18. Structure Functions

1

18. Structure Functions
Revised August 2023 by E.C. Aschenauer (BNL), R.S. Thorne (UCL) and R. Yoshida (ANL).

18.1 Deep inelastic scattering
High-energy lepton-nucleon scattering plays a key role in determining the partonic structureof the proton. The process `N → `0 X is illustrated in Fig. 18.1. The filled circle in this figure represents the internal structure of the proton which can be expressed in terms of structure
functions.

k
k
P, M

q
W

Figure 18.1: Kinematic quantities for the description of deep inelastic scattering. The quantities
k and k 0 are the four-momenta of the incoming and outgoing leptons, P is the four-momentum of
a nucleon with mass M , and W is the mass of the recoiling system X. The exchanged particle isa γ, W ± , or Z; it transfers four-momentum q = k − k 0 to the nucleon.
Invariant quantities:0ν = q·P
M = E − E is the lepton’s energy loss in the nucleon rest frame (in earlier literature sometimes
ν = q · P ). Here, E and E 0 are the initial and final lepton energies in the nucleon rest
frame.
→− →−Q2 = −q 2 = 2(EE 0 − k · k 0 ) − m2` − m2`0 where m` (m`0 ) is the initial (final) lepton mass. If
EE 0 sin2 (θ/2)  m2` , m2`0 , then≈ 4EE 0 sin2 (θ/2), where θ is the lepton’s scattering angle with respect to the lepton beam
direction.
2Q
x = 2Mν where, in the parton model, x is the fraction of the nucleon’s momentum carried by the
struck quark. Beyond leading order the equation remains the definition of x, but thisis no longer identical to nucleon momentum fraction.
q·P= Eν is the fraction of the lepton’s energy lost in the nucleon rest frame.
y = k·PW 2 = (P + q)2 = M 2 + 2M ν − Q2 is the mass squared of the system X recoiling against the
scattered lepton.
222
s = (k + P )2 = Q
xy + M + m` is the center-of-mass energy squared of the lepton-nucleon system.

The process in Fig. 18.1 is called deep (Q2  M 2 ) inelastic (W 2  M 2 ) scattering (DIS). In
what follows, the masses of the initial and scattered leptons, m` and m`0 , are neglected.
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:20am

18. Structure Functions

2

18.1.1 DIS cross sections
The double-differential cross section for deep inelastic scattering can be expressed in terms ofkinematic variables in several ways.
d2 σ
2π M ν
d2 σ
d2 σ
=
.
= x (s − M 2 )dx dy
dx dQ2E 0 dΩNrest dE 0(18.1)

In lowest-order perturbation theory, the cross section for the scattering of polarized leptons
on polarized nucleons can be expressed in terms of the products of leptonic and hadronic tensors
associated with the coupling of the exchanged bosons at the upper and lower vertices in Fig. 18.1
(see Refs. [1–4])d2 σ2πyα2 Xj=
ηj Lµν
j Wµν .
dxdy
Q4
j(18.2)For neutral-current processes, the summation is over j = γ, Z and γZ representing photon andZ exchange and the interference between them, whereas for charged-current interactions there is
only W exchange, j = W . (For transverse nucleon polarization, there is a dependence on theazimuthal angle of the scattered lepton.) The lepton tensor Lµν is associated with the coupling of
the exchange boson to the leptons. For incoming leptons of charge e = ±1 and helicity λ = ±1,Lγµν = 2 kµ kν0 + kµ0 kν − (k · k 0 − m2` )gµν − iλεµναβ k α k 0β ,
e
e
γ
LγZ
µν =(gV + eλgA ) Lµν ,

e
e 2 γ
LZ
µν = (gV + eλgA ) Lµν ,2 γ
LWµν =(1 + eλ) Lµν ,(18.3)e = −1 .
where gVe = − 21 + 2 sin2 θW ,gA
2
Although here the helicity basis is adopted, an alternative approach is to express the tensors in
Eq. (18.3) in terms of the polarization of the lepton.
The factors ηj in Eq. (18.2) denote the ratios of the corresponding propagators and couplings
to the photon propagator and coupling squaredηγ = 1 ; ηγZ =
ηZ =

2
ηγZ

; ηW =GF MZ2
√
2 2πα

!

Q2
Q2 + MZ2!

2
GF M WQ2
24πα Q2 + MW

1
2

;
!2

.

(18.4)The hadronic tensor, which describes the interaction of the appropriate electroweak currents withthe target nucleon, is given by
Wµν =

1
4π

Z

d4 z eiq·z

D

h

i

P, S Jµ† (z), Jν (0)E

P, S ,(18.5)

where Jα is the hadronic contribution to the electromagnetic, or weak current and S denotes thenucleon-spin 4-vector, with S 2 = −M 2 and S · P = 0.
31st May, 2024

18. Structure Functions

3

18.2 Structure functions of the proton
The structure functions are defined in terms of the hadronic tensor (see Refs. [1–3])qµ qν
P̂µ P̂νWµν = −gµν +F1 (x, Q2 ) +F2 (x, Q2 )
2
q
P ·q
qαP β
− iεµναβF3 (x, Q2 )2P · q




qα
S·q β
β
2
β
2
+ iεµναβ
S g1 (x, Q ) + S −P
g2 (x, Q )
P ·q
P ·q
 


S·q
1
1
−
P̂
Ŝ
+
Ŝ
P̂
+
P̂µ P̂ν g3 (x, Q2 )
µ
ν
µ
ν
2
P ·q
P ·q


S·q
+
P ·q



"

P̂µ P̂νqµ qν
g4 (x, Q2 ) + −gµν +
P ·q
q2




#

g5 (x, Q2 )(18.6)

whereP̂µ = Pµ −

P ·q
qµ ,
q2

Ŝµ = Sµ −

S·q
qµ .
q2

(18.7)

In [2],the definition of Wµν with µ ↔ ν is adopted, which changes the sign of the εµναβ terms
in Eq. (18.6), although the formulae given below are unchanged. Ref. [1] tabulates the relation
between the structure functions defined in Eq. (18.6) and other choices available in the literature.
The cross sections for neutral- and charged-current deep inelastic scattering on unpolarized
nucleons can be written in terms of the structure functions in the generic form
d2 σ i4πα2 i
=
η
dxdy
xyQ2

(

x2 y 2 M 2
1 − y −Q2!

F2i

+ y xF1i
2

∓

y2
y −
2

!

)

xF3i

,

(18.8)

where i = NC, CC corresponds to neutral-current (eN → eX) or charged-current (eN → νX or
νN → eX) processes, respectively. For incoming neutrinos, LWµν of Eq. (18.3) is still true, but withe, λ corresponding to the outgoing charged lepton. In the last term of Eq. (18.8), the − sign istaken for an incoming e+ or ν and the + sign for an incoming e− or ν. The factor η NC = 1 for
unpolarized e± beams, whereasη CC = (1 ± λ)2 ηW(18.9)
with ± for `± ; and where λ is the helicity of the incoming lepton and ηW is defined in Eq. (18.4);
for incoming neutrinos η CC = 4ηW . The CC structure functions, which derive exclusively from W
exchange, are
F1CC = F1W , F2CC = F2W , xF3CC = xF3W .
(18.10)
The NC structure functions F2γ , F2γZ , F2Z are, for e± N → e± X, given by [5],ee2eF2NC = F2γ − (gVe ± λgA)ηγZ F2γZ + (gVe 2 + gA
± 2λgVe gA) ηZ F2Z(18.11)and similarly for F1NC , whereaseee2
xF3NC = −(gA± λgVe )ηγZ xF3γZ + [2gVe gA± λ(gVe 2 + gA)]ηZ xF3Z .

(18.12)The polarized cross-section difference∆σ = σ(λn = −1, λ` ) − σ(λn = 1, λ` ) ,31st May, 2024

(18.13)

18. Structure Functions

4

where λ` , λn are the helicities (±1) of the incoming lepton and nucleon, respectively, may be
expressed in terms of the five structure functions g1,...5 (x, Q2 ) of Eq. (18.6). Explicitly,(2d2 ∆σ i8πα2 i
2 2M
η
−λ
y
2
−
y
−
2x
y
=
`
dxdy
xyQ2
Q2

M2
M2
+ λ` 4x3 y 2 2 g2i + 2x2 y 2
Q
Q

−

M2
1 + 2x y 2
Q

!"

2

!

M2
1 − y − x2 y 2 2QM2
1 − y − x2 y 2 2
Q

xg1i

!

g3i

!

#)

g4i + xy 2 g5i(18.14)

with i = NC or CC as before. The Eq. (18.13) corresponds to the difference of antiparallel minus
parallel spins of the incoming particles for e− or ν initiated reactions, but the difference of parallel minus antiparallel for e+ or ν initiated processes. For longitudinal nucleon polarization, thecontributions of g2 and g3 are suppressed by powers of M 2 /Q2 . These structure functions give an
unsuppressed contribution to the cross section for transverse polarization [1], but in this case thecross-section difference vanishes as M/Q → 0.
Because the same tensor structure occurs in the spin-dependent and spin-independent parts of
the hadronic tensor of Eq. (18.6) in the M 2 /Q2 → 0 limit, the differential cross-section differenceof Eq. (18.14) may be obtained from the differential cross section Eq. (18.8) by replacing
F1 → −g5 ,F2 → −g4 ,F3 → 2g1 ,(18.15)and multiplying by two, since the total cross section is the average over the initial-state polarizations. In this limit, Eq. (18.8) and Eq. (18.14) may be written in the formid2 σ i2πα2 i h
i
i
2 i
=
η
Y
F
∓
Y
xF
−
y
F
+ 2
−
3
L ,
dxdy
xyQ2
i
d2 ∆σ i4πα2 i h
i
i
2 i
η
−Y
g
∓
Y
2xg
+
y
g
=
+ 4
−
1
L ,
dxdy
xyQ2

(18.16)

with i = NC or CC, where Y± = 1 ± (1 − y)2 andFLi = F2i − 2xF1i ,gLi = g4i − 2xg5i .

(18.17)In the naive quark-parton model, the analogy with the Callan-Gross relations [6]FLi = 0, are the
Dicus relations [7] gLi = 0. Therefore, there are only two independent polarized structure functions:g1 (parity conserving) and g5 (parity violating), in analogy with the unpolarized structure functions
F1 and F3 .
18.2.1 Structure functions in the quark parton model
In the naive quark-parton model [8,9], contributions to the structure functions F i and g i can be
expressed in terms of the quark distribution functions q(x, Q2 ) of the proton, where q = u, u, d, detc. The quantity q(x, Q2 )dx is the number of quarks (or antiquarks) of designated flavor thatcarry a momentum fraction between x and x + dx of the proton’s momentum in a frame in which
31st May, 2024

18. Structure Functions

5

the proton momentum is large. One of the most striking predictions of the quark-parton model isthat the structure functions Fi , gi scale, i.e., Fi (x, Q2 ) → Fi (x) in the Bjorken limit that Q2 and ν→ ∞ with x fixed [10]. This property is related to the assumption that the transverse momentumof the partons in the infinite-momentum frame of the proton is small.
For the neutral-current processes ep → eX,hih

F3γ , F3γZ , F3Z

i

=

h

i

= 21

F2γ , F2γZ , F2Z = x

Xh

i

q2
e2q , 2eq gVq , gVq 2 + gA
(q + q) ,

q

X

q
q
[0, 2eq gA, 2gVq gA] (q − q) ,qhg1γ , g1γZ , g1Zig5γ , g5γZ , g5Z =

Xh

X

i

q2
e2q , 2eq gVq , gVq 2 + gA(∆q + ∆q) ,qqq[0, eq gA, gVq gA
] (∆q − ∆q) ,(18.18)qqwhere gVq = ± 21 − 2eq sin2 θW and gA= ± 21 , with ± according to whether q is a u− or d−typequark respectively. The quantity ∆q is the difference q ↑ −q ↓ of the distributions with the quark
spin parallel and antiparallel to the proton spin.
For the charged-current processes e− p → νX and νp → e+ X, the structure functions are:

F2W

−

W−

F3

W−

g1

= 2x(u + d + s + c . . .) ,= 2(u − d − s + c . . .) ,= (∆u + ∆d + ∆s + ∆c . . .) ,−g5W = (−∆u + ∆d + ∆s − ∆c . . .) ,(18.19)where only the active flavors have been kept and where CKM mixing has been neglected. For e+ p →++νX and νp → e− X, the structure functions F W , g W are obtained by the flavor interchanges−−d ↔ u, s ↔ c in the expressions for F W , g W . The structure functions for scattering on a neutron
are obtained from those of the proton by the interchange u ↔ d. For both the neutral- andcharged-current processes, the quark-parton model predicts 2xF1i = F2i and g4i = 2xg5i .
Neglecting masses, the structure functions g2 and g3 contribute only to scattering from transversely polarized nucleons, and have no simple interpretation in terms of the quark-parton model.
They arise from off-diagonal matrix elements hP, λ0 |[Jµ† (z), Jν (0)]|P, λi, where the proton helicities
satisfy λ0 6= λ. In fact, the leading-twist contributions to both g2 and g3 are both twist-2 andtwist-3, which contribute at the same order of Q2 . The Wandzura-Wilczek relation [11] expresses
the twist-2 part of g2 in terms of g1 as
g2i (x) = −g1i (x) +

Z 1
x

dy i
g (y) .
y 1

(18.20)However, the twist-3 component of g2 is unknown. Similarly, there is a relation expressing thetwist-2 part of g3 in terms of g4 . A complete set of relations, including M 2 /Q2 effects, can be foundin [12].
18.2.2 Structure functions and QCD
In QCD, there are perturbative corrections to the partonic cross sections defining structure
functions. Also, the radiation of hard gluons from the quarks violates the assumption that thetransverse momentum of the partons is small, leading to logarithmic scaling violations, which are
31st May, 2024

18. Structure Functions

62

F2(x,Q )particularly large at small x, see Fig. 18.2. The radiation of gluons produces the evolution of thepartons and structure functions. As Q2 increases, more and more gluons are radiated, which inturn split into qq pairs. This process leads both to the softening of the initial quark momentum
distributions and to the growth of the gluon density and the qq sea as x decreases.

1.6

1.4

1.2

1

0.8

0.6

0.4

H1+ZEUSBCDMS (0.98)
NMC (1.00)
SLAC (1.00)
E665 (1.00)

0.2

10

-4

10

-3

10

-2

10

-1

1

x

Figure 18.2: The proton structure function F2p given at two Q2 values (6.5 GeV2 and 90 GeV2 ),which exhibit scaling at the ‘pivot’ point x ∼ 0.14. See the captions in Fig. 18.10 and Fig. 18.12 forthe references of the data. The various data sets have been renormalized by the factors shown inbrackets in the key to the plot, which were globally determined in a previous HERAPDF analysis[13]. The curves were obtained using the PDFs from the HERAPDF analysis [14]. In practice, data
for the reduced cross section, F2 (x, Q2 ) − (y 2 /Y+ )FL (x, Q2 ), were fitted, rather than F2 and FL
separately. The agreement between data and theory at low Q2 and x can be improved by a positive
higher-twist correction to FL (x, Q2 ) [15,16] (see Fig. 8 of Ref. [16]), or small-x resummation [17,18].
31st May, 2024

18. Structure Functions

7

For Q2  M 2 , the structure functions satisfy the factorization theorem [19],

Fi =

X

Cia ⊗ fa + O(M 2 /Q2 ),(18.21)awhere ⊗ denotes the convolution integral

C ⊗ f =Z 1xdy
C(y) f
y

x
y

 

,

(18.22)and where the coefficient functions Cia , where a = g or q, are given as a power series in αs . The
scale-dependent parton distribution fa (x, µ2 ) corresponds, at a given x, to the density of parton ain the proton integrated over transverse momentum kt up to the factorization scale µ. Typically,µ is the scale of the probe Q. For parton distributions x always refers to the nucleon momentum
fraction of the parton, whereas for structure functions it retains the definition in Sec. 18.1. The
parton evolution in µ is described in QCD by a DGLAP equation (see Refs. [20–23]) which has theschematic form∂fa
αs (µ2 ) X∼(Pab ⊗ fb ) ,∂ ln µ22πb(18.23)where the Pab , which describe the parton splitting b → a, are also given as a power series in αs .
Although perturbative QCD can predict, via Eq. (18.23), the evolution of the parton distributionfunctions from a particular scale, µ0 , these DGLAP equations cannot predict them a priori at
any particular µ0 . Thus they must be measured at a starting point µ0 before the predictions of
QCD can be compared to the data at other scales, µ. In general, all observables involving a hard
hadronic interaction (such as structure functions) can be expressed as a convolution of calculable,
process-dependent coefficient functions and these universal parton distributions, e.g. Eq. (18.21).
It is often convenient to write the evolution equations in terms of the gluon, non-singlet (q N S )and singlet (q S ) quark distributions, such thatq N S = qi − q i(or qi − qj ),

qS =

X

(qi + q i ) .

(18.24)i

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
∂q N Sαs (µ2 )=Pqq ⊗ q N S ,2∂ ln µ2π∂∂ ln µ2qSg!

αs (µ2 )=2π

Pqq 2nf Pqg
Pgq
Pgg

!

!

⊗

qS
,
g
(18.25)31st May, 2024

18. Structure Functions

8

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

Pqq = 34##"

1 + x2
1 + x2= 34+ 2δ(1 − x) ,(1 − x) +(1 − x)+"#

1 + (1 − x)2
x + (1 − x)
,
x
 


11 nf
1−x
x
+
δ(1 − x),
Pgg = 6+ x(1 − x) +−x
(1 − x)+2
3
Pqg = 21

h

2

2

i

(18.26), Pgq = 34

(18.27)(18.28)where the notation [F (x)]+ defines a distribution such that for any sufficiently regular test function,f (x),
Z 1

Z 1

dxf (x)[F (x)]+ =0dx (f (x) − f (1))F (x) .

(18.29)0

In general, the splitting functions can be expressed as a power series in αs . The series contains
both terms proportional to ln µ2 and to ln(1/x) and ln(1 − x). The leading-order DGLAP evolution
sums up the (αs ln µ2 )n contributions, while at next-to-leading order (NLO) the sum over theαs (αs ln µ2 )n−1 terms is included [24, 25]. The NNLO contributions to the splitting functions andthe DIS coefficient functions are also all known [26–28], as are the N3 LO corrections to the coefficient
functions [29]. Some Mellin moments of the N3 LO splitting functions are also known [30], 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), independentof the value of ln µ2 . At leading order, LLx, this is done by the BFKL equation for the unintegrated
distributions (see Refs. [31,32]). The leading-order (αs ln(1/x))n terms result in a power-like growth,−ωxwith ω = (12αs ln2)/π, at asymptotic values of ln 1/x. The next-to-leading ln 1/x (NLLx)
contributions are also available [33, 34]. They are so large (and negative) that the results initiallyappeared 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 stablehierarchy [35–38], and this has been used as the basis for a framework for including the corrections
in phenomenological studies [39,40]. There are some limited indications that small-x resummations
become necessary for sufficient precision for x . 10−3 at low scales [17, 18]. At sufficiently small x
it is expected that evolution should also be affected by non-linear/saturation effects due to a highdensity of partons, i.e. the gluon density would be so high that gluon-gluon recombination effects
would become significant, see [41–43] for reviews. However, there is not yet any very convincing indication for a non-linear regime for Q2 & 2 GeV2 in hadron DIS. Better evidence for saturation may
be found in more exclusive final states such as dihadron correlations in proton-nucleus collisions,e.g. [44, 45].
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 theleading order, it is necessary to specify, and to use consistently, both a renormalization and afactorization scheme. The renormalization scheme used almost universally is the modified minimal
subtraction (MS) scheme [46,47]. The most popular choices for the factorization scheme is also MS[48]. Historically, sometimes the DIS [49] scheme was adopted, in which there are no higher-ordercorrections to the F2 structure function, and this was extended to a specific definition appropriate
to small x [50]. assimilated in the parton distribution functions.
31st May, 2024

918. Structure Functions

The discussion above relates to the Q2 behavior of leading-twist (twist-2) contributions to thestructure 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/Qn−2 . Provided a cut,
say W 2 > 15 GeV2 is imposed, the higher-twist terms appear to be numerically unimportant forQ2 above a few GeV2 , except possibly for very small x and more definitely for x close to 1 [51–53],
though it is important to note that they are likely to be larger in xF3 (x, Q2 ) than in F2 (x, Q2 ) (seee.g. [54]) due to a lack of a constraining sum rule for xF3 (x, Q2 ).

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. [55–61] for reviews. Table 18.1 highlights some of the processes and their primary sensitivity to PDFs. LHC data are playing an increasing role [62], 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 andµ2 . As more precise LHC data for W ± , Z, γ, jet, bb̄ and tt̄ 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 [63], 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 [63], NNPDF [66], CT [67], ATLAS [68] and ABMP [69]. The CJ group produce
PDFs at NLO and concentrate on the high x, low Q2 regime, including data with lower kinematic
cuts [70]. All groups start evolution at Q20 = 1−4 GeV2 . Most groups use input PDFs of the form
xf = xa (...)(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 theirbest fit values, and orthogonal eigenvector sets of PDFs depending on linear combinations of theparameter 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 theprobability measure in the space of PDFs with the use of neural networks. Fits are performed toa 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 [71] and vice versa [72]. Recently
accumulated information on N3 LO calculation has been used by MSHT in order to produce afirst set of approximate N3 LO global PDFs [73] together with theoretical uncertainties, and analternative approach to theoretical uncertainty has been made in [74].
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, mQ , differs, and has a long history, which may be traced from Refs. [75–86]. The
MSHT, CT, NNPDF and HERAPDF analyses use different variants of the General-Mass VariableFlavour-Number Scheme (GM-VFNS). This combines fixed-order contributions to the coefficient
functions (or partonic cross sections) calculated with the full mQ dependence, with the all-orderresummation 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 [78], which provide the boundary conditions between nf and nf + 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 mQ
dependence. The GM-VFNS and FFNS approaches yield different results: in particular αs (MZ2 )31st May, 2024

18. Structure Functions

10

Table 18.1: The main processes relevant to global PDF analyses, orderedin three groups: fixed-target experiments, HERA and the pp̄ 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.
Process

Subprocess

Partonsx range`± {p, n} → `± Xγ∗q → qq, q̄, gx & 0.01`± n/p → `± X

d/u

x & 0.01

pp → µ+ µ− X

γ ∗ d/u → d/uuū, dd¯ → γ ∗0.015 . x . 0.35

pn/pp → µ+ µ− X¯(ud)/(uū)
→ γ∗

q̄
¯
d/ū

ν(ν̄) N → µ− (µ+ ) X

W ∗q → q0q, q̄

0.01 . x . 0.5

ν N → µ− µ+ X

W ∗s → cs

0.01 . x . 0.2

ν̄ N → µ+ µ− X

W ∗ s̄ → c̄s̄

0.01 . x . 0.2

e± p → e± X

γ∗q → qg, q, q̄10−4 . x . 0.1

e+ p → ν̄ X

W + {d, s} → {u, c}d, sx & 0.01

e± p → e± cc̄X, e± bb̄Xγ ∗ c → c, γ ∗ g → cc̄

c, b, g10−4 . x . 0.01

e± p → jet+Xγ ∗ g → q q̄g0.01 . x . 0.1

pp̄, pp → jet(dijet)+Xg, q

0.00005 . x . 0.5

pp̄ → (W ± → `± ν) X

gg, qg, qq → 2j
ud → W + , ūd¯ → W −x & 0.05

pp → (W ± → `± ν) X

ud¯ → W + , dū → W −¯ s̄
u, d, s, ū, d,¯ s̄, gu, d, s, ū, d,

x & 0.001

pp̄(pp) → (Z → `+ `− )Xuu, dd, ..(uū, ..) → Z

u, d, s, ..(g)x & 0.001

pp → W − c, W + c̄s, s̄

x ∼ 0.01

pp → (γ ∗ → `+ `− )X

gs → W − c¯ .. → γ ∗uū, dd,q̄, g

x & 10−5

pp → (γ ∗ → `+ `− )Xuγ, dγ, .. → γ ∗γ

x & 10−2

pp → bb̄ X, tt̄X

gg → bb̄, tt̄

gx & 10−5 , 10−2

pp → t(t̄) X,

bu(b̄d) → td(t̄u)

b, d/ux & 10−2

pp → exclusive J/ψ, Υγ ∗ (gg) → J/ψ, Υgx & 10−5 , 10−4

pp → γ X

gq → γq, g q̄ → γ q̄gx & 0.005

0.015 . x . 0.35

and the large-x gluon PDF at large Q2 are both significantly smaller in the FFNS. It has beenargued [52, 53, 85] that the difference is due to the slow convergence of the lnn (Q2 /m2Q ) terms incertain regions in a FFNS. The final HERA combination of heavy flavour structure function data
has been published [87], and the evolution of these measurements and their interpretation may be
31st May, 2024

11

18. Structure Functions

Figure 18.3: Kinematic domains in x and Q2 probed by fixed-target and collider experiments,
where here Q2 can refer either the literal Q2 for deep inelastic scattering, or the hard scale ofthe process in hadron-hadron collisions, e.g. invariant mass or transverse momentum p2T . Someof the final states accessible at the LHC are indicated in the appropriate regions, where y is the
rapidity. The incoming partons have x1,2 = (Q/14 TeV)e±y where Q is the hard scale of the process
shown in blue in the figure. For example, open charm production [64] and exclusive J/ψ and Υ
production [65] at high |y| at the LHC may probe the gluon PDF down to x ∼ 10−5 .

traced in [88].
31st May, 2024

12

18. Structure Functions

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 [67], MSHT2018 [63], and
NNPDF3.1 [89], have been combined [90] using the Monte Carlo approach [71] mentioned above.
The single combined set of PDFs is discussed in detail in Ref. [91].
For illustration, we show in Fig. 18.4 the PDFs obtained in the NNLO MSHT analysis [63]at scales µ2 = 10 and 104 GeV2 (and an equivalent plot for polarized PDFs from NNPDF [92]).
The values of αs found by MSHT [93] may be taken as representative of those resulting from the
GM-VFNS analyses
NLO : αs (MZ2 ) = 0.1203 ± 0.0015,
NNLO : αs (MZ2 ) = 0.1174 ± 0.0013,
where the error (at 68% C.L.) corresponds to the uncertainties resulting from the data fitted (theuncertainty 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 αs (MZ2 ) = 0.1185 ± 0.0012 at NNLO andthe CT group [67], who obtain αs (MZ2 ) = 0.1164 ± 0.0026. The ABMP analysis [69],which uses a
FFNS, finds αs (MZ2 ) = 0.1147 ± 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 witha photon contribution were considered in Refs. [95–97]. However, due to weak data constraints, theuncertainty 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 theoreticalprecision, 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 oftens 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 fromcorrelation functions. However, there remain a number of sources of systematic uncertainty; discretization effects, pion mass dependence, finite volume effects, excited state contamination andrenormalization, 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 arein [113–117].
31st May, 2024

18. Structure Functions

13
1.2

MSHT20NNLO, Q2 = 10 GeV2

1.2

MSHT20NNLO, Q2 = 10 GeV2

xf (x, Q ) 1.2
2

MSHT20NNLO, Q2 = 104 GeV2

xf (x, µ2 = 104 GeV2 )<latexit sha1_base64="v1KnjzBAa5iynjGfQz6UQHPh3L0=">AAACB3icbVDLSsNAFJ34rPUVdSnIYBEqlJKUom6UggtdVrAPaNIymU7aoTNJmJlIS+jOjb/ixoUibv0Fd/6N0zYLbT1w4XDOvdx7jxcxKpVlfRtLyyura+uZjezm1vbOrrm3X5dhLDCp4ZCFoukhSRgNSE1RxUgzEgRxj5GGN7ie+I0HIiQNg3s1iojLUS+gPsVIaaljHg2hnx8WHB63S5e21S47hcQRHN6Q+rhdOu2YOatoTQEXiZ2SHEhR7ZhfTjfEMSeBwgxJ2bKtSLkJEopiRsZZJ5YkQniAeqSlaYA4kW4y/WMMT7TShX4odAUKTtXfEwniUo64pzs5Un05703E/7xWrPwLN6FBFCsS4NkiP2ZQhXASCuxSQbBiI00QFlTfCnEfCYSVji6rQ7DnX14k9VLRPiuW78q5ylUaRwYcgmOQBzY4BxVwC6qgBjB4BM/gFbwZT8aL8W58zFqXjHTmAPyB8fkDmGWXOg==</latexit>2

xf (x, µ2 = 10 GeV2 ) xf (x, Q )<latexit sha1_base64="UKA1T/bDU55Nm3UlXSqAPhO9Vko=">AAACBXicbVDLSsNAFJ34rPUVdamLwSJUKCUpRd0oBRe6rGAf0KRlMp20Q2eSMDORltCNG3/FjQtF3PoP7vwbp20W2nrgwuGce7n3Hi9iVCrL+jaWlldW19YzG9nNre2dXXNvvy7DWGBSwyELRdNDkjAakJqiipFmJAjiHiMNb3A98RsPREgaBvdqFBGXo15AfYqR0lLHPBpCPz8sODxuly5tyykkjuDwhtTH7dJpx8xZRWsKuEjslORAimrH/HK6IY45CRRmSMqWbUXKTZBQFDMyzjqxJBHCA9QjLU0DxIl0k+kXY3iilS70Q6ErUHCq/p5IEJdyxD3dyZHqy3lvIv7ntWLlX7gJDaJYkQDPFvkxgyqEk0hglwqCFRtpgrCg+laI+0ggrHRwWR2CPf/yIqmXivZZsXxXzlWu0jgy4BAcgzywwTmogFtQBTWAwSN4Bq/gzXgyXox342PWumSkMwfgD4zPH1/UlpQ=</latexit>

xf (x, Q2 )

0.8

0.8
g/10

0.8

g/10

uV
c

0.4
c

g/10

dV

su

d

u0.4
V

0.001

0.01

0.1

s
dV
u

0.4

0
0.0001

uV

b

s

1

0

d0.0001
V

0.001

0.01

d

0.1

1

c
2 and2 Q2 = 10
4 GeV
2 =
2 , associatedFigure1: MSHT20
NNLO
PDFsPDFs
at Q2at=Q10
GeV, with
68% confidenceFigure
1: MSHT20
NNLO
10 GeVand Q2 =104 2GeV
with associated
68% confidencelevellevel
uncertainty
bands.
0.4
0.4
uuncertainty
bands.
d0

NNPDFpol1.1 (NLO)
xf(x,µ 2=10 GeV2)

xf(x,µ 2=104 GeV 2)0.0001
0.3

0.001
0.01
0.1 0.3
1
consider:CMS
13 TeV
datadata
on W
production[29], [29],which
teststests
predictions
particularly
consider:CMS
13 TeVon +Wc +c production
which
predictions
particularly
uvat Q2 = 10 GeV2 and Q2 = 104 GeV2 , with associated 68
Figure1:
MSHT20
NNLO
PDFs
dependentonthestrangequark;
theratios
of
Z
and
t
t̄
cross
sections
at
8
TeV
and
13
dependent on the strange quark; the ratios of Z and tt̄ cross sections at 8 TeV andTeV
13 TeV0.2
0.2
uv
uncertainty
bands.
atlevel
ATLAS
[30];
the
CMS
measurementsofsingle-top
production
[31,
32];
the
potential
impactat ATLAS [30]; the CMS measurements of single-top production [31, 32]; the potential impactof LHCb
exclusiveJ/ J/
production
datadata[33, 34],
as accounted
for infortheof [35],and andof LHCb
exclusiveproduction[33, 34],
as accounted
in analysis
the analysisof [35],
LHCb
datadata
on 0.1D
production[33, 36,
as accounted
for infortheof [38].
In In
0.1
LHCb
on mesonD mesonproduction[33,37],36, 37],as accounted
in analysis
the analysisof [38].
Section11 we
our our
MSHT
PDFs
withwith
thosethoseof themostmost
recentglobalanalysesof ofSection11 compare
we compare
MSHT
PDFs
of otherthe otherrecentglobalanalysesg u
gu
consider:
CMS
TeV
data
+with
c production[29],
tests predictionss W
PDFs– NNPDF3.1[2] 13andCT18[3], and
also
with
older
sets setsof PDFsof othercollaborations.
s collaborations.
PDFs– NNPDF3.1[2] andCT18[3],on
and
also
older
of PDFs
ofwhichother
00c quark; ofthe
Independent
Section12 wesummarise
the theavailability
MSHT20sets sets
and
theirdelivery.
In at 8 TeV
b PDF
strangeofPDF
Zandtct̄ crosssections
In Section
12on
wethe
summariseavailabilitytheof ratios
the MSHT20
andd their delivery. IndSection13 we
our our
13 present
we[30];
present
conclusions.
atSection
ATLAS
theconclusions.
CMS
measurements of single-top production [31, 32]; the pote
d
v

−0.1

−0.1

dv

of LHCb exclusive J/ production data [33, 34], as accounted for in the analysis
10
10
10
1
1
2LHCb
the
theoreticalprocedures
data10oninDin
meson
production[33,1036, 37],10 as xaccounted
forin the analysi
2 Changes
ChangesthetheoreticalproceduresxSection 11 we compare our MSHT PDFs with those of the other most recent globa
FigureThe bands
are x times
theparton(x)(wheref =As incasecaseof18.4:
MMHT14,
we present
PDFPDFsetsunpolarized
at LO,NLO
and distributions
NNLO
in ↵fSin. In
thelatterAstheinutheof
MMHT14,we
present
sets
at
LO,
NLO
and
NNLO
↵
.
In
the
S
PDFs
–
NNPDF3.1
[2]
and
CT18
[3],
and
also
with
older
sets
of
PDFs
of other co
,
d
,
u,
d,
s
'
s̄,
c
=
c̄,
b
=
b̄,
g)
obtained
in
the
NNLO
MSHT20globalanalysis[63](top)
atlatterv v
casecase
we usethesplittingfunctionscalculated
in
[39,
40]
and
for
structurefunction
data,
the
2
2
2
4
2
2
we
use
the
splittingfunctionscalculated
in
[39,
40]
and
for
structurefunctiondata,
scales µ = 10 GeV (left) and µ = 10 GeV (right), with αs (MZ ) = 0.118. The polarized parton the
In Section
12 functions
wesummarisetheavailabilityof (bottom).
the MSHT20
PDF
sets and theirmasslesscoefficient
calculated
in [41–46].
There
are
however,a significant
numberdistributions
f (x)functions
obtained
incalculated
the NLO
NNPDFpol1.1
fit[92]
masslesscoefficientin [41–46].
There
are however,a significant
numberofSectionchanges in13
ourwe
theoreticaldescriptionof the data, compared to that used in the MMHT14
present
ourconclusions.
of changes in our theoretical description of the data, compared to that used in the MMHT14
−3

−2

−1

−3

−2

−1

analysis. We present these in this section, and when appropriate we also mention some of theanalysis.Comprehensive
We present thesethisaresection,and
when
appropriate
we[118],alsowhichmentionsomeof thesets ofin
PDFs
available
fromthe LHAPDF
library
can belinkedmain e↵ects on the PDFs resulting from these improvements.
main directlye↵ects intoon the
PDFs
resultingtofrom
these
improvements.
a user’s
programme
provideaccessto recent PDFs in a standard format. This also2

includes many nuclear and polarized PDFs.

Changes in the theoretical procedures

2.1 Input distributions
2.1 Input distributions

InAs
MMHT14began
use parameterisations
theinput distributions
on Chebyshev
31stforMay,
2024
thewecase
oftoMMHT14,
we present
PDF
setsdistributionsat LO,based
NLO
NNLO in ↵S .
In in
MMHT14
we beganto use parameterisations
forthe input
basedandon Chebyshev
polynomials. Following the detailed study in [47], we take for most PDFs a parameterisation
polynomials.
the detailed
study incalculated[47], we takein
for[39,most40]PDFs
case
we use Following
the splitting
functionsanda parameterisation
for structure functi

18. Structure Functions

14

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 functionof 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 comparisonof the nuclear modification factors for nCTEQ15HQ [120], EPPS21 [122] and nNNPDF3.0 [123] isshown in Fig. 18.5 where, for example, for the up quark RuPb = (ZP b uP b +(AP b −ZP b )dP b )/(ZP b up +(AP b − ZP b )dp ) i.e. it involves “physical” nuclear PDFs and is normalised such that RuPb = 1
corresponds to no nuclear effects.
1.6
nNNPDF3.0
EPPS21nCTEQ15HQ

1.2
1.0
0.8

1.2

1.4

RdPb(x, Q)

1.4

Q = 10 GeV
RuPb(x, Q)

RgPb(x, Q)

1.4

1.0
0.8
0.6

0.6
10−4

10−3

10−2

10−1

90% CL PDF errors

1.2
1.0
0.8
0.6

10−4

10−3

10−2

10−1

10−4

10−3

10−4

10−3

10−2

10−1

10−2

10−1

1.75
1.4

1.4

1.2

1.2

1.0
0.8

1.0
0.8

0.6

0.6

0.4

0.4

10−4

10−3

10−2

x

10−1

RsPb
+ (x, Q)RdPb
¯ (x, Q)

RūPb(x, Q)

1.50
1.25
1.00
0.75
0.50
10−4

10−3

10−2

x

10−1

0.25

x

Figure 18.5: 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).
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.
p/A
fi (x, Q2 ) = RiA (x, Q2 )fip (x, Q2 ).
(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 approch 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 usedalong with DIS data in most fits, though the most recent data [132] shows some tendency torequire 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,
31st May, 2024

15

18. Structure Functions

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 improvedby 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, andeven moreso for planning of higher energy colliders. It has a long theoretical history [140–144], andin recent years experimental studies have started. DPS is often supressed, but this is less the casein 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 supressed 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 vectorbosons or mesons. Recent developments have been involved in putting the theoretical foundationsof 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 caseof two partons [145–148, 157–159]. The current knowledge of DPDs is still limited, though there
are a number of attempts to impose theoretical constrains as a limit on the forms [160–162]. The
DPDs can be calculated quite accurately when the tranverse separtion 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 beendevised [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 rangeof Q2 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 datausing NLO global analyses which include measurements of the g1 structure function in inclusive
polarized DIS, ‘flavour-tagged’ semi-inclusive DIS data, open–charm production in DIS and resultsfrom polarized pp scattering at RHIC. There are recent results on DIS from JLAB [164] (for g1n /F1n ),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 thesemi-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 [179], dijet [180] and inclusive π 0 -production [181], but not
yet the single spin asymmetries in W ± , Z 0 production [182, 183]. A determination [184], using
the NNPDF methodology, concentrates just on the inclusive polarized DIS data, and finds theuncertainties on the polarized gluon PDF have been underestimated in the earlier analyses. An
update to this [92], where jet and W ± data from pp collisions 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] andinclusive π 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 longitudinallypolarized scattering data [178]. A simultaneous extraction of spin-dependent parton distributions
and fragmentation functions has recently also been performed [189]. Calculations of polarised
31st May, 2024

18. Structure Functions

16

0.4
x∆g(x,µ2=10 GeV2)

0.3
0.2
0.1
0
-0.1

DSSV14and 68% C.L. contours-0.2

MC-replicas
MC-average
NNPDFpol1.1-0.3

and 1-σ contours

-0.4
0.001

0.003

0.01

0.03

0.1

0.3 0.5

1x
Figure 18.6: Ensemble of replicas (dotted blue lines) for the NLO gluon helicity density ∆g(x, Q2 )at Q2 = 10 GeV2 shown along with its statistical average (solid blue line) and variance (dot-dashedblue lines). The corresponding results from the DSSV14 fit (black lines) [172] and the NNPDFpol1.1
analysis (green lines) [92] are shown for comparison. Figure taken from Ref. [178].

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, thanthe 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 [92]and DSSV [178] is shown in Fig. 18.6 at scale µ2 = 10 GeV2 . The world data of the inclusive
structure function g1 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 γ → qq. While in this state, the partonic content of the photon may beresolved, for example, through the process e+ e− → e+ e− γ ? γ → e+ e− X, where the virtual photon
emitted by the DIS lepton probes the hadronic structure of the quasi-real photon emitted by theother lepton. The perturbative LO QED contributions to this process with γ → qq in conjunction
with γ ? q(q̄) → q(q̄), are subject to QCD corrections due to the radiation of gluons from thesequarks.
Often the equivalent-photon approximation is used to express the differential cross section fordeep inelastic electron–photon scattering in terms of the structure functions of the transverse quasi31st May, 2024

18. Structure Functions

17

real photon times a flux factor NγT (for these incoming quasi-real photons of transverse polarization)
2 h

i
d2 σ
γ
2
2 γ
2
T 2πα
2
F
(x,
Q
)
−
y
F
(x,
Q
)
,
=
N
1
+
(1
−
y)
γ
2
L
dxdQ2
xQ4

(18.31)

where we have used F2γ = 2xFTγ + FLγ (where FT is the transverse structure function), not to be
confused with F2γ of Sec. 18.2. Complete formulae are given, for example, in the comprehensive
review of [191].
The hadronic photon structure function, F2γ , evolves with increasing Q2 from the ‘hadronlike’ 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 evolutionof F2γ with Q2 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 γ → q q̄ coupling.
The logarithmic evolution was first predicted in the quark–parton model (γ ? γ → q q̄) [192,193], andthen an improved expression was obtained using QCD corrections in the limit of large Q2 [194].
The evolution is now known to NLO [195–197]. The NLO data analyses to determine the partondensities of the photon can be found in Refs. [198–200].

18.5 Diffractive DIS (DDIS)
Some 10% of DIS events are diffractive, γ ∗ p → X + p, in which the slightly deflected proton
and the cluster X of outgoing hadrons are well-separated in rapidity [201]. Besides x and Q2 , two
extra variables are needed to describe a DDIS event: the fraction xIP 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 F2D satisfies collinear factorization, and can be expressed as the convolution [208]
F2D =

X

D
C2a ⊗ fa/p,

(18.32)a=q,gwith the same coefficient functions as in DIS (see Eq. (18.21)), and where the diffractive parton
D (a = q, g) satisfy DGLAP evolution. Second, Regge factorization is assumed [209],
distributions fa/pD
fa/p(xIP , t, z, µ2 ) = fIP /p (xIP , t) fa/IP (z, µ2 ),(18.33)where fa/IP are the parton densities of the Pomeron, which itself is treated like a hadron, andz ∈ [x/xIP , 1] is the fraction of the Pomeron’s momentum carried by the parton entering the hard
subprocess. The Pomeron flux factor fIP /p (xIP , 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 partondensities may be found in [210].
Although collinear factorization holds as µ2 → ∞, there are non-negligible corrections for finiteµ2 and small xIP . 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 calculatediffractive hadron-hadron cross sections, since account must first be taken of “soft” rescattering
effects.

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18. Structure Functions

xIP z f(z)

18

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

z
D , obtained from fitting to the ZEUS data
Figure 18.7: Diffractive parton distributions, xIP zfa/p
with Q2 > 5 GeV2 [202],H1 data with Q2 > 8.5 GeV2 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 xIP =0.003 chosen here. The H1 2007 Jets distribution [205] is similar to H1 2006 Fit B.

18.6 Three-dimensional structure of hadrons
Generalized parton distributions (GPDs) and their compliment, 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
hP, λ|Ô|P, λi, where P and λ are the 4-momentum and helicity of the proton, and Ô 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 hP 0 , λ0 |Ô|P, λi; 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.
√
The physical significance of GPDs is best seen using light-cone coordinates, z ± = (z 0 ± z 3 )/ 2,
and in the light-cone gauge, A+ = 0. It is conventional to define the generalised quark distributions
31st May, 2024

18. Structure Functions

19

Figure 18.8: Schematic diagrams of the three distinct kinematic regions of the imaginary part of
Hq . The proton and quark momentum fractions refer to P̄ + , and x covers the interval (-1,1). In the
ERBL domain the GPDs are generalisations of distribution amplitudes which occur in processes
such as pp̄ → J/ψ.
in terms of quark operators at light-like separation
1
Fq (x, ξ, t) =
2

Z

1
=
2P̄ +

dz − ixP̄ + z − 0ehP |ψ̄(−z/2)γ + ψ(z/2)|P i
2πz + =z 1 =z 2 =0iσ +α ∆αu(P )Hq (x, ξ, t) ū(P )γ u(P ) + Eq (x, ξ, t) ū(P )2m
0!

(18.34)0+with P̄ = (P + P 0 )/2 and ∆ = P 0 − P , and where we have suppressed the helicity labels of the
protons and spinors. We now have two extra kinematic variables:ξ = −∆+ /(P + P 0 )+ .

t = ∆2 ,(18.35)We see that −1 ≤ ξ ≤ 1. Similarly, we may define GPDs H̃q and Ẽq with an additional γ5 between
the quark operators in Eq. (18.34); and also an analogous set of gluon GPDs, Hg , Eg , H̃g and Ẽg .
After a Fourier transform with respect to the transverse components of ∆, we are able to describe
the spatial distribution of partons in the impact parameter plane in terms of GPDs [220, 221].
For P 0 = P, λ0 = λ the matrix elements reduce to the ordinary PDFs of Sec. 18.2.1
Hq (x, 0, 0) = q(x),Hq (−x, 0, 0) = −q̄(x),Hg (x, 0, 0) = xg(x),(18.36)H̃q (x, 0, 0) = ∆q(x), H̃q (−x, 0, 0) = ∆q̄(x), H̃g (x, 0, 0) = x∆g(x),(18.37)

where ∆q = q ↑ −q ↓ as in Eq. (18.18). No corresponding relations exist for E, Ẽ as they decouple
in the forward limit, ∆ = 0.
The functions Hg , Eg are even in x, and H̃g , Ẽg are odd functions of x. We can introduce valenceand ‘singlet’ quark distributions which are even and odd functions of x respectively. For example
HqV (x, ξ, t) ≡ Hq (x, ξ, t) + Hq (−x, ξ, t) = HqV (−x, ξ, t),(18.38)HqS (x, ξ, t) ≡ Hq (x, ξ, t) − Hq (−x, ξ, t) = −HqS (−x, ξ, t).

(18.39)All the GPDs satisfy relations of the form
H(x, −ξ, t) = H(x, ξ, t)and

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H(x, −ξ, t)∗ = H(x, ξ, t),(18.40)18. Structure Functions

20

and so are real-valued functions. Moreover, the moments of GPDs, that is the x integrals of xn Hq
etc., are polynomials in ξ of order n+1. Another important property of GPDs are Ji’s sum rule [214]
1
2

Z 1
−1

dx x (Hq (x, ξ, t) + Eq (x, ξ, t)) = Jq (t),(18.41)

where Jq (0) is the total angular momentum carried by quarks and antiquarks of flavour q, with asimilar relation for gluons.
To visualize the physical content of Hq , we Fourier expand ψ and ψ̄ in terms of quark, antiquark
creation (b, d) and annihilation (b† , d† ) operators, and sketch the result in Fig. 18.8. There are
two types of domain: (i) the time-like or ‘annihilation’ domain, with |x| < |ξ|, where the GPDs
describe the wave functions of a t-channel q q̄ (or gluon) pair and evolve according to modified
ERBL equations [222, 223]; (ii) the space-like or ‘scattering’ domain, with |x| > |ξ|, where the
GPDs generalise the familiar q̄, q (and gluon) PDFs and describe processes such as ‘deeply virtualCompton scattering’ (γ ∗ p → γp), γp → J/ψp, etc., and evolve according to modified DGLAP
equations. The splitting functions for the evolution of GPDs are known to NLO [224–226].
GPDs describe new aspects of proton structure and must be determined from experiment. We
can parametrise them in terms of ‘double distributions’ [227, 228], which reduce to diagonal PDFs
as ξ → 0. Alternatively, flexible SO(3)-based parametrisations have been used to determine GPDs
from DVCS data [229, 230]; a more recent summary may be found in Ref. [231, 232].
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 andtransversely polarized quark parton distributions [233, 234]. The novel TMD PDFs have physicalinterpretations. For example, the Sivers function [235] represents the distribution of unpolarized

4

1

3

10-1

2

10-2

1

10-3
10-4
0

1

2

3

4

5

6

7

Figure 18.9: (Left) Sivers function in the momentum space for u quark [236] at x = 0.1 asa function of transverse momentum kT for different values of Q. The bands are 68%CI. (Right)
Momentum space density function as defined in [236] 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 ofthe distribution along the x-direction due to the Sivers function can be seen. The width of thepatterns in the plot roughly indicates the uncertainty of the extraction.
31st May, 2024

21

18. Structure Functions

partons inside a transversely polarized hadron. For (pseudo)scalar particles, such as kaon andpions, there are two independent leading-twist TMD FFs, one being the ordinary unpolarizedfragmentation function and the other the Collins FF [237] which is related to the probability of apolarized quark fragmenting into an unpolarized hadron.
Factorization of TMDs has been shown for semi-inclusive DIS, for the Drell-Yan process aswell 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 arehighlighted 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 [236].
Because TMD PDFs encode nonperturbative information about transverse momentum andpolarization 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 canhave 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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18. Structure Functions

29

109

ProtonH1+ZEUS107

F2p × 2 i

105

E665
x = 8.5e 06 (i=26)
x = 1.6e 05
HERMESx = 4e 05
x = 0.0001
NMC
x = 0.0002
BCDMS
x = 0.00032
SLAC
x = 0.00051
JLab
x = 0.00076
x = 0.0013
x = 0.002
x = 0.0036
x = 0.0052
x = 0.0089
x = 0.018
x = 0.026103

x = 0.035
x = 0.05
x = 0.08
x = 0.13
x = 0.19
x = 0.25

101

x = 0.4
x = 0.56
x = 0.65

10 1
x = 0.75
x = 0.85 (i=1)

10 3 1
10

100

101

102
2
Q (GeV 2)

103

104

105

Figure 18.10: The proton structure function F2p measured in electromagnetic scattering of electronsand 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 F2p with a HERAPDF
NLO fit, for all measured points where the predicted ratio of F2p to reduced cross-section was within 10% ofunity. The data are plotted as a function of Q2 in bins of fixed x. Some points have been slightly offset in
Q2 for clarity. The H1+ZEUS combined binning in x is used in this plot; all other data are rebinned to thex values of these data. For the purpose of plotting, F2p has been multiplied by 2ix , where ix is the numberof the x bin, ranging from ix = 1 (x = 0.85) to ix = 26 (x = 0.0000085). Only data with W 2 > 3.5 GeV2 isincluded. Plot from CJ collaboration (Shujie Li – private communication).
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, nuclex/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).
31st May, 2024

18. Structure Functions

30

106
x = 0.00076 (i=19)

105
104

F2d × 2 i

103

Deuteron

E665
HERMES
NMC
BCDMS
SLAC
JLab

x = 0.0013
x = 0.002
x = 0.0036
x = 0.0052
x = 0.0089
x = 0.018
x = 0.026
x = 0.035
x = 0.05
x = 0.08102

x = 0.13
x = 0.19
x = 0.25

101

x = 0.4

100

x = 0.56
x = 0.65

10 1

x = 0.75

10 2
x = 0.85 (i=1)

10 3 1
10

100

101
2
Q (GeV 2)

102

103

Figure 18.11: The deuteron structure function F2d measured in electromagnetic scattering of electrons/positrons (SLAC,HERMES,JLAB) and muons (BCDMS, E665, NMC) on a fixed target, shown as a functionof Q2 for bins of fixed x. Statistical and systematic errors added in quadrature are shown. For the purposeof plotting, F2d has been multiplied by 2ix , where ix 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 GeV2 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).
31st May, 2024

18. Structure Functions
2

F2(x,Q )2

F2(x,Q )+c(x)

31

1.2

1.8

1.6

1.4

1

H1+ZEUS
SLAC
BCDMS
NMC

1.2
0.8
1
0.6
0.8

0.6

0.4

0.4
0.2
CCFR
NMC
NuTeV

0.2

0
1

10

10

2

10
2

3

2

Q (GeV )

0
10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

1

x

Figure 18.12: a) The deuteron structure function F2 measured in deep inelastic scattering of muons on
a fixed target (NMC) is compared to the structure function F2 from neutrino-iron scattering (CCFR and
NuTeV) using F2µ = (5/18)F2ν − x(s + s)/6, where heavy-target effects have been taken into account. The
data are shown versus Q2 , 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) = 0.05ix is added to F2 , where ix is the number of thex bin, ranging from 0 (x = 0.75) to 7 (x = 0.175). For ix = 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. Yanget al., Phys. Rev. Lett. 86, 2741 (2001); NuTeV—M. Tzanov et al., Phys. Rev. D74, 012008 (2006).
b) The proton structure function F2p mostly at small x and Q2 , measured in electromagnetic scattering ofelectrons and positrons (H1, ZEUS), electrons (SLAC), and muons (BCDMS, NMC) on protons. Lines are
ZEUS Regge and HERAPDF parameterizations for lower and higher Q2 , respectively. The width of the bins
can be up to 10% of the stated Q2 . Some points have been slightly offset in x for clarity. The H1+ZEUS
combined values for Q2 ≥ 3.5 GeV2 are obtained from the measured reduced cross section and converted toF2p with a HERAPDF NLO fit, for all measured points where the predicted ratio of F2p to reduced crosssection was within 10% of unity. A turn-over is visible in the low-x points at medium Q2 (3.5 GeV2 and6 GeV2 ) for the H1+ZEUS combined values. In order to obtain F2p from the measured reduced cross-section,
FL must be estimated; for the points shown, this estimate is obtained from HERAPDF2.0. No FL value
consistent with the HERA data can eliminate the turn-over. This may indicate that at low x and Q2 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 Q2 < 3.5 GeV2 ),H. Abramowicz et al., Eur. Phys. J. C75, 580 (2015) (data for Q2 ≥ 3.5 GeV2 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.

31st May, 2024

18. Structure Functions(x,Q ) + c(Q)2

H1+ZEUS5

H1+ZEUS

0.5

bb

EMC

F2

cc

2

F2 (x,Q ) + c(Q)

32

4
0.4

3
0.3

2

0.2

1

0.1

0
10

0
-7

10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

1

x

10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

1

x

Figure 18.13: a) The charm-quark structure function F2cc (x), i.e. that part of the inclusive structure

function F2p 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 F2cc
using the PDFs from the MMHT NNLO fit) and muons on iron (EMC). For the purpose of plotting, a constantc(Q) = 0.07iQ 1.7 is added to F2cc where iQ is the number of the Q2 bin, ranging from 1 (Q2 = 2.5 GeV2 ) to12 (Q2 = 2000 GeV2 ). 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).
b) The bottom-quark structure function F2bb (x). For the purpose of plotting, a constant c(Q) = 0.01i1.6
Q is
added to F2bb where iQ is the number of the Q2 bin, ranging from 1 (Q2 = 2.5 GeV2 ) to 12 (Q2 = 2000 GeV2 ).
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 afunction of x in bins of Q2 . Points may have been slightly offset in x for clarity. Some data have been
rebinned to common Q2 values. Also shown is the MMHT2014 parameterization given at several Q2 values(L. A. Harland-Lang et al., Eur. Phys. J. C75, 204 (2015)).

31st May, 2024

18. Structure Functions

1

γZ

1
2

x F3

x F3

γZ

33

Q =40-180 GeV

2

0.5

0.5

0.25

0.25

0

0
10

2

0.75

2

Q =1000 GeV

0.75

-2

10

-1

0

1

0.2

0.4

0.6

x
CCFR
NuTeV

2

xF3(x,Q ) + c(x)x65CHORUS

4

3

2

1

0
1

10

10
2

2

1
2

Q (GeV )

10

10
2

2
2

Q (GeV )Figure 18.14: The structure function xF3γZ 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).
c) The structure function xF3 of the nucleon measured in ν-Fe scattering. The data are plotted as a functionof Q2 in bins of fixed x. For the purpose of plotting, a constant c(x) = 0.5(ix − 1) is added to xF3 , whereix is the number of the x bin as shown in the plot. The NuTeV and CHORUS points have been shiftedto the nearest corresponding x bin as given in the plot and slightly offset in Q2 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.
31st May, 2024

18. Structure Functions
2

FL (x,Q )

34

1

0

1

0

-4

10

-2

1 10

-4

10

-2

1 10

-4

10

-2

1 10

-4

10

-2

1 10

-4

10

-2

1 10

-4

10

-2

1

x

2

FL (x,Q )

10

1

0

10

-3

10

-2

1

x

BCDMS
NMC
SLAC
H1

0

ZEUS

2

FL (Q )

10

-3

10

-2

10

-3

10

-2

10

-3

10

-2

10

-3

10

-2

10

-3

10

-2

x

0.5
H1
ZEUS

0

1

10

10

2
2

2

Q (GeV )Figure 18.15: Top panels: The longitudinal structure function FL as a function of x in bins of fixed Q2
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 FL by using the BDCMS parameterization of F2 (A.C. Benvenuti
et al., Phys. Lett. B223, 485 (1989)). It is assumed that the Q2 dependence of the fixed-target data is small
within a given Q2 bin. Some of the other data may have been rebinned to common Q2 values. Some points
have been slightly offset in x for clarity. Also shown is the MSTW2008 parameterization given at threeQ2 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).
Bottom panel: The longitudinal structure function FL as a function of Q2 . Some points have been slightly
offset in Q2 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 F2
structure function in order to extract FL . Statistical and systematic errors added in quadrature are shownfor both plots.
31st May, 2024

18. Structure Functions

35(i = 0)1pg (x, Q2) + ci

x=0.0036
x=0.004512

x=0.0055
x=0.007

10

EMC

SMC

E143

E155

HERMESCOMPASS 160 GeV

CLAS W>2.5 GeV

COMPASS 200 GeVCOMPASS NLO fit

x=0.009x=0.012ci = 12.1 - 0.7 ⋅ i

x=0.0178x=0.024x=0.035x=0.0496x=0.077(i = 10)x=0.12x=0.174

x=0.22x=0.292

x=0.41x=0.57x=0.74

0
1

10

102

Q2 (GeV2/ c 2)
Figure 18.16: World data on the spin-dependent structure function g1p as a function of Q2 for various

values of x The lines represent the Q2 dependence for each value of x, as determined from a NLO QCD
fit. The dashed ranges represent the region with W 2 < 10 (GeV/c2 )2 . References: EMC—J. Ashman
et al., Phys. Lett. B206, 363 (1988); Nucl. Phys. B328, 1 (1989); E143—K. Abe et al., Phys. Rev. D58,
112003 (1998); SMC—B. Adeva et al., Phys. Rev. D58, 112001 (1998); HERMES—A. Airapetian et al.,Phys. Rev. D75, 012007 (2007); E155—P.L. Anthony et al., Phys. Lett. B493, 19 (2000); COMPASS—
M.G. Alekseev et al., Phys. Lett. B690, 466 (2010), C. Adolph, et al., Phys. Lett. B753, 18 (2016); CLAS—
K.V. Dharmawardane et al., Phys. Lett. B641, 11 (2006) (which also includes resonance region data notshown on this plot — there is also low W 2 CLAS data in Y. Prok et al., Phys. Rev. C90, 025212 (2014) and
N. Guler et al., Phys. Rev. C92, 055201 (2015)).

31st May, 2024

18. Structure Functions

12
x=0.0045(i = 1)x=0.0055
x=0.007

10

SMC

E143

E155

HERMESCOMPASS

CLAS W>2.5 GeV

COMPASS NLO fit

x=0.009x=0.0121gd(x, Q2) + 12.1 - 0.7 ⋅ i

36

x=0.0178x=0.024x=0.035x=0.0496(i = 10)x=0.077x=0.124x=0.17x=0.22x=0.292

x=0.41x=0.57x=0.74

0
1

10

102

Q2 (GeV2/ c 2)
Figure 18.17: World data on the spin-dependent structure function g1d as a function of Q2 for various

values of x The lines represent the Q2 dependence for each value of x, as determined from a NLO QCD fit.
The dashed ranges represent the region with W 2 < 10 (GeV/c2 )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);

31st May, 2024

18. Structure Functions
14

2

F2 (x,Q )/α + c(x)

37

γ

12

10

8

6

4
ALEPH
DELPHI
L3
OPAL
AMY
JADE
PLUTO
TASSO
TOPAZ
TPC

2

0
10

-1

1

10

10

2
2

10

3

2

Q (GeV )Figure 18.18: The hadronic structure function of the photon F2γ divided by the fine structure constant

α measured in e+ e− scattering, shown as a function of Q2 for bins of x. Data points have been shiftedto the nearest corresponding x bin as given in the plot. Some points have been offset in Q2 for clarity.
Statistical and systematic errors added in quadrature are shown. For the purpose of plotting, a constantc(x) = 1.5ix is added to F2γ /α where ix 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).
31st May, 2024