Quarks, Gluons, and the Strong Interaction
8.3 Scaling Violations of the Structure Functions
110 8 Quarks, Gluons, and the Strong Interaction
From (7.13) we can see that the Q2-dependence of the coupling strength corresponds to a dependence on separation. For very small distances and correspondingly high values of Q2, the interquark coupling decreases, vanishing asymptotically. In the limit Q2 ! 1, quarks can be considered “free”, this is calledasymptotic freedom. By contrast, at large distances, the interquark coupling increases so strongly that it is impossible to detach individual quarks from hadrons (confinement).
8.3 Scaling Violations of the Structure Functions 111
F2(x,Q 2)
1
0.5
x = 0 .008 (x 4.0)
x = 0 .0125 (x 3.2)
x = 0 .0175 (x 2.5)
x = 0 .025 (x 2.0)
x = 0 .035 (x 1.5)
x = 0 .050 (x 1.2)
x = 0 .070 (x 1.0)
1 10
Deuteron NMC SLAC BCDMS
Q2 [(GeV/c)2] F2(x,Q2)
1 10
x = 0 .009 (x 7.5)
x=0.11 (x 5.2)
x = 0 .14 (x 3.7)
x = 0 .18 (x 2.5)
x = 0 .225 (x 1.7)
x = 0 .50 (x 1.0) x = 0 .275 (x 1.2)
x = 0 .35 (x 1.0) 1
0.1 Deuteron NMC SLAC BCDMS
Q2 [(GeV/c)2]
Fig. 8.5 Structure functionF2 of the deuteron as a function ofQ2 at different values ofxon a logarithmic scale. The results shown are from muon scattering at CERN (NMC and BCDMS collaboration) [5,9] and from electron scattering at SLAC [21]. For clarity, the data at the various values ofxare multiplied by constant factors. Thesolid lineis a QCD fit, taking into account the theoretically predicted scaling violation. The gluon distribution and the strong coupling constant are free parameters here
outgoing electron proton
remnants
struck quark
p e
Fig. 8.6 A deep-inelastic scattering event in the H1 detector. The proton beam enters from the right, the electron beam from the left
112 8 Quarks, Gluons, and the Strong Interaction
10-1 1 10 102 103 104 105 106 107 108 109
10-1 1 10 102 103 104 105 106
x = 0.00005, j = 45
x = 0.0002, j = 39 x = 0.00032, j = 37
x = 0.0005, j = 35 x = 0.0008, j = 33
x = 0.0013, j = 31 x = 0.0021, j = 29
x = 0.0032, j = 27 x = 0.005, j = 26 x = 0.008, j = 25
x = 0.013, j = 24 x = 0.021, j = 23
x = 0.032, j = 22
x = 0.85, j = 13 HERMES
SLAC NMC E665 BCDMS HERA
Q2[ (GeV/c )2 ]
x = 0.75, j = 14
x = 0.65, j = 15 x = 0.40, j = 16 x = 0.25, j = 17 x = 0.13, j = 19 x = 0.08, j = 20
x = 0.18, j = 18 x = 0.05, j = 21
pj
x = 0.00008, j = 43 x = 0.00013, j = 41
Fig. 8.7 The proton structure function Fp2 as a function ofQ2 in intervals of x. Shown is a combined data set of the two HERA experiments H1 and ZEUS together with data from deep- inelastic scattering experiments of the first and second generation with a stationary proton target
In Fig.8.7 a combined data set of both experiments is presented for the structure functionF2p.x;Q2/measured in deep-inelastic positron-proton scattering at a centre-of-mass energyp
sD319GeV. Also shown are data from deep-inelastic scattering experiments with electrons, positrons or muons using stationary proton targets. Only data in theQ2 region aboveQ2min D 0:8 .GeV=c/2are shown, where the structure functionF2 can be interpreted in the framework of the quark-parton model. For better visibility the data at eachxvalue have been multiplied by constant factors and a fewxintervals have been omitted.
8.3 Scaling Violations of the Structure Functions 113
Q2=15 (GeV /c)2
Q2=3.5 (GeV /c)2 Q2=650 (GeV /c)2
100 Momentum fraction x
Proton structure function F2
2
1.5
1
0.5
0
10-1 10-2 10-3 10-4 10-5 10-6
Fig. 8.8 Schematic presentation of the proton structure functionF2as function ofxat three values ofQ2
The HERA data cover the impressive large kinematic range106 < x < 0:7 and0:04 .GeV=c/2 < Q2 < 105.GeV=c/2. This is many times larger than for the experiments of the first and second generation using a stationary proton target. Also here we see at low values ofxa considerable increase ofF2 withQ2; the smaller x, the steeper the increase. At large values ofxa somewhat weaker decrease of the structure function withQ2is observed.
The variation of thexdependence of the proton structure function with Q2 is schematically shown in Fig.8.8for three values of Q2. The lines are a fit to the experimental data of all the experiments of the three generations taking into account the scaling violation predicted by QCD. The experimental points were omitted as to clearly show the decrease of the structure function with increasingQ2 at large values ofx.
The DGLAP equations This violation of scaling is not caused by a finite size of the quarks. In the framework of QCD, it can be traced back to fundamental processes in which the constituents of the nucleon continuously interact with each other (Fig.8.1). Quarks can emit or absorb gluons, gluons may split into qq pairs, or emit gluons themselves. Thus, the momentum distribution between the constituents of the nucleon is continually changing.
Figure8.9is an attempt to illustrate how this alters the measurements of structure functions at different values ofQ2. A virtual photon can resolve dimensions of the order of„=p
Q2. At smallQ2DQ20, quarks and any possibly emitted gluons cannot be distinguished and a quark distribution q.x;Q20/is measured. At largerQ2 and higher space-time resolution, emission and splitting processes must be considered.
114 8 Quarks, Gluons, and the Strong Interaction
q ( x )
q ( y )
e e
Q02 Q >2 Q02
P (x/y)qq q ( x )
Fig. 8.9 With increasingQ2the space-time resolution of the virtual photon increases, allowing to resolve more partons
A quark with the momentum fractionxcan originate from a parent quark with a larger momentum fractionythat has radiated a gluon (cf. Fig.8.1a). The probability that this happens is proportional to˛s.Q2/Pqq.x=y/, wherePqq.x=y/is a so-called splitting function.
But a quark with momentum fractionxcan also originate from a gluon with higher momentum fraction y (cf. Fig.8.1b). The probability for this process is proportional to another splitting functionPqg.x=y/. Similarly the gluon distribution g.x;Q2/ is modified by radiation of gluons from a quark (Pgq.x=y/) or from another gluon (Pgg.x=y/), respectively (cf. Fig.8.1c, d). The splitting functions can be calculated in QCD. Thus, with increasingQ2the number ofresolvedpartons sharing the nucleon’s momentum increases. The quark distribution q.x;Q2/at small momentum fractions x, therefore, is larger than q.x;Q20/, whereas the effect is reversed for largex. This is the origin of the increase of the structure function with Q2at small values ofxand its decrease at largex.
The change in the quark distribution and in the gluon distribution withQ2at fixed values ofxis proportional to the strong coupling constant˛s.Q2/and depends upon the size of the quark and gluon distributions at all larger values ofx. The mutual dependence of the quark and gluon distributions can be described by a system of coupled integral-differential equations [3,12,16], named after their authors as DGLAP equations:
dq.x;Q2/
d lnQ2 D ˛s.Q2/ 2
Z 1
x
dy y
Pqq.x=y/q.y;Q2/CPqg.x=y/g.y;Q2/
; (8.2) dg.x;Q2/
d lnQ2 D ˛s.Q2/ 2
Z 1
x
dy y
"
X
q
Pgq.x=y/q.y;Q2/CPgg.x=y/g.y;Q2/
# : (8.3) In higher orders of ˛s.Q2/ one obtains similar expressions with more and different splitting functions which take into account more complicated processes, as for example those sketched in Fig.8.10, where the gluon or an antiquark that
8.3 Scaling Violations of the Structure Functions 115
q_ q q_
q_ q
q_ q
Quark q
Electron Electron
Quark Fig. 8.10 Examples of higher-order contributions to deep-inelastic scattering
already originates from gluon splitting radiate another gluon that then produces another generation of sea quarks.
If˛s.Q2/and the shape of q.x;Q20/and g.x;Q20/are known at a given valueQ20, then q.x;Q2/and g.x;Q2/can be predicted via the DGLAP equations for all other values ofQ2. Alternatively, the coupling˛s.Q2/and the gluon distribution g.x;Q2/, which cannot be directly measured, can be determined from the observed scaling violation of the structure functionF2.x;Q2/.
The solid lines in Fig.8.5show a fit to the scaling violation of the measured structure functions from a QCD calculation [6]. The fit value of 250MeV=c corresponds to a coupling constant:
˛s.Q2D100 .GeV=c/2/0:16 : (8.4) Also the data shown in Fig.8.7can be excellently described by QCD. They allow a determination of˛sat a much largerQ2than was possible from the experiments with a stationary target. In a so-called double-logarithmic perturbative calculation one obtains for the strong coupling constant
˛s.Q2D104.GeV=c/2/0:12 : (8.5) As predicted by theory,˛sdecreases with increasingQ2.
Conclusion Scaling violation in the structure functions is a highly interesting phenomenon. It is not unusual that particles which appear point-like turn out to be composite when studied more closely (e.g., atomic nuclei in Rutherford scattering with low-energy˛particles or in elastic high-energy electron scattering). In deep- inelastic scattering, however, a new phenomenon is observed. With increasing resolution, quarks and gluons turn out to be composed of quarks and gluons;
which themselves, at even higher resolutions, turn out to be composite as well. The quantum numbers (spin, flavour, colour,. . . ) of these particles remain the same; only
116 8 Quarks, Gluons, and the Strong Interaction the mass, size, and the effective coupling˛schange. Hence, there appears to be in some sense a self similarity in the internal structure of strongly interacting particles.