avail- able data from all sources, $e^+e^- \to h + X$, $ep \to e'h + X$ and $pp \to h + X$.

We note that recently a ‘hybrid’ type of high-$p_T$ jet/hadron observable has also been considered both theoretically [163–169] and experimentally [170–177]. It is defined by an identified spe- cific hadron found inside a fully reconstructed jet. This gives rise to a *same-side* hadron-jet momentum correlation that may be addressed using perturbative methods. One of several rele- vant kinematical variables (see [168] for an overview) is $z_h \equiv (\vec{p}_T^h \cdot \vec{p}_T^{jet})/(p_T^{jet})^2$, where $\vec{p}_T^h$ and $\vec{p}_T^{jet}$ are the transverse mo- menta of the hadron and the jet, respectively. The observable provides an alternative window on fragmentation functions in a more exclusive setting, enabling novel tests of the universality of fragmentation functions. Varying $z_h$ and/or the hadron species, one can map out the fragmentation functions ‘locally’ as func- tions of $x$. This is in contrast to the single-inclusive observable $pp \to h + X$, which inevitably samples over a broad range of $x$. Although hadron-in-jet data are not yet routinely included in analyse of fragmentation functions, a ‘proof-of-principle’ analy- sis does exist [178] that shows the potential of the observable in providing constraint on fragmentation functions.

A comparison of recent NLO fits of fragmentation functions for $\pi^+ + \pi^- \to$ obtained by the updated version of BDSS21 [128], JAF20 [129] and NNPDF1.0 [130] is shown in Fig. 19.7. Differ- ences among the functions for these sets are visible, especially for the gluon fragmentation function over the full range of $x$ and for the quark functions at large momentum fractions. The differences are even larger for other species of hadrons like kaons and protons [131, 132, 136, 140]. Recent analyses [128–130, 136, 138, 139, 179, 180] estimate the uncertainties involved in the extraction of fragmen- tation functions.

Photonic fragmentation functions play a relevant role in the the- oretical understanding of inclusive photon production in (leptonic and hadronic) high energy processes. In the sprit of the anal- ogy between parton fragmentation functions and parton distri- bution functions, also photonic fragmentation functions are anal- ogous to the photon structure function $F_2^\gamma$ and to the proton’s photonic parton distributions (see review on structure functions in Section 18 of this Review). Since photons have a pointlike coupling to quarks [181], the corresponding fragmentation func- tions obey inhomogeneous evolution equations and are generally decomposed into a perturbative and a non-perturbative compo-

nent [135, 182, 183]. The hadronic part, sometimes approximated by the Vector Meson Dominance Model, can in principle be ob- tained by performing a global analysis to the available prompt photon production data [7, 12, 15, 19–21, 85, 147, 184, 185], although in practice this has not been done. We note that also the cross section for photons produced in fully reconstructed jets has been proposed [186] as a new tool for obtaining access to photon frag- mentation functions, in analogy to the hadron-in-jet cross section discussed above.

## 19.6 Identified particles in $e^+e^-$ and semi-inclusive DIS

There is a great wealth of measurements of $e^+e^-$ fragmenta- tion into identified particles. A collection of references for data on fragmentation into identified particles is provided in Table 53.1 of this Review. As a representative example, Figure 19.8 shows dif- ferential charged-hadron spectra as functions of the scaled hadron momentum at several CM energies.

Quantitative results of studies of scaling violations in $e^+e^-$ fragmentation have been reported in [7, 21, 187, 188]. Scaling viola- tions may be used to extract a value of $\alpha_s$; the values obtained are consistent with the world average (see review on QCD in Section 9 of this Review).

Many studies have been made of production of identified particles in lepton-hadron scattering, although fewer particle species have been measured than in $e^+e^-$ collisions. References [149, 150, 152–154, 189–191] and [151, 192–197] are representative of the data from fixed target and ep collider experiments, respec- tively. QCD calculations performed at NLO provide an overall good description of the HERA data [33, 34, 58, 197–199], both for SIDIS [200] and for the hadron transverse momentum distribu- tion [53, 201] in the kinematic regions in which the calculations are predictive. Threshold corrections to SIDIS have been com- puted at NNLO [202] and even at one order higher [203]. Very recently, the full set of NNLO QCD corrections to the correspond- ing coefficient functions were obtained in [204, 205].

Fig. 19.9(a) compares lower-energy fixed-target and HERA data on strangeness production, showing that the HERA spectra have substantially increased multiplicities, albeit with statistical pre- cision that is insufficient to study scaling violations. The fixed- target data show that the $\Lambda$ rate substantially exceeds the $\bar{\Lambda}$ rate in the remnant region, owing to the conserved baryon number from the baryon target. Fig. 19.9(b) shows $1/N \cdot dn/dz$ for neu- tral and charged pion production, where $z$ is defined as the ratio of the pion energy to that of the exchanged boson, both measured in the laboratory frame. Results are shown from the HERMES and the EMC experiments, where the HERMES data have been evolved to $\langle Q^2 \rangle = 25 \text{ GeV}^2$ at NLO QCD, in order to be compa- rable with the EMC data. Each of the experiments uses various kinematic cuts to ensure that the measured particles lie in the region that is expected to be associated with the struck quark. In the DIS kinematic regime accessed at these experiments, and over the range in $z$ shown in Fig. 19.9, the $z$ and $x_F$ variables
<figure><img src="image_107.png" /><figcaption>Figure 19.11: (a) Selection of inclusive charged-particle transverse momentum spectra [114, 117, 119, 121, 161, 206, 207], normalized to the non-single diffractive cross section (NSD). (b)-(d) Selection of identified charged-particle transverse momentum spectra [208–213] normalized to the NSD cross section. All spectra are scaled to the NSD cross-section using measurements of total, inelastic, elastic, single, or non-single diffractive cross sections from [214–218, 218–220]. The overall normalization uncertainty of about 3% is not shown. Superimposed are fits of the Tsallis distribution in Eq. (19.13).</figcaption></figure>

have similar values [30]. The precision data on identified particles can be used in the study of the quark flavor content of the proton [179, 221, 222].

Data on identified particle production can aid the investigation of the universality of jet fragmentation in $e^{+}e^{-}$ and DIS. The strangeness suppression factor $\gamma_s$, as derived principally from tuning the Lund string model [223] within JETSET [224], is typically found to be around 0.3 in $e^{+}e^{-}$ experiments [75], although values closer to 0.2 [225] have also been obtained. A number of measurements of so-called $V^0$-particles ($K^0$, $\Lambda^0$) and the relative rates of $V^0$'s and inclusively produced charged particles have been performed at HERA [151, 192, 226] and fixed target experiments [149]. These typically favour a stronger suppression ($\gamma_s \approx 0.2$) than usually obtained from $e^{+}e^{-}$ data, although values close to 0.3 have also been obtained [227, 228].

However, when comparing the description of QCD-based models for lepton-hadron interactions and $e^{+}e^{-}$ collisions, it is important to note that the overall description by event generators of inclusively produced hadronic final states is more accurate in $e^{+}e^{-}$ collisions than in lepton-hadron interactions [229]. Predictions of particle rates in lepton-hadron scattering are affected by uncertainties in the modelling of the parton composition of the proton and photon, the extended target remnant, and initial and final-state QCD radiation. Furthermore, the tuning of event generators for $e^{+}e^{-}$ collisions is typically based on a larger set of parameters and uses more observables [75] than are used when optimizing models for lepton-hadron data [230].

## 19.7 Fragmentation in hadron-hadron collisions

An extensive set on high-transverse momentum ($p_T$) single-inclusive hadron data has been collected in $h_1h_2 \to hX$ scattering processes, both at high energy colliders and fixed-target experiments [155–162, 185, 231–253]. Fig. 19.10 shows the invariant cross sections $Ed^3\sigma/dp^3$ for a compilation of neutral-pion and charged-hadron production data for energies in the range $\sqrt{s} \approx 23-7000$ GeV.

The differential cross section for high-transverse momentum hadron production has been computed to NLO accuracy in perturbative QCD [254]. The NLO corrections are typically large and can even double the prediction for the cross section at fixed-target energies. Nevertheless, the NLO calculations significantly under-predict the cross-section for several fixed-target energy data sets [249, 255, 256]. Different strategies have been developed to ameliorate the theoretical description at fixed-target energies. A possible phenomenological approach involves the introduction of a non-perturbative intrinsic partonic transverse momentum [160, 249, 257, 258]. Furthermore, the resummation of the dominant higher order corrections at threshold produces an enhancement of the theoretical calculation that significantly improves the description of the data [259, 260].

Data collected at high energy colliders are either included in global fit analyses or used as a test for the universality of fragmentation functions. A certain tension has been observed between data sets from RHIC and the LHC [261]. The tension can be largely resolved [128] by taking properly into account the theoretical scale dependence in the global analysis, allowing factorization and renormalization scales to variate according to the kinematics of each experiment within a conventional range.

Transverse momentum distributions can usually be fit by power laws [262]. An approach to describe the low $p_T$ particle spectra is the Tsallis distribution [263–265], which is based on a non-extensive generalization of the Boltzmann-Gibbs statistics. The functional form [266]

$$ \frac{d^2N}{dp_T dy} = p_T \frac{dN}{dy} \frac{(n-1)(n-2)}{nT(nT + m_0(n-2))} \left[ 1 + \frac{m_T - m_0}{nT} \right]^{-n} \quad (19.13) $$

is frequently used to fit the transverse momentum spectra, where $dN/dy$ is the particle's multiplicity, $T$ and $n$ are fit parameters of the Tsallis distribution, $m_0$ is the either the mass of the most abundant particle, i.e. the pion for inclusive spectra, or the mass of an identified particle, and $m_T = \sqrt{p_T^2 + m_0^2}$. The parameter $n$ is related to the non-extensive parameter $q = n/(n-1)$ of the original Tsallis formula [267], and $T$ is connected to the temperature in the Boltzmann-Gibbs statistics. The Tsallis distribution has been very successfully fit to measured transverse momentum distributions of both inclusive charged particles and identified particle spectra for hadron-hadron collisions, see for example [268–270], for collisions of heavy nuclei, see for example [271], and also for $e^{+}e^{-}$ collisions, see for example [272]. The energy dependence of the fitted Tsallis parameters has also been investigated in detail, see [266, 273]. Fig. 19.11 shows examples of hadron production data in pp and $p\bar{p}$ collisions compared to Tsallis distributions.

Hadron production provides a critical observable for probing the high energy-density matter produced in heavy-ion collisions. Measurements at colliders show a suppression of inclusive hadron yields at high transverse momentum for AA collisions compared to pp scattering, indicating the formation of a dense medium opaque to quark and gluons, see e.g. [274].

## 19.8 Heavy quark fragmentation

It was recognized very early [275] that a heavy flavored meson should retain a large fraction of the momentum of the primordial heavy quark, and therefore its fragmentation function should be much harder than that of a light hadron. In the limit of a very heavy quark, one expects the fragmentation function for a heavy quark to go into any heavy hadron to be peaked near $x = 1$.

When the heavy quark is produced at a momentum much larger than its mass, one expects important perturbative effects, enhanced by powers of the logarithm of the transverse momentum
over the heavy quark mass, to intervene and modify the shape of the fragmentation function. In leading logarithmic order (i.e., including all powers of $\alpha_s \log(m_Q/p_T)$), the total (i.e., summed over all hadron types) perturbative fragmentation function is simply obtained by solving the leading evolution equation for fragmentation functions, Eq. (19.5), with the initial condition due to the finite mass of the heavy quark given by $D_Q(x, \mu^2)|_{\mu^2=m_Q^2} = \delta(1-x)$ and $D_i(x, \mu^2)|_{\mu^2=m_Q^2} = 0$ for $i \neq Q$ (here $D_i(x, \mu^2)$, stands for the probability to produce a heavy quark Q from parton $i$ with a fraction $x$ of the parton momentum).

Several extensions of the leading logarithmic result have appeared in the literature. Next-to-leading-log (NLL) order results for the perturbative heavy quark fragmentation function have been obtained in [279]. The resummation of the dominant logarithmic contributions at large $x$ was performed in [280] to next-to-leading-log accuracy. Fixed-order calculations of the fragmentation function at order $\alpha_s^2$ in $e^+e^-$ annihilation have appeared in [281] while the initial condition for the perturbative heavy quark fragmentation function has been extended to NNLO in [282].

Inclusion of non-perturbative effects in the calculation of the heavy-quark fragmentation function is done by convoluting the perturbative result with a phenomenological non-perturbative form. This form follows from the simple kinematical consideration that the formation of a hadron by attaching light quarks/antiquarks to the heavy quark will slightly decelerate the heavy quark. Thus, its shape will show a peak that becomes increasingly centered next to $x=1$ the higher the quark mass. Among the most popular parameterizations we have the following:

$$ \text{Peterson et al. [283]: } D_{np}(x) \propto \frac{1}{x} \left(1 - \frac{1}{x} - \frac{\epsilon}{1-x}\right)^{-2}, \qquad (19.14) $$

$$ \text{Kartvelishvili et al. [284]: } D_{np}(x) \propto x^{\alpha}(1-x), \qquad (19.15) $$

$$ \text{Collins \& Spiller [285]: } D_{np}(x) \propto \left( \frac{1-x}{x} + \frac{(2-x)\epsilon_C}{1-x} \right) \times (1+x^2) \times \left( 1 - \frac{1}{x} - \frac{\epsilon_C}{1-x} \right)^{-2} \qquad (19.16) $$

$$ \text{Colangelo \& Nason [286]: } D_{np}(x) \propto (1-x)^{\alpha} x^{\beta} \qquad (19.17) $$

$$ \text{Bowler [287]: } D_{np}(x) \propto x^{-(1+bm_{h,\perp}^2)} \times (1-x)^a \exp\left(-\frac{bm_{h,\perp}^2}{x}\right) \qquad (19.18) $$

$$ \text{Braaten et al. [288]: } (\text{see Eqs. (31), (32) in [288]}) \qquad (19.19) $$

where $\epsilon$, $\epsilon_C$, $a$, $bm_{h,\perp}^2$, $\alpha$, and $\beta$ are non-perturbative parameters that depend on the heavy hadron considered. The parameters entering the non-perturbative forms are fitted together with some model of hard radiation, which can be either a shower Monte Carlo, a leading-log or NLL calculation (which may or may not include Sudakov resummation), or a fixed order calculation. In [281], for example, the Peterson et al. [283] $\epsilon$ parameter for charm and bottom production is fitted from the measured distributions of Refs. [289, 290] for charm, and of [291] for bottom. If the leading-logarithmic approximation (LLA) is used for the perturbative part, one finds $\epsilon_c \approx 0.05$ and $\epsilon_b \approx 0.006$; if a second order calculation is used one finds $\epsilon_c \approx 0.035$ and $\epsilon_b \approx 0.0033$; if a NLL improved fixed order $O(\alpha_s^2)$ calculation is used instead of NLO $O(\alpha_s)$ one finds $\epsilon_c \approx 0.022$ and $\epsilon_b \approx 0.0023$. The larger values found in the LL approximation are consistent with what is obtained in the context of parton shower models [292], as expected. The $\epsilon$ parameter for charm and bottom scales roughly with the inverse square of the heavy flavor mass. This behavior can be justified by several arguments [275, 293, 294]. It can be used to relate the non-perturbative parts of the fragmentation

functions of charm and bottom quarks [281, 286, 295].

A more conventional approach [296] involves the introduction of a unique set of heavy quark fragmentation functions of non-perturbative nature that obey the usual massless evolution equations in Eq. (19.5). Finite mass terms of the form $(m_Q/p_T)^n$ are kept in the corresponding short distance coefficient function for each scattering process. Within this approach, the initial condition for the perturbative fragmentation function provides the term needed to define the correct subtraction scheme to match the massless limit for the coefficient function (see e.g. [297]). Such an implementation is in line with the variable flavor number scheme introduced for parton distributions functions, as described in Section 18 of this Review.

High statistics data for charmed-meson production near the $\Upsilon$ resonance (excluding decay products of B mesons) have been published [276, 277]. They include results for D and $D^*$, $D_s$ (see also [298, 299]) and $\Lambda_c$. Shown in Fig. 19.12(a) are the CLEO and BELLE inclusive cross-sections times branching ratio $B$, $sBd\sigma/dx_p$, for the production of $D^0$ and $D^{*+}$. The variable $x_p$ approximates the light-cone momentum fraction $x$, but is not identical to it. The two measurements are consistent with each other.

The branching ratio $B$ represents $D^0 \to K^-\pi^+$ for the $D^0$ results and for the $D^{*+}$ the product of the branching fractions for $D^{*+} \to D^0\pi^+$ and $D^0 \to K^-\pi^+$. Given the high precision of CLEO's and BELLE's data, a superposition of different parametric forms for the non-perturbative contribution is needed to obtain a good fit [52]. Older studies are reported in Refs. [290, 300, 301]. Charmed meson spectra on the $Z$ peak have been published by OPAL and ALEPH [302, 303].

Charm quark production has also been extensively studied at HERA by the H1 and ZEUS collaborations. Measurements have been made of $D^{*\pm}$, $D^\pm$, and $D_s^\pm$ mesons and the $\Lambda_c$ baryon. See, for example, Refs. [304, 305].

Experimental studies of the fragmentation function for b quarks, shown in Fig. 19.12(b), have been performed at LEP and SLD [278, 291, 306]. Commonly used methods identify the B meson through its semileptonic decay or based upon tracks emerging from the B secondary vertex. Heavy flavor contributions from gluon splitting are usually explicitly removed before fitting for the fragmentation functions. The studies in [278] fit the B spectrum using a Monte Carlo shower model supplemented with non-perturbative fragmentation functions yielding consistent results.

The experiments measure primarily the spectrum of B mesons. This defines a fragmentation function that includes the effect of the decay of higher mass excitations, like the $B^*$ and $B^{**}$. In the literature (cf. details in Ref. [307]), there is sometimes ambiguity in what is defined to be the bottom fragmentation function. Instead of using what is directly measured (i.e., the B meson spectrum), in some cases corrections are applied to account for $B^*$ or $B^{**}$ production.

Heavy-flavor production in $e^+e^-$ collisions is the primary source of information for the role of fragmentation effects in heavy-flavor production in hadron-hadron and lepton-hadron collisions. The QCD calculations tend to underestimate the data in certain regions of phase space. Some experimental results from LHC summarized in [308] show such deviations e.g. at high transverse jet momentum and also at low di-jet separation angles, see [309] for details, and were already theoretically investigated in [310].

Both bottomed- and charmed-meson spectra have been measured at the Tevatron with unprecedented accuracy [311]. The measured spectra are in good agreement with QCD calculations (including non-perturbative fragmentation effects inferred from $e^+e^-$ data [312]).

The HERA collaborations have produced a number of measurements of beauty production; see, for example, Refs. [304, 313–316]. As for the Tevatron data, the HERA results are described well by QCD-based calculations using fragmentation models optimized with $e^+e^-$ data.

Besides degrading the fragmentation function by gluon radiation, QCD evolution can also generate soft heavy quarks, increasing in the small $x$ region as $\sqrt{s}$ increases. Several theoretical studies are available on the issue of how often $b\bar{b}$ or $c\bar{c}$ pairs are
<figure><img src="image_108.png" /><figcaption>Figure 19.12: (a) Efficiency-corrected inclusive cross-section measurements for the production of $D^0$ and $D^{*+}$ in $e^{+}e^{-}$ measurements at $\sqrt{s} \approx 10.6$ GeV, excluding B decay products [276] [277]. (b) Measured $e^{+}e^{-}$ fragmentation function of b quarks into B hadrons at $\sqrt{s} \approx 91$ GeV [278].</figcaption></figure>

produced indirectly via a gluon splitting mechanism [317–319]. Experimental results from studies on charm and bottom production via gluon splitting, given in [303, 320–324], yield weighted averages of $\bar{n}_{g \to c\bar{c}} = 3.05 \pm 0.45\%$ and $\bar{n}_{g \to b\bar{b}} = 0.277 \pm 0.072\%$, respectively. The production of bottom-antibottom quark pairs via gluon splitting has also been investigated at hadron colliders, see for example [325–327].

## 19.9 Spin-dependent and transverse-momentum dependent fragmentation functions

The fragmentation functions we have considered so far apply to the spin-averaged case in which the polarization of the produced hadron is not observed, or the hadron has spin-0. We have also only considered ‘collinear’ fragmentation functions $D_i^h(x, \mu^2)$ which carry only one kinematical variable, the momentum fraction $x$. New insights into fragmentation and hadronization become available when also the dependence of fragmentation functions on the spin of the produced hadron and/or its relative transverse momentum with respect to the fragmenting parton are considered. In the latter case, one refers to the fragmentation functions as ‘transverse-momentum dependent (TMD)’ fragmentation functions.

Staying first with collinear fragmentation functions, two types of spin-dependent fragmentation functions to spin-1/2 hadrons can be considered. The helicity-dependent fragmentation function measures the transfer of longitudinal spin from the fragmenting parton to the hadron [38, 328–331]. It is given by

$$ \Delta D_i^h(x, \mu^2) \equiv D_{i+}^{h+}(x, \mu^2) - D_{i+}^{h-}(x, \mu^2), \qquad (19.20) $$

where the superscripts $\pm$ refer to the helicities of the parton and hadron. $\Lambda$ hyperons are ideally suited for measurements of the $\Delta D_i^h$, thanks to their self-analyzing weak decay $\Lambda \to \pi p$. Measurements of the longitudinal spin transfer to Lambda hyperons have been presented in $e^{+}e^{-}$ (on the Z resonance), $\ell p$, and pp scattering in Refs. [332–338]. One may readily extend Eq. (19.20) to the case of transverse polarization of hadrons and quarks [339], where the corresponding fragmentation functions are known as ‘transversity’ fragmentation functions. There are also measurements constraining these fragmentation functions [333, 340, 341]. A recent phenomenological analysis was presented in [342].

If the transverse-momentum ($k_T$) dependence of fragmentation functions is considered, there are eight types of leading-twist functions, defined by the correlations among the hadronic and partonic spin vectors and transverse-momentum vectors they represent. (For review, see [5]). We note that the eight fragmentation functions given in the table below exist separately for each quark and antiquark flavor, and a similar set may be introduced for gluons. Upon integration over the transverse momentum $k_T$ the collinear unpolarized, helicity, and transversity fragmentation functions are

reproduced.

The various fragmentation functions may be obtained from spin asymmetries and angular distributions in hadron production processes. There is a large body of precision data by now on transverse-momentum distributions in $e^{+}e^{-}$ annihilation [343] and SIDIS [190, 344] that provide constraints on the unpolarized TMD fragmentation functions $D_i^h$, which have been analyzed theoretically, partly also including TMD evolution effects and high orders of perturbation theory [345–352].

Besides the unpolarized functions $D$ most of the attention in experiment and theory has been on the function $H^\perp$ which describes the production of unpolarized (or spin-0) hadrons by transversely polarized quarks. This function is known as the ‘Collins function’ [353]. Its importance also derives from the fact that it may be used to probe the quark transversity PDF of the nucleon [354] which gives the probability of finding a transversely polarized quark with its spin aligned or anti-aligned with the spin of a transversely polarized nucleon. The transversity function is chiral-odd, and therefore not accessible through measurements of inclusive lepton-hadron scattering. The Collins effect in semi-inclusive DIS, on the other hand, provides an avenue for accessing transversity. The Collins fragmentation function is chiral-odd and T-odd, leading to a characteristic single-spin asymmetry in the azimuthal angular distribution of the produced hadron in the hadron scattering plane. A number of SIDIS [355–366] and $e^{+}e^{-}$ experiments [367–371] have performed measurements of the Collins effect, for charged pions and kaons. These have been analyzed theoretically [372, 373], leading to an extraction of the nucleon’s transversity distributions [373]. The Collins effect has also been studied in pp scattering, where one considers azimuthal transverse single-spin asymmetries for distributions of hadrons inside jets [177, 374, 375].

In the context of extractions of transversity PDFs also fragmentation functions for same-side pairs of hadrons with small invariant mass, dihadrons, have been introduced and studied [376–384]. Compared to the Collins effect, dihadron fragmentation functions have the advantage that they may be defined purely in collinear factorization. The relevant spin-dependent dihadron fragmentation function exploits a correlation between the transverse polarization of the fragmenting quark and the relative momentum of the two hadrons. In SIDIS with a transversely polarized hadron beam, the dihadron cross section then contains a specific modulation in the azimuthal orientation of the plane containing the momenta of the two hadrons. The coefficient of this modulation is a product of the spin-dependent dihadron fragmentation function and the target’s transversity PDF. The dihadron fragmentation functions may be separately extracted from measurements in $e^{+}e^{-}$ annihilation, and the Belle experiment has presented data [385] that have been analyzed theoretically [386, 387].
In lepton scattering, HERMES [388] and COMPASS [389, 390] have reported data sensitive to the spin-dependent dihadron fragmentation functions, and recently the STAR experiment at RHIC has presented data in the azimuthal distribution of $\pi^{+}\pi^{-}$ pairs produced in pp scattering with one transversely polarized proton [391]. The results have been successfully used for the extraction of transversity PDFs [387, 392–394].

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## 20. High Energy Soft QCD and Diffraction

Revised August 2023 by V.A. Khoze (Durham U.), M.G. Ryskin (Petersburg Nuclear Phys. Inst.) and M. Taševský (Prague, Inst. Phys.).

### 20.1 Introduction

Despite the enormous successes of Quantum Chromodynamics (QCD) (see Section 9 in [1] and [2]) there remain a number of deep questions to be answered in the domain of strong interaction physics. These concern first of all small momentum transfer processes which are generically called soft interactions.

One of the most challenging problems is the high-energy behaviour of hadronic scattering processes. At high collision energies, $\sqrt{s}$, soft interactions play a dominant role. Unfortunately, soft interactions cannot be described in terms of perturbative QCD. These are non-perturbative phenomena related to confinement which are generally considered in the context of the analytic $S$-matrix, based on first principles, such as analyticity, crossing symmetry and unitarity of partial waves, see e.g. [3, 4]. At high energies the most self-consistent way to perform the calculations and to describe the data is the Regge approach (see for example [5–7]), which will be considered below. As discussed in Section 20.5, this formalism could be smoothly matched with perturbative QCD calculations at larger transverse momenta. Therefore, here we will concentrate on the properties of high energy soft interactions that can be expected from the extension of the perturbative QCD domain.

The main aim of this review is to present the well-established theoretical framework, based on Regge theory and QCD, used for describing high-energy collisions. A limited number of some new experimental results, mainly from the LHC, are shown in order to demonstrate that the gross features of the data are in agreement with this approach. We are not focussing on any particular phenomenological or Monte Carlo model, which are covered in the dedicated reviews and books, see e.g. Section 43 in [1], [2, 8–14] and Chapter 2 in [15].

Typically, in multiparticle production, the secondaries<fnref n="1" /> fill the whole available rapidity interval.<fnref n="2" /> However, there exists an important class of events in which a large interval of rapidity (typically at least 4 units) is devoid of any hadronic activity. Such an interval is called a Large Rapidity Gap (LRG). The most frequent case with a LRG is elastic scattering. There are also events in which one of the incoming protons (or both) is transformed (dissociates) into a set of two or more final state particles with the mass $M \ll \sqrt{s}$ and proton quantum number. All these events have properties similar to those of the well-known from optics pattern of diffraction of a beam of light on an obstacle. By analogy, in high-energy physics, the corresponding processes are usually called diffractive. The classic example is the elastic scattering of hadrons on nuclei (see e.g. [16]), which manifests an angular distribution with a series of minima and maxima, analogous to the diffraction of light on a black disk. At LHC energies diffractive processes constitute up to 40% of the total (pp) cross section, $\sigma_{tot}$. Therefore, we will pay special attention to the description of the elastic scattering amplitude and proton diffractive dissociation. Diffraction dissociation can be considered as a quantum mechanical process caused by the fact that different components of the incoming hadron wave function have different probabilities for interaction with a target [17]. This feature allows us to probe the transverse size of the interaction region.

Note that besides being of a fundamental interest in their own right for understanding the high energy behaviour of the QCD amplitude, there are several reasons why it is important to study soft and diffractive processes. Firstly, soft interactions unavoidably give an underlying component to rare ‘hard’ events, from which we hope to extract signals for New Physics. Secondly, we should be able to estimate the probability that rapidity gaps, which occur

in ‘hard’ diffractive events, survive rescattering effects, that is, survive the population of the gaps by the secondary particles from the underlying event. Thirdly, an understanding of diffractive processes is very important for evaluation of pile-up backgrounds in high-luminosity pp collisions, which have a direct impact on various experimental measurements. Pile-up corresponds to soft independent interactions in the same bunch crossing whose number rises with increasing instantaneous luminosity. And, finally, studies of diffractive processes should help in the understanding of the structure of high-energy cosmic ray cascades, which requires a very detailed knowledge of the spectra of particles carrying a large fraction $x$ of the incoming momentum in proton-air and nucleus-air interactions, see for instance [18].

Experimentally, diffractive processes are selected using two distinct features:

1. large regions (typically at least $\Delta\eta > 4$) in the detector are devoid of hadronic activity (LRG) and/or

2. one or both incoming particles stay intact after collision and are registered by the dedicated forward detectors placed a few hundred meters from the interaction point. The momentum loss of the initial particle, $\xi = 1 - x$, is typically smaller than 0.15.

Thus, in the case of proton-proton collisions, diffractive events correspond to elastic $pp \to pp$ scattering and to $pp \to p + X$ (Single Dissociation, SD) and $pp \to X + Y$ (Double Dissociation, DD) processes, where the + sign denotes a large rapidity gap. Note that strictly speaking in high energy physics it is impossible to define (and select) rigorously purely diffractive events. We can always have some admixture of events of different origin. As a rule we call ‘diffractive’ the events with sufficiently large gap (with say $\Delta y > 4$, see above) and the vacuum quantum numbers transferred across the gap. Typically at the LHC the integrated cross sections of diffractive dissociation, $\sigma_{SD}$, $\sigma_{DD}$, are of the order of 5–10 mb depending on the gap size. Schematic diagrams of all discussed processes are shown in Fig. 20.1.

### 20.2 Regge pole approach

In pre-QCD times, in order to describe the behaviour of scattering amplitudes at high energy, $\sqrt{s}$, and small momentum-transfer squared, $-t$, Regge theory was developed and successfully applied in a wide range of energies. The Regge approach [5–7] is based on the singularities of amplitudes in the complex angular momentum, $j$, plane.

For instance, the measured $\pi^- p \to \pi^0 n$ amplitude behaves as

$$T_{\pi p}(s, t) \propto s^{\alpha_{\rho}(t)}, \qquad (20.1)$$

where the process is described by the exchange of the $\rho$-trajectory, $j = \alpha_{\rho}(t) \simeq 0.5 + 0.9t$ (with $t = (p_{\pi^-} - p_{\pi^0})^2$ in GeV$^2$). This trajectory passes through the spin-1 $\rho$-meson resonance in the ‘crossed’ $t$-channel $\pi^- \pi^0 \to \bar{p}n$; that is, $\alpha_{\rho}(t = m_{\rho}^2) = 1$. The corresponding cross section decreases with increasing $s$.

On the other hand, high-energy total and elastic $pp$ cross sections are observed to grow slowly with energy (see e.g. Section 53 in [1]) and in terms of Regge theory are dominated by the exchange of a trajectory with vacuum quantum numbers, $\sigma_{tot} \propto s^{j-1}$. The simplest possibility is to assume that the rightmost singularity in the $j$-plane, which drives the high-energy behaviour of the cross section, is the leading (at $t \le 0$) Regge pole at $j = \alpha(t)$. Then the $pp$ elastic amplitude reads

$$T_{\text{el}}(s, t) \propto s^{\alpha_{\text{P}}(t)}. \qquad (20.2)$$

The total cross section can then be conveniently expressed using the so called optical theorem which states that

$$s\sigma_{\text{tot}} = \text{Im}T_{\text{el}}(s, t = 0), \qquad (20.3)$$

as illustrated in the upper part of Fig. 20.2, and thus

$$\sigma_{\text{tot}} \propto s^{\alpha_{\text{P}}(0)-1}. \qquad (20.4)$$

<footnote n="1">¹Here and in what follows, we call secondaries the new particles produced in the course of the interaction.</footnote>

<footnote n="2">²For definition of particle rapidity (pseudorapidity), see Section 49.5.2 in [1]; $y = \frac{1}{2} \ln \frac{E+p_z}{E-p_z}$ ($\eta = -\ln(\tan(\theta/2))$; the correct variable is the rapidity $y$, however, experimentally it is simpler to use the pseudorapidity $\eta$ which does not require identifying the particles, setting $m = 0$. For $p_T \gg m$, $\eta \simeq y$.</footnote>
<figure><img src="image_109.png" /><figcaption>**Figure 20.1:** Schematic diagrams of soft pp processes. (a) non-diffractive processes, (b) elastic scattering, (c) single dissociation and (d) double dissociation. The double line corresponds to the Pomeron exchange.</figcaption></figure>

<figure><img src="image_110.png" /><figcaption>**Figure 20.2:** Illustration of the optical theorem for the total cross section and for high-mass diffractive dissociation in the absence of absorptive corrections.</figcaption></figure>

The pole with the largest intercept, originally assumed to be $\alpha_P(0) = 1$ since high-energy total cross sections were thought to have a constant asymptotic behaviour, is called the Pomeron<fnref n="3" />

Prior to the LHC, the energy behaviour of $pp, p\bar{p}, \pi p, Kp$ cross sections was satisfactorily reproduced by the sum of the Pomeron and secondary Reggeons (the poles at lower values of $j$, typically with $\alpha_P(0) \simeq 0.5$, see [22, 23] and Section 51 in [24]). However, above Tevatron energies the secondary Reggeon contributions (which all have intercepts $\alpha(0) \simeq 0.5$) are highly suppressed, which enables us to study the properties of the Pomeron only.

A popular parameterization of the elastic pp-scattering amplitude by Donnachie-Landshoff (DL) is the Regge form [25]

$$T_{\text{el}}(s, t) = \eta_P \sigma_0 F_1^2(t) s^{\alpha_P(t)}, \quad (20.5)$$

where $\sigma_0 = 21.7$ mb [26] and $\eta_P$ is the signature factor

$$\eta_P = \frac{1 + \exp(-i\pi\alpha_P(t))}{\sin(-\pi\alpha_P(t))}, \quad (20.6)$$

$F_1$ is the Dirac electromagnetic form factor of the proton and the effective Pomeron trajectory

$$\alpha_P(t) = 1 + \Delta + \alpha' t \simeq 1 + 0.0808 + 0.25t, \quad (20.7)$$

with $t$ given in GeV². The intercept $\alpha_P(0)$ just above 1 reproduces the observed slow growth of the total hadron-hadron cross sections at high energies.

However, this simple parameterization is becoming increasingly deficient at higher energies. This is because due to unitarity we have to take into account not only Regge poles, but also the cuts in the $j$-plane [27, 28], which correspond to the multiple exchange of Regge poles in the $t$-channel, see for instance [29–31]. A powerful technique to evaluate Reggeon diagrams was developed by Gribov [7, 32] (Reggeon calculus or Reggeon Field Theory (RFT)), which allows us to calculate the multi-Pomeron contributions.

## 20.3 Theoretical description of high-energy diffraction

Diffractive processes (see e.g. reviews [33–37]) represent a rich testing ground for the dynamics of soft interactions as well as Monte Carlo models for soft hadron-hadron physics (see for reviews e.g. [8], Section 43 in [1] and Chapter 2 in [15]).

There is no universally agreed definition of diffractive processes. Theoretically, diffraction is the effect caused by the absorption of the incoming plane-wave in some region of impact parameter, $b$. After a decomposition of the distorted plane-wave over the outgoing momentum, $q$, due to absorption we arrive at some set of plane-waves with non-zero transverse momentum, $q_t \neq 0$. Experimentally, we call diffractive the events with large rapidity gaps (LRG) in the distribution of the final state particles. However, this definition is appropriate only for the events with very large gap sizes ($\Delta\eta > 4-5$); otherwise gaps can also be caused by fluctuations in the hadronization process [38].

In the case of proton-proton collisions, diffraction corresponds to elastic $pp \to pp$ scattering and to the $pp \to p + X$ and $pp \to X_1 + X_2$ processes where one or both protons are allowed to dissociate into a system $X$ with the quantum numbers of the proton. The $p \to X$ dissociation is caused by the fact that the individual components of the incoming proton wave function interact differently with the target (see Section 20.3.1).

Theoretically, high-energy diffraction may be studied from either the $s$-channel or the $t$-channel viewpoint.

### 20.3.1 Diffraction from the $s$-channel viewpoint

Unitarity plays a central role in diffractive processes. To discuss unitarity effects it is convenient to work in terms of impact parameter, $b$. The total cross section is closely related to the elastic scattering amplitude and the scattering into inelastic final states via the $s$-channel unitarity of the $S$-matrix (see Sections 50 and 53 in [1]), $SS^\dagger = I$, or

$$\mathrm{disc} \, T \equiv T - T^{\dagger} = i T^{\dagger} T \quad (20.8)$$

<footnote n="3">³Pomeron pole was named after I. Y. Pomeranchuk. The history of the Pomeron is discussed in [19–21].</footnote>
<figure><img src="image_111.png" /><figcaption>**Figure 20.3:** Two-Pomeron exchange in the *t* channel expressed as a sum over all diffractive intermediate states in the *s*-channel. The crosses indicate that the particles are on the mass shell.</figcaption></figure>

with $S = I + iT$. If we were to focus, for example, on the unitarity for elastic and quasielastic processes, then *disc T* would simply denote a cut in *s*-channel between incoming and outgoing particles as visualized by crosses in Fig. 20.3.

At high energies, the *s*-channel unitarity relation is diagonal in the *b* basis such that

$$2\operatorname{Im} T_{\text{el}}(s, b) = |T_{\text{el}}(s, b)|^2 + G_{\text{inel}}(s, b) \quad (20.9)$$

with

$$\sigma_{\text{tot}} = 2 \int d^2 b \operatorname{Im} T_{\text{el}}(s, b) \quad (20.10)$$

$$\sigma_{\text{el}} = \int d^2 b |T_{\text{el}}(s, b)|^2 \quad (20.11)$$

$$\sigma_{\text{inel}} = \int d^2 b [2 \operatorname{Im} T_{\text{el}}(s, b) - |T_{\text{el}}(s, b)|^2] . \quad (20.12)$$

The general solution of Eq. (20.9) is

$$T_{\text{el}}(b) = i(1 - e^{-\Omega(b)/2}) \quad (20.13)$$

and

$$G_{\text{inel}}(s, b) = 1 - e^{-\text{Re}\Omega(b)} = 1 - P_{\text{nointer}}(s, b), \quad (20.14)$$

where $G_{\text{inel}}$ is the sum over all inelastic intermediate states and $P_{\text{nointer}}$ is a probability to have no inelastic interactions. $G_{\text{inel}}(s, b)$ describes the *b*-profile of inelastic particle collisions. It satisfies the condition $0 \le G_{\text{inel}} \le 1$ and determines how absorptive the interaction region is at a given impact parameter (with $G_{\text{inel}}=1$ for full absorption and $G_{\text{inel}}=0$ for the complete dominance of elastic scattering). As seen from Eq. (20.14), $\exp(-\text{Re}\Omega(s, b))$ is the probability that no inelastic interactions occur at impact parameter $b$. $\Omega$ ($\text{Re}\Omega \ge 0$) is called the opacity (optical density) or eikonal. The quantity

$$S^2(b) \equiv e^{-\text{Re}\Omega(b)} = P_{\text{nointer}}(b) \quad (20.15)$$

is the so-called survival factor, which enables us to calculate the probability that the LRG survives soft rescattering.

In terms of the opacity the elastic cross section takes the form

$$\begin{aligned} \frac{d\sigma_{\text{el}}}{dt} &= \frac{1}{16\pi s^2} |T_{\text{el}}(s, t)|^2 = \frac{1}{4\pi} \left| \int d^2 b \, e^{i\vec{q}_t \cdot \vec{b}} (1 - e^{-\Omega(b)/2}) \right|^2 \\ &= \pi \left| \int bdb \, J_0(q_t b) (1 - e^{-\Omega(b)/2}) \right|^2, \end{aligned} \quad (20.16)$$

where $q_t = \sqrt{|t|}$ and $J_0$ is the zeroth-order Bessel function.

To describe the elastic scattering at one fixed energy we can always find an appropriate parameterization for the opacity $\Omega(b)$ and tune the parameters to reproduce the observed $d\sigma_{\text{el}}/dt$ cross

section. Moreover, we can fix the form of the parameterization, but choose, at each particular energy, the corresponding values of parameters; see, e.g. [39]. Alternatively, we may simply take the Fourier-Bessel transform from the experimental data [33, 40, 41]

$$\operatorname{Im} T_{\text{el}}(b) = \int \frac{q_t dq_t}{4\pi} \sqrt{\frac{d\sigma_{\text{el}}}{dt} \frac{16\pi}{1 + \rho^2}} J_0(q_t b), \quad (20.17)$$

where the square root represents $\operatorname{Im}T_{\text{el}}(q_t)$, with $\rho \equiv \operatorname{Re}T_{\text{el}}/\operatorname{Im}T_{\text{el}}$. In this way, we first determine $T_{\text{el}}$ from the data for $d\sigma_{\text{el}}/dt$, and then calculate $\Omega(b)$ using Eq. (20.13), assuming in accordance with data that $\rho$ is small (or $\rho(t) = \text{constant}$).

At high energies $\rho^2 \ll 1$, which is usually well justified except in the diffractive dip region (see Section 20.3.3.1 for discussion of the dip region).

The value of $\rho$ can be derived via the dispersion relation, see [3]:

$$\frac{1}{s} \operatorname{Re} T_{\text{el}}(s) = \frac{1}{\pi} \int_{-\infty}^{+\infty} \frac{ds'}{s' - s} \sigma_{\text{tot}}(|s'|) = \frac{1}{\pi} \int_{0}^{\infty} \sigma_{\text{tot}}(s') \frac{2sds'}{s'^2 - s^2} \quad (20.18)$$

Since we consider just the charge-parity *C*-even amplitude, here for negative $s'$ we put $\sigma_{p\bar{p}} = \sigma_{pp}$. That is, for negative $s'$, which corresponds to the interaction with an *antiparticle*, we use the same $\sigma_{pp}(|s'|)$. The major contribution comes from $s' \simeq s$. Thus, with a good accuracy we can evaluate $\rho(t=0)$ as

$$\rho \approx \frac{\pi}{2} \frac{\partial \ln \sigma_{\text{tot}}(s)}{\partial \ln s}. \quad (20.19)$$

### 20.3.2 Diffractive dissociation

The elastic cross section probes the optical density of the proton. The well known example of scattering on a black disk, with $G_{\text{inel}} = 1$ for $b < R$, gives $\sigma_{\text{el}} = \sigma_{\text{inel}} = \pi R^2$ and $\sigma_{\text{tot}} = 2\pi R^2$. In general, the absorption of the initial wave (due to inelastic channels) leads, via *s*-channel unitarity, to elastic scattering.

Inelastic diffraction (i.e. proton dissociation) is a consequence of the *internal structure of hadrons*. This can be conveniently described at high energies, where the lifetimes of each particular Fock component of the incoming hadron/proton wave function (the hadronic fluctuations) are large, $\tau \sim E/m^2$, and during these time intervals the corresponding Fock states can be considered as ‘frozen’. Each hadronic constituent can undergo a scattering with its own probability and thus destroys coherence of the fluctuations<fnref n="4" />. As a result, the outgoing superposition of states will be different from the incident particle, and will most likely contain multiparticle states, so we will have *inelastic*, as well as elastic scattering.

To calculate diffractive dissociation we can enlarge the set of intermediate states $(p, N_a^*)$, from just the single elastic channel, and introduce a multichannel eikonal. However, it is more convenient

<footnote n="4">⁴At high energies the configurations with different transverse separation, $r$, between the quarks (valence partons) can serve as an example of such Fock states. An interaction with the QCD Pomeron does not change the value of $r$, while the cross section $\sigma \propto \alpha_s^2 r^2$ (see Section 20.4.2 and [42–44]).</footnote>