4.2Deep inelastic scatteringcftp.tecnico.ulisboa.pt/~gernot.eichmann/2013-hadron... · 2013-07-08 · distributions of quarks and gluons inside the nucleon. Phase space in inelastic

108 Currents and scattering

4.2 Deep inelastic scattering

So far we have dealt with elastic eN scattering from where we can extract the nucleon’selectromagnetic form factors in the spacelike region. In a general inelastic scatteringprocess the nucleon will break up and produce all possible hadronic final states. De-pending on the invariant mass of the hadronic end product, the inelastic cross sectionwill exhibit nucleon resonance peaks and contain nucleon-meson continua. Moreover,deep inelastic scattering probes the composite nature of the nucleon: it gives us ac-cess to the parton distribution functions which measure the longitudinal momentumdistributions of quarks and gluons inside the nucleon.

Phase space in inelastic scattering. We continue to work with the variables de-fined in Eq. (D.4). In the limit of massless electrons (k2

i = k2f = 0) we have now three

independent Lorentz invariants: the spacelike momentum transfer τ ≥ 0, the inelastic-ity ω ≥ 0, and the crossing variable ν. It is experimentally convenient to work withthe variables E, θ and W , where E is the energy of the incoming lepton in the labframe, θ is the lepton scattering angle, and W = M

√1 + 4ω is the invariant mass of

the hadrons in the final state (see Eqs. (D.7) and (D.14)). In the one-photon exchangeapproximation, the cross section factorizes again in a leptonic and a hadronic part,where the hadronic subprocess depends only on τ and ω.

The resulting phase space in the (ω, τ) plane at fixed E is sketched in Fig. 4.4. Ifwe solve the relations (D.14) for τ(ω, θ, E) we obtain

τ = (ε− ω)4ε sin2 θ

2

1 + 4ε sin2 θ2

, ε :=E

2M. (4.51)

For constant lepton energy E, the physically allowed region is then bounded by ω = 0(elastic scattering), θ = 0 ⇒ τ = 0 (forward angles), and backward angles θ = π ⇒τ = (ε− ω) 4ε/(1 + 4ε) ≈ ε− ω, which is the triangular blue area in the plot. Its sizeis characterized by the external control parameter E: if we increase the energy of thelepton beam we can reach higher τ and ω values. The invariant mass W is directlyrelated to the variable ω: the elastic threshold ω = 0 corresponds to W = M , and0 < ω . 1 is the region where nucleon resonances appear at fixed W , starting with the∆(1232) peak; see Fig. 4.5. Above W ∼ 2 GeV, there is no visible resonance structureleft in the cross section. The limit τ + ω → ∞ and ω/τ = const. defines the Bjorkenlimit: this is the region of deep inelastic scattering (DIS) where scaling occurs (moreon that below). From Eqs. (D.7) and (D.8) we have

ω + τ =ν ′

2M, 1 +

ω

τ=

1

x, (4.52)

and hence the Bjorken limit means ν ′ → ∞ and constant Bjorken-x. The lines ofconstant x are shown in Fig. 4.4; the limit ω = 0 ⇔ x = 1 corresponds to elasticscattering.4

4It is more common to plot Fig. 4.4 in terms of ν′ and τ . The upper right quadrant in the (ω, τ)plane is then squeezed in the (ν′, τ) plane; the elastic limit ω = 0 becomes the line τ = ν′/(2M).

4.2 Deep inelastic scattering 109

Resonanceregion

(elastic)

Scalingregion

𝜔

𝜏

𝑥 ¹₃

𝑥 ≪ 1

𝑥 ¹₂

𝑥 1

𝜃 180°

𝜃 45°

𝜃 30°

𝜃 15°

𝜃 0°𝐸/2𝑀

18. Structure functions 1

NOTE: THE FIGURES IN THIS SECTION ARE INTENDED TO SHOW THE REPRESENTATIVE DATA.

THEY ARE NOT MEANT TO BE COMPLETE COMPILATIONS OF ALL THE WORLD’S RELIABLE DATA.

Figure 18.8: The proton structure function Fp2 measured in electromagnetic scattering of electrons and

positrons on protons (collider experiments H1 and ZEUS for Q2 ≥ 2 GeV2), in the kinematic domain of theHERA data (see Fig. 18.10 for data at smaller x and Q2), and for electrons (SLAC) and muons (BCDMS,E665, NMC) on a fixed target. Statistical and systematic errors added in quadrature are shown. The dataare 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 the x values ofthese data. For the purpose of plotting, F

p2 has been multiplied by 2ix , where ix is the number of the x bin,

ranging from ix = 1 (x = 0.85) to ix = 24 (x = 0.00005). References: H1 and ZEUS—F.D. Aaron et al.,JHEP 1001, 109 (2010); BCDMS—A.C. Benvenuti et al., Phys. Lett. B223, 485 (1989) (as given in [66]) ;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).

Figure 4.4: Phase space in inelastic eN scattering in the variables τ and ω at fixed leptonenergy E. The various regions illustrate: the physically allowed phase space (triangular, blue),the resonance region (brown) and the deep inelastic region (red). Lines of constant scatteringangle θ and Bjorken-x are also shown.

Cross section and structure functions. Let’s work out the cross section for inelasticeN scattering. In an inclusive measurement only the outgoing electron is detected butnot the remnants of the proton. The cross section in the one-photon approximation hasstill the generic form of Eqs. (4.18–4.19) with the same leptonic tensor (4.25); however,the hadronic contribution to the invariant matrix element |M|2 and to the phase spacefactor now sums over all possible final states. We can absorb the integral over d3pfand the δ−function for energy-momentum conservation into a generic hadronic tensorWµν , whose explicit form we will discuss later in Eq. (4.79):

dσ =1

4ME

d3kf(2π)3 2E′

e4

q4Lµν 4πM Wµν ⇒ dσ

dΩ dE′=α2

q4

E′

ELµνW

µν . (4.53)

Since current conservation must still hold because the sum of the outgoing chargesmust equal the nucleon charge, Wµν must be transverse in its Lorentz indices. Themost general transverse tensor that we can construct according to these constraints isgiven by5

Wµν = −W1(τ, ω)Tµνq +W2(τ, ω)

M2pµT p

νT , (4.54)

5In principle there would be three more tensor structures apart from the two written here, namelyqµqν and pµT q

ν ± qµpνT , but the transversality constraints qµWµν = 0, Wµνqν = 0 ensure that theirdressing functions vanish. Moreover, we only deal with unpolarized scattering; in general there wouldbe two further spin-dependent structure functions ∼ g1, g2 and also another structure in the leptontensor which only contribute to polarized scattering.


Figure 4.5: Double-differential inelastic eN cross section from Eq. (4.57) at fixed leptonenergy E and scattering angle θ. At large invariant masses, the resonance peaks are washedout. (Halzen and Martin, Quarks and Leptons: An Introductory Course in Modern ParticlePhysics, Wiley, 1984.)

where the transverse projector and momenta were defined in Eq. (4.26). The tworesponse functions W1 and W2 depend on the Lorentz invariants τ and ω. Combiningthis with the leptonic tensor yields

LµνWµν = 4

[W2

M2

((p · k)2 +

q2

4p2T

)−W1

(k2 +

3

4q2

)]

= 4M2[W2

(ν2 − (τ + ω)2 − τ

)+ 2W1 τ

]

= 4EE′ cos2 θ

2

[W2 + 2W1 tan2 θ

2

].

(4.55)

In going from the first to the second line we used p · k = M2 ν, k2 = M2τ and

p2T = p2 − (p · q)2

q2= M2

(1 + 2ω + τ +

ω2

τ

)=M2

τ

(τ + (τ + ω)2

), (4.56)

and to obtain the third line we exploited Eqs. (D.14) and (D.11). The resulting crosssection, shown in Fig. 4.5, is

d2σ

dΩ dE′=

α2 cos2 θ2

4E2 sin4 θ2

[W2 + 2W1 tan2 θ

2

]. (4.57)

How does this compare to the limit of elastic scattering? From (4.21) and (4.29)we can write down the double-differential cross section for a pointlike fermion in theelastic case:

d2σ

dΩ dE′=|M|24ME

1

(4π)2

E′ δ(ω)

2M2=

α2

4M2τ2

E′

E

δ(ω)

2M(ν2 + τ2 − τ)

=α2 cos2 θ

2

4E2 sin4 θ2

δ(ω)

2M

(1 + 2τ tan2 θ

2

).

(4.58)


Hence, the response functions reduce in the elastic limit to

W1(τ, ω) = τδ(ω)

2M, W2(τ, ω) =

δ(ω)

2M. (4.59)

The elastic peak in the cross section is clearly visible in Fig. 4.5. From comparing thetwo cross sections we also find that W1 encodes the spin of the target and vanishes for aspinless particle. For scattering on a composite nucleon, the results (4.59) in the elasticlimit have to be multiplied with the Sachs form factor combinations in the Rosenbluthcross section (4.33). Finally, we can trade the dependence on ω by a dependence onthe Bjorken variable x if we use the relations

τ =ν ′

2Mx , ω =

ν ′

2M(1− x) ⇒ 2MW1 = x δ(1− x) =: 2F1 ,

ν ′W2 = δ(1− x) =: F2 .(4.60)

Here we defined the dimensionless structure functions6 F1(τ, x) and F2(τ, x). For elas-tic scattering on a pointlike particle they are only functions of x; in addition, theδ−function enforces x = 1 in the elastic limit.

Bjorken scaling and the parton model. One might expect that for inelastic scatter-ing processes (x 6= 1), away from the nucleon resonance region, the structure functionsF1 and F2 are complicated functions of τ and x. However, it turns out that in the deepinelastic domain (τ and ω large) they are almost independent of τ and only functionsof ω/τ , or equivalently x:

F1,2(τ, x) ≈ F1,2(x) , F2(x) = 2xF1(x) . (4.61)

This is called Bjorken scaling and visible in the left plot of Fig. 4.6. The second relationis the Callan-Gross relation which entails that F1 and F2 are not independent of eachother. The origin of scaling is simply a dimensional argument that follows from thenear scale invariance of massless perturbative QCD (up to logarithmic corrections).A dimensionless function can only depend on dimensionless variables; τ and ω aredimensionless because we scaled the momenta with the nucleon mass, which requiredthe presence of a nonperturbative nucleon bound-state mass to begin with. If wescatter instead on (nearly) massless quarks, no such scale is available and therefore thedimensionless structure functions cannot depend on τ and ω individually but only ontheir dimensionless combination τ/ω ∼ q2/p · q. Hence, the observation of scaling isan indication for the composite nature of the nucleon in terms of pointlike, essentiallymassless quarks and gluons.

The experimental fact of Bjorken scaling has led to the development of the partonmodel. The proton is viewed as a collection of ’partons’, namely valence quarks, seaquarks and gluons. The incoming momentum pi of the proton (mass M) is the sum ofthe parton momenta: pi =

∑k pk, where pk is the four-momentum of a single onshell

parton with mass mk. The basic assumption we need in the following is collinearity:

6Try not to confuse them with the Dirac and Pauli form factors from the last section. In fact, eventheir physical meanings are reversed: the structure function F1 carries the spin dependence, whereas inthe form factor case it is rather the Pauli form factor (or the magnetic form factor GM ) that containsthe nucleon spin.


pk = ξk pi, which can be justified in the infinite momentum frame. If we genericallywrite

pi =

(√p2 +M2

p

), pk =

(√ξ2k p

2 + (p⊥k )2 +m2k

ξk p+ p⊥k

), (4.62)

then ξk defines the longitudinal momentum fraction of parton k in the direction ofthe proton’s three-momentum p. In the infinite-momentum frame (|p| → ∞) we canneglect the transverse components and masses:7

|p⊥k | |p| , mk ≈ ξkM |p| , ⇒ pk ≈ ξk pi ,∑

k

ξk = 1 . (4.63)

The collinearity assumption allows for a simple interpretation of the Bjorken scalingvariable. We have seen in Appendix D that elastic scattering on the nucleon correspondsto x = −q2/(2pi · q) = 1. In the inelastic process (x 6= 1), elastic scattering on a singleparton k then entails that

xk := − q2

2pk · q=

x

ξk= 1 ⇒ ξk = x , (4.64)

so that the Bjorken variable x assumes the meaning of the parton’s longitudinal mo-mentum fraction in the infinite-momentum frame. The photon only couples to thosepartons whose momentum fraction is ξk = x, hence a measurement of the structurefunction F2(x) allows us to ’see’ how the parton momenta are distributed inside theproton. In elastic scattering we have x = 1 and the photon couples to the whole protonsince it carries the full momentum.

Let us define the momentum distribution fk(ξ) of a parton in the hadron (the partondistribution function or PDF), so that fk(ξ) dξ is the probability density that a partoncarries a momentum fraction between ξ and ξ + dξ. Momentum conservation implies

∑

k

1∫

0

dξ ξ fk(ξ) = 1 . (4.65)

Now suppose we scatter on spin-12 quarks. Using the relations (4.60) together with

xk = x/ξk and mk = ξkM , the structure functions F ki for the parton k are

2F k1 = 2MW k1 =

M

mk2mkW

k1 =

M

mkxk δ(1− xk) = δ(ξk − x) ,

F k2 = ν ′W k2 = δ(1− xk) =

ξ2k

xδ(ξk − x) = x δ(ξk − x) ,

(4.66)

and integrating over all partons yields

Fi(x) =∑

k

e2k

∫dξ fk(ξ)F

ki (ξ, x) ⇒

F1(x) = 12

∑k e

2k fk(x) ,

F2(x) = x∑

k e2k fk(x) .

(4.67)

7The relation mk = ξkM is a bit nonsensical as it would imply that the ’mass’ of a parton changeswith its momentum fraction, but we need it for consistency of the naive parton model.

4.2 Deep inelastic scattering 11318. Structure functions 1







12 18. Structure functions

0

0.2

0.4

0.6

0.8

1

1.2

10 -4 10 -3 10 -2 10 -1

x

x f(x

)

0

0.2

0.4

0.6

0.8

1

1.2

10 -4 10 -3 10 -2 10 -1

x

x f(x

)

Figure 18.4: Distributions of x times the unpolarized parton distributions f(x)(where f = uv, dv, u, d, s, c, b, g) and their associated uncertainties using the NNLOMSTW2008 parameterization [13] at a scale µ2 = 10 GeV2 and µ2 = 10, 000 GeV2.

18.4. The hadronic structure of the photon

Besides the direct interactions of the photon, it is possible for it to fluctuate into ahadronic state via the process γ → qq. While in this state, the partonic content of thephoton may be resolved, 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 ofthe quasi-real photon emitted by the other lepton. The perturbative LO contributions,γ → qq followed by γ∗q → q, are subject to QCD corrections due to the coupling ofquarks to gluons.

Often the equivalent-photon approximation is used to express the differential crosssection for deep inelastic electron–photon scattering in terms of the structure functionsof the transverse quasi-real photon times a flux factor NT

γ (for these incoming quasi-realphotons of transverse polarization)

d2σ

dxdQ2= NT

γ2πα2

xQ4

[(1 + (1 − y)2

)Fγ2 (x,Q2) − y2F

γL(x,Q2)

],

where we have used Fγ2 = 2xF

γT +F

γL , not to be confused with F

γ2 of Sec. 18.2. Complete

formulae are given, for example, in the comprehensive review of Ref. 68.

The hadronic photon structure function, Fγ2 , evolves with increasing Q2 from

the ‘hadron-like’ behavior, calculable via the vector-meson-dominance model, to thedominating ‘point-like’ behaviour, calculable in perturbative QCD. Due to the point-likecoupling, the logarithmic evolution of F

γ2 with Q2 has a positive slope for all values of x,

June 18, 2012 16:19

Figure 4.6: Left panel: scaling behavior in the structure function F2(Q2, x). Source: PDG,Beringer et al., Phys. Rev. D86, 010001 (2012). Right panels: experimental data for the ratioand difference of proton and neutron structure functions in Eq. (4.73). Source: Halzen andMartin, see Fig. 4.5.

Hence we have shown that in the parton model F1 and F2 are indeed only functions ofx, and we can confirm the Callan-Gross relation (4.61). The latter is an experimentalindication for the spin-1

2 nature of the quarks: if quarks had spin zero, F1(x) wouldvanish.

Due to the Callan-Gross relation only the structure function F2(x) will be relevantin the following. What will it look like? If the proton consisted of a single ’quark’that carried all of its momentum, the structure function would have a single peak atx = 1. If it consisted of three non-interacting quarks, the quarks would all carry thesame momentum fraction and F2(x) would have a peak at x = 1

3 . If the three quarksinteract with each other, they can exchange momentum and hence the momentumfraction carried by each quark will fluctuate; the resulting structure function is a smoothdistribution peaked near x = 1

3 . Finally, the presence of sea quarks will lead to anenhancement at small x because sea quarks are created in Bremsstrahlung-like processes

which are typically enhanced at small momenta and lead to xf(x)x→0−−→ const. Note

that gluons will also contribute to the momentum sum rule (4.65) whereas the structurefunction only probes electrically charged partons (quarks).


Parton distribution functions. Now let’s see how much information on the PDFswe can gather from experimental data on F2(x). There is no sensible way to distinguishtwo identical partons within a proton, but we can still group them according to thevarious quark and antiquark flavors: fk(x) = u(x), u(x), d(x), d(x), etc., so that wehave

F p2 (x)

x= q2

u ((u(x) + u(x)) + q2d

(d(x) + d(x)

)+ q2

s (s(x) + s(x)) + . . . (4.68)

It is usually sufficient to stop at the strange quark because the probability for findingcharm in the proton is negligible. u(x) is the probability distribution for up quarks inthe proton, u(x) that of anti-up quarks, and so on. One can also measure the structurefunction Fn2 of the neutron via electron-deuteron scattering. Charge symmetry entailsthat the d distribution in the neutron is identical to the u distribution in the proton:u = up = dn, d = dp = un, s = sp = sn, and analogously for the antiquark PDFs:

Fn2 (x)

x= q2

d ((u(x) + u(x)) + q2u

(d(x) + d(x)

)+ q2

s (s(x) + s(x)) + . . . (4.69)

In the following it will be more convenient to work with valence- and sea-quark distri-butions, defined via

u = uv + us ,

d = dv + ds ,

u = us ,

d = ds ,

s = ss ,

s = ss ,(4.70)

because antiquarks and strange quarks can only appear in the sea. Now, since thePDFs are number densities defined on the momentum fraction x, the integrals overthis range are just the total flavor numbers of each quark type:

1∫

0

dxuv(x) = 2 ,

1∫

0

dx dv(x) = 1 ,

1∫

0

dx[fs(x)− fs(x)

]= 0 . (4.71)

The third relation expresses fermion number conservation for each flavor f = u, d, s:by summing over all individual partons, we must recover charge 1, baryon number 1and strangeness 0 of the proton.

Can we extract the valence and sea distributions from the data for F p,n2 (x)? We havetwo measured quantities but too many unknowns. Let’s make the further simplifyingassumption that all sea-quark distributions are identical: fs(x) = fs(x) =: S(x). Thenthe structure functions for the proton and neutron become

F p2 (x)

x= q2

u uv(x) + q2d dv(x) + (q2

u + q2d + q2

s) 2S(x) ,

Fn2 (x)

x= q2

d uv(x) + q2u dv(x) + (q2

u + q2d + q2

s) 2S(x) ,

(4.72)

from where we can form their ratio and their difference:

R =Fn2F p2

=uv + 4dv + 12S

4uv + dv + 12S, F p2 − Fn2 =

x

3(uv − dv) . (4.73)


18. Structure functions 1








0

0.2

0.4

0.6

0.8

1

1.2

10 -4 10 -3 10 -2 10 -1

x

x f(x

)

0

0.2

0.4

0.6

0.8

1

1.2

10 -4 10 -3 10 -2 10 -1

xx

f(x)

Figure 18.4: Distributions of x times the unpolarized parton distributions f(x)(where f = uv, dv, u, d, s, c, b, g) and their associated uncertainties using the NNLOMSTW2008 parameterization [13] at a scale µ2 = 10 GeV2 and µ2 = 10, 000 GeV2.





d2σ

dxdQ2= NT

γ2πα2

xQ4

[(1 + (1 − y)2

)Fγ2 (x,Q2) − y2F

γL(x,Q2)

],


γT +F







June 18, 2012 16:19


0

0.2

0.4

0.6

0.8

1

1.2

10 -4 10 -3 10 -2 10 -1

x

x f(x

)

0

0.2

0.4

0.6

0.8

1

1.2

10 -4 10 -3 10 -2 10 -1

x

x f(x

)Figure 18.4: Distributions of x times the unpolarized parton distributions f(x)(where f = uv, dv, u, d, s, c, b, g) and their associated uncertainties using the NNLOMSTW2008 parameterization [13] at a scale µ2 = 10 GeV2 and µ2 = 10, 000 GeV2.





d2σ

dxdQ2= NT

γ2πα2

xQ4

[(1 + (1 − y)2

)Fγ2 (x,Q2) − y2F

γL(x,Q2)

],


γT +F







June 18, 2012 16:19

Figure 4.7: Valence, sea-quark and gluon PDFs shown at two different resolution scales.Source: PDG, see Fig. 4.6.

The ratio satisfies the Nachtmann inequality 14 ≤ R(x) ≤ 4: in a region of x where

the up (down) quarks dominate, we have R = 14 (R = 4); if the sea quarks dominate

we will find R = 1. The ratio is plotted in Fig. 4.6 and reveals that the sea quarksare indeed dominant at small x whereas valence up quarks are important at large x.The difference in (4.73) is also shown in the figure: it measures only the valence-quarkcontribution and shows a peak around x = 1/3, as we had expected. Finally, the sum

9

5(F p2 + Fn2 ) = x

(uv + dv +

24

5S

)(4.74)

can be plugged into the momentum sum rule (4.65) which now takes the form∫dxx (uv + dv + 6S) + ε = 1 , (4.75)

where ε is the gluon contribution to the proton’s longitudinal momentum. From theexperimental data we can roughly estimate

9

5

∫dx (F p2 + Fn2 ) ≈ 0.54 ≈ 1− ε , (4.76)

which entails that the gluons carry almost half of the proton’s momentum. In fact, thegluon PDFs dominate at small values of x, see Fig. 4.7.

How good is the assumption that all sea-quark distributions are identical? If we goback to the original equations (4.68) and (4.69), take their difference and integrate overx, we have∫dx

F p2 − Fn2x

=1

3

∫dx (uv − dv + us + us − ds − ds)

(4.71)=

1

3+

2

3

∫dx (us − ds)


which should equal 13 if us = ds = S (this is the Gottfried sum rule). Instead, the

experimental value is ∼ 0.23 ⇒∫dx (ds − us) ∼ 0.15, which entails that the light

quark sea is indeed flavor-asymmetric.

Scaling violations. The left plot in Fig. 4.6 demonstrates that scaling is not exactbecause the structure functions exhibit a Q2 dependence, which is most pronounced atsmall and large values of x. In terms of the PDFs, this implies that their x−dependenceis not completely independent of the resolution scale Q2 but also evolves with Q2, whichcan be seen in Fig. 4.7. We can intuitively understand this as follows: a photon withintermediate Q2 does not resolve the full spatial structure of the proton and mainlysees three interacting quarks, together with parts of the sea. In contrast, a high-Q2 photon can resolve small distances and will reveal more and more of the quarksea which contains short-distance processes such as gluon emission from a quark orgluon splitting into qq pairs. As a result, the sea-quark contributions will be moreprominent at higher Q2. On the other hand, since the photon can resolve more partons,momentum conservation implies that each parton carries now a smaller fraction of thetotal momentum, and hence the PDFs will be shifted to smaller x. The resultingstructure function F2(x) that sums up the individual quark PDFs will rise with higherQ2 at small x and fall with higher Q2 at large x.

The short-distance dynamics depend on the resolution scale through the couplingαs(Q

2). As a consequence, the individual quark structure functions F ki will no longer bemere δ−functions as in Eq. (4.66) but also inherit a Q2 dependence from the coupling.Since the coupling is dimensionless, it also introduces a scale µ (the factorization scale),so that Eq. (4.67) becomes

Fi(x,Q2) =

∑

k

e2k

∫dξ fk(ξ, µ)F ki

(ξ, x, Q

2

µ2

). (4.77)

The F ki encode the short-distance splitting processes and are calculable in perturbativeQCD. The PDFs fk, which now also depend on µ, are inherently nonperturbative andhave to be fitted to experimental data or calculated with nonperturbative methods.Since the nucleon structure function must be independent of the factorization scale µ,its total derivative with respect to µ must vanish. Similarly to the Callan-Symanzikequation (1.87), one then derives the DGLAP equations (Dokshitzer, Gribov, Lipatov,Altarelli, Parisi) dFi/dµ = 0 which relate PDFs at different µ with each other andthereby allow one to calculate the scaling violations using QCD perturbation theory.

Compton amplitude. How can PDFs be calculated nonperturbatively? The keyidea is to relate the hadronic tensor Wµν that enters the inelastic eN cross section tothe nucleon’s forward Compton scattering amplitude. The latter is measured in theprocess γN → γN and given by

Tµν(p, q) := i

∫d4z eiqz 〈N(pi)|TV µ

em

(z2

)V ν

em

(− z2)|N(pi)〉 , (4.78)

where V µem is the electromagnetic current operator from Eq. (2.183). The hadronic


tensor in Eq. (4.53) has the form

4πM Wµν(p, q) =∑

X

d3pf(2π)3 2EX

〈N(pi)|V µem(0) |X(pf )〉〈X(pf )|V ν

em(0) |N(pi)〉

× (2π)4 δ4(q + pi − pf )

(4.79)

and encodes the electromagnetic transition from the nucleon to all possible final states.If we write the δ−function in momentum space,

(2π)4 δ4(q + pi − pf ) =

∫d4z ei(q+pi−pf )z , (4.80)

use Eq. (2.75) to shuffle the z−dependence of the phase factor ei(pi−pf )z into the currentoperators, and sum over the complete set of states X, we obtain:

4πM Wµν(p, q) =

∫d4z eiqz 〈N(pi)|V µ

em

(z2

)V ν

em

(− z2)|N(pi)〉

=

∫d4z eiqz 〈N(pi)|

[V µ

em

(z2

), V ν

em

(− z2)]|N(pi)〉 .

(4.81)

In the second line we have replaced the product of the currents by their commutatorbecause the matrix element of ∼ V ν

em

(− z2)V µ

em

(z2

)is zero: it gives rise to a δ−function

δ(q−pi+pf ) which cannot be saturated by any intermediate state. Energy conservationwould require EX = M−E+E′ = M−ν ′ ≤M , but the nucleon is the lightest ground-state baryon.

By means of the optical theorem, Eq. (4.81) can be written as the imaginary partof the forward Compton scattering amplitude: 4πM Wµν(p, q) = 2 ImTµν(p, q). Inthe forward limit, the Lorentz-invariant dressing functions of the Compton amplitudedepend on the same variables as the inelastic cross section, τ and x. However, whereasτ ≥ 0 for real or virtual photons, the variable x is no longer constrained to the in-terval x ∈ [0, 1] but can be arbitrary. The Compton amplitude has non-analyticitiesarising from intermediate baryon resonances and baryon-meson continua in the s andu channels, extending from8

M2 ≤ s <∞ , M2 ≤ u <∞ ⇔ −1 ≤ x ≤ 1 , (4.83)

and the hadronic tensor is proportional to the imaginary part along the cut x ∈ [0, 1] inthe complex x plane. Hence, a theoretical handle on the nucleon Compton scatteringamplitude allows us also to compute the nucleon’s structure functions in DIS.

8If we apply the kinematics of App. D to the Compton scattering process directly, then in theforward limit (vanishing momentum transfer q = 0) we have p = pi = pf and the photon momentumis k = ki = kf , so that k2 and the crossing variable ν are the independent Lorentz-invariants. Thevariables τ and x, as originally defined in DIS, are then given by

τ = − k2

4M2, x = − k2

2p · k =2τ

ν⇒ s, u = M2 + 4M2τ

(−1± 1

x

)(4.82)

and therefore s, u ≥M2 implies |x| ≤ 1.


𝑝

𝑥𝑝

𝑞 2

~ 𝑝𝑝

𝑥𝑝𝑥𝑝

𝑞𝑞

PDF

Figure 4.8: Hadronic tensor Wµν in the parton model, and its relation with the forwardCompton scattering amplitude and its factorized handbag structure.

Light cone expansion. What about the PDFs? To begin with, it is important torealize that in the Bjorken limit the Fourier transform in Eq. (4.81) is dominated bythe behavior close to the light cone z2 → 0, i.e., where the two interaction points areseparated by a lightlike distance. This is easiest seen using light-cone variables:

a± :=1√2

(a0 ± a3) , a⊥ = (a1, a2) ⇒ a · b = a+b− + a−b+ − a⊥ · b⊥ . (4.84)

Then the integral (4.81) becomes schematically:

W (p, q) =

∫dz− eiq+z−

∫dz+ e

iq−z+

∫

z2⊥<2z+z−

d2z⊥ e−iq⊥·z⊥W (p, z) . (4.85)

The domain of the z⊥ integration is restricted since the current commutator vanishesoutside the light cone (z2 = 2z+z− − z2

⊥ < 0) due to causality. In light-cone variables,the Bjorken limit ν ′ →∞, x = const. corresponds to q+ →∞ and q− = const:

√2 q± = q0 ± q3 (D.9)

= ν ′ ±√ν ′2 − q2 = ν ′

(1±

√1 +

2Mx

ν ′

)≈

2ν ′ +Mx+ . . .

−Mx+ . . .

For q+ → ∞ and q− = const, the integral (4.85) is determined by the behavior of theintegrand for z− → 0 and z+ finite; this is the area with the least oscillations accordingto the Riemann-Lebesgue lemma. The condition z2

⊥ < 2z+z− then implies z2 → 0+

but zµ 6= 0, which is the light cone.In order to proceed one has to work out the current commutator in Eq. (4.81). We

derived equal-time current commutators earlier in Eq. (2.42) using the anticommutationrelations (1.46) for the quark fields. For free fields one can generalize that formula tounequal times x0 6= y0 with the generalized anticommutation relations

ψ(x), ψ(y)

= S(x− y),

ψ(x), ψ(y)

=ψ(x), ψ(y)

= 0 , (4.86)

where S(z) := (i/∂+m) ∆(z), and ∆(z) is the causal propagator9 that vanishes outside

9In contrast to the Feynman propagator (2.82), the causal propagator sums up the positive- andnegative-energy pole residues of a free scalar propagator.


the light cone, i.e., for spacelike distances z2 < 0:

∆(z) :=

∫d3p

2Ep

e−ipz − eipz(2π)3

∣∣∣p0=Ep

=

∫d4p

(2π)3e−ipz ε(p0) δ(p2 −m2) , (4.87)

and ε(a) = a/|a| = Θ(a) − Θ(−a) is the sign function. At equal times z0 = 0, thecausal propagators reduce to ∆(z) = 0, ∂0∆(z) = −iδ3(z) and S(z) = γ0 δ

3(z) whichreproduces the equal-time (anti-)commutation relations for scalar and fermion fields.Rederiving the current commutator relation in this case gives the result10

[jΓa (x), jΓ′

b (y)]

= ifabc j+c (x, y) + dabc j

−c (x, y) +

δabN

j−(x, y) , (4.88)

which depends on the bilocal currents

j±a (x, y) :=1

2

(ψ(x) ΓS(x− y) Γ′ ta ψ(y)± ψ(y) Γ′ S(y − x) Γ ta ψ(x)

). (4.89)

Here we can already recognize the handbag structure from Fig. 4.8 when putting theresult back in the hadronic tensor Wµν ; for the electromagnetic current commutator wehave Γ = γµ and Γ′ = γν . The light-cone singularities come from the free propagatorS(z) which for a massless fermion reduces to

S(z)m=0−−−−→ 1

2π/∂(ε(z0) δ(z2)

). (4.90)

It represents the hard part of the process, namely the scattering of the photon on asingle perturbative quark which was the underlying assumption of the parton model.

The soft part is expressed through the remaining matrix element of bilocal quark-antiquark currents which is closely related to the quantity in Eq. (4.2). One can workout the Dirac structures for ΓS(z)Γ′ and Γ′S(−z)Γ and expand the resulting currentsin Taylor series about z = 0. This leads to the operator product expansion (OPE),schematically written as

j(z2 ,−

z2

)=∑

i

ci(z)Oi(0) , (4.91)

where the Oi(0) are local operators and the ci(z) are the respective Wilson coefficients.The operators which are most important at high Q2 are those for which the ci(z) aremost singular as z2 → 0. This allows for a rigorous definition of PDFs that enter inEq. (4.77) and makes them accessible for nonperturbative calculations.

Finally, the relation with the Compton amplitude also allows one to define non-forward generalized parton distributions (GPDs). They encode the transverse structureof the proton which is related to the orbital momentum carried by the quarks andgluons. In contrast to the PDFs they are no longer connected with deep inelasticscattering because nonvanishing momentum transfer implies pf 6= pi. Hence, they haveto be extracted directly from deeply virtual Compton scattering (DVCS) or relatedprocesses.

10Extra care should be taken with regard to Schwinger terms, which include higher derivatives ofthe δ−function and do not show up in commutators of zero components of currents.

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4.2Deep inelastic scatteringcftp.tecnico.ulisboa.pt/~gernot.eichmann/2013-hadron... · 2013-07-08 · distributions of quarks and gluons inside the nucleon. Phase space in inelastic