Deep-Inelastic Scattering
7.5 Interpretation of Structure Functions in the Parton ModelModel
Structure functions describe the internal composition of the nucleon. We now assume the nucleon to be built from different types of quarks q carrying an electrical charge zq e. The cross-section for electromagnetic scattering from a quark is proportional to the square of its charge, and hence toz2q.
We denote the distribution function of the quark momenta by q.x/, i.e., q.x/dxis the expectation value of the number of quarks of type q D u;d;s in the nucleon whose momentum fraction lies within the interval Œx;xC dx. The momentum distribution of the valence quarks we denote by qv.x/ and correspondingly the distribution of the antiquarks in the “sea” by qNs.x/. The proton consists of two valence u-quarks and one valence d-quark. Therefore we have
Z 1
0 uv.x/dx D2;
Z 1
0 dv.x/dx D1 : (7.14)
The structure functionF2is the sum of the momentum distributions weighted byx andz2q. Here the sum is over all types of quarks and antiquarks:
F2.x/Dx X
qDu;d;s
z2qŒq.x/C Nqs.x/ ; (7.15) with q.x/Dqv.x/Cqs.x/for u- and d-quarks and q.x/Dqs.x/for s-quarks.
7.5 Interpretation of Structure Functions in the Parton Model 97 Structure functions of proton and neutron Much detailed information about the distribution functions of quarks can be obtained by the study of combinations of the structure functionsFp2andFn2of proton and neutron. In the absence of free neutron targets, information aboutF2n must be obtained from deep-inelastic scattering of deuterons. By convention in scattering off nuclei, the structure function is always given per nucleon. Except for small corrections due to the Fermi motion of the nucleons, the structure function of the deuteronFD2 is equal to the proton-neutron average structure functionF2N
F2D F2pCFn2
2 DWFN2 ; (7.16)
and hence we haveF2n2F2DFp2.
According to (7.15), the structure functionsF2of the proton and the neutron are given by
Fp2.x/Dx 4 9
upvCupsC Nups C 1
9
dpvCdpsC Ndps C1
9
spsCNsps
Fn2.x/Dx 4 9
unvCunsC Nuns C 1
9
dnvCdnsC Ndns C1
9
snsCNsns
; (7.17) where up;nv .x/denotes the distribution of valence u-quarks in the proton and the neutron, respectively, and us.x/p;nthe distribution of the sea u-quarks etc.
Fromisospin symmetrywe obtain for the quark distributions upv;s.x/Ddnv;s.x/DWuv;s.x/ ;
dpv;s.x/Dunv;s.x/DWdv;s.x/ : (7.18) Ratio of the neutron and proton structure functions The effective quantum numbers of the sea quarks average out to zero and we therefore have qs.x/D Nqs.x/.
We assume that the distributions of s-quarks in the proton and the neutron are identical (Nsps.x/ D Nsns.x/), and also that the contributions of the two light u- and d-quarks to the “sea” are equal (Nus.x/D Nds.x/). (Below we will see that this relation is only approximately true.) Because of the larger mass of s-quarks, fluctuations into quark-antiquark pairs of this flavour have a smaller probability and we have uNs.x/ >Nss.x/.
Summing up thez2q-weighted contributions of all sea quarks we can define S.x/D10uNs.x/C2Nss.x/ : (7.19) Then we obtain for the ratio of the neutron and proton structure functions:
Fn2.x/
Fp2.x/ D Œuv.x/C4dv.x/CS.x/
Œ4uv.x/Cdv.x/CS.x/: (7.20)
98 7 Deep-Inelastic Scattering
x
0.001 0.01 0.1 1
2
p /F2n F
0.2 0.4 0.6 0.8 1 1.2
= 4 (GeV/c)2
Q2
Fig. 7.7 The structure function ratioFn2=F2pas a function ofx[2]. The data were obtained from muon scattering with beam energies of 90 and 280 GeV. Shown are results atQ2D4 .GeV=c/2
Figure 7.7shows the ratio Fn2=F2p as a function of x. Plotted are data of one of the second-generation muon experiments [1,2]. This experiment has a beam energy that is more than an order of magnitude higher than the experiments done at SLAC, and therefore the data cover much smaller values ofx. Since the proton is composed of two valence u-quarks and one valence d-quark we could assume that their distributions are related by uv.x/ D 2dv.x/. For a vanishing contribution of the sea quarks (S.x/D 0),Fn2=F2pwould obtain the value2=3independent ofx. In reality, however, the ratio approaches unity forx!0and decreases withxdown to a value of approximately1=4forx!1. We can interpret this behaviour as follows:
for small values ofxthe distribution of sea quarksS.x/is much larger than the two valence quark distributions, the ratio is mainly determined by the last term in the numerator and denominator of (7.20). Asx !1, the situation is reversed: the sea quarks no longer play a role and we obtain the value1=4for the ratio by neglecting in (7.20) bothS.x/and dv.x/compared to uv.x/. The distribution of d-quarks drops much faster withxthan the u-quark distribution. This implies that large momentum fractions in the proton (neutron) are carried by u-quarks (d-quarks).
Difference of the proton and neutron structure functions The difference of the proton and neutron structure functions is given by
F2p.x/F2n.x/Dx 1
3.uv.x/dv.x//C2
3.uNs.x/ Nds.x//
: (7.21)
7.5 Interpretation of Structure Functions in the Parton Model 99
x
0.001 0.01 0.1 1
2N F
0.1 0.2 0.3 0.4 0.5
= 4 (GeV/c)2
Q2
2n - F2
p F
0 0.05 0.1
Fig. 7.8 The structure functionFN2 for an “average” nucleon (closed symbols,left scale) and the difference of the proton and neutron structure functionsFp2Fn2 (open symbols,right scale) as function ofx[2]. The data were obtained from muon scattering with beam energies of 90 and 280 GeV. Shown are results atQ2D4 .GeV=c/2
Thus if and only if the “sea” is symmetric in the two light-quark flavours, i.e., uNs.x/ D Nds.x/, then the contributions from sea quarks drop out and the difference (7.21) is a pure valence quark distribution. In Fig.7.8data from the same muon experiment are shown forF2pF2n(open symbols, right scale) as a function ofx. The distribution has a maximum nearx 1=3and drops down to zero for x! 0andx !1. This supports our assumption, made above, that at low values ofxmainly sea quarks contribute to the structure function. Also at large values of xthe distribution becomes very small. Thus it is very unlikely thatonequark alone carries the major part of the momentum of the nucleon.
The observed behaviour has often been interpreted as resulting from three valence quarks, each of them carrying on average one third of the nucleon’s momentum and the sharply defined momentum atxD 1=3is then washed out by the Fermi motion of the quarks inside the nucleon. This interpretation is incorrect.
As we will see below, quarks carry only about half of the nucleon’s momentum. The distributions uv.x/and dv.x/both have a maximum nearx0:17and the maximum ofF2pFn2 nearx D 1=3accidentally arises from the differentxdependencies of these two distributions.
When we divide (7.21) byxand integrate overx, we obtain SGD
Z 1
0
1 x
Fp2.x/Fn2.x/
dx D 1
3.21/C2 3
Z 1
0 .uNs.x/ Nds.x//dx: (7.22)
100 7 Deep-Inelastic Scattering
ForuNs.x/D Nds.x/the last term drops out and we getSGD 13. This is theGottfried Sum Rule[8]. Experimentally, however, the integral amounts to [2]
SGD0:235˙0:026 : (7.23) This leads to the conclusion thatdNs.x/ > uNs.x/and that consequently the quark- antiquark “sea” is not symmetric in the two light-quark flavours. We will come back to this finding in Sect.8.4.
Quark charges All of the quantitative statements made in the present chapter confirm the assignment of the fractional quark chargeseuD2e=3andedD 1e=3.
In addition, a convincing confirmation comes from the comparison of the nucleon structure functions measured in deep-inelastic scattering of electrons or muons and of neutrinos that we will discuss in Sect.10.6. Thus we can conclude:
Quarks carry fractional charges of2e=3and1e=3.
Structure function for an “average” nucleon Finally, after having discussed the ratio and the difference of the proton and neutron structure functions, we can get another important information about the structure of the nucleon by looking at their average. The structure function for an “average” nucleon (7.16) reads:
F2N.x/D 5 18x X
qDd;u
Œq.x/C Nqs.x/ C 1 9x
ss.x/C Nss.x/
D 5
18x X
qDd;u;s
Œq.x/C Nqs.x/ 1
3x Nss.x/ : (7.24) The last term in the equation is small, since s-quarks occur only as sea quarks.
To a good approximation, FN2 is therefore given by the product of the average squared charges5=18of u- and d-quarks (in units ofe2) and the sum over all quark distributions.
The integral of F2N.x/ is taken over all quark momenta weighted by their distribution functions and the average squared quark charges. Therefore, the integral should yield the value5=18, provided that the whole nucleon momentum is carried by its charged constituents, the quarks.
However, integration of the data shown in Fig.7.8only yields the value Z 1
0 FN2.x/dx 0:55 5
18 : (7.25)
Thus we have to conclude:
Quarks carry only about half of the nucleon’s momentum.
The other half must be carried by uncharged particles interacting neither electro- magnetically nor weakly. This finding was the starting point for the development of
References 101 QCD, the field theory of the strong interaction. The electrically neutral constituents have been identified with the field quanta of this interaction, thegluons.
Problems
1. Deep-inelastic scattering
Derive the Callan-Gross relation (7.11). Which value for the mass of the target must be used?
2. Parton momentum fractions andx
Show that in the parton model of deep-inelastic scattering, if we donotneglect the masses of the nucleonMand of the partonm, the momentum fraction of the scattered parton in a nucleon with momentumPis given by
Dx 1C m2c2M2c2x2 Q2
:
In the deep-inelastic domain x2MQ22c2 1andmQ2c22 1. (Hint: for small","0we can approximatep
1C".1C"0/1C "2.1C"04"/.)
References
1. P. Amaudruz et al., Nucl. Phys.B371, 3 (1992) 2. M. Arneodo et al., Phys. Rev.D50, 1 (1994)
3. W.B. Atwood,Lectures on Lepton Nucleon Scattering and Quantum Chromodynamics. Progress in Physics, vol. 4 (Birkhäuser, Boston/Basel/Stuttgart, 1982)
4. W. Bartel et al., Phys. Lett.B28, 148 (1968) 5. E.D. Bloom et al., Phys. Rev. Lett.23, 930 (1969) 6. M. Breidenbach et al., Phys. Rev. Lett.23, 935 (1969) 7. C.G. Callan Jr., D.J. Gross, Phys. Rev. Lett.22, 156 (1969) 8. K. Gottfried, Phys. Rev. Lett.18, 1174 (1967)
9. D.H. Perkins,Introduction to High Energy Physics, 4th edn. (Addison-Wesley, Wokingham, 2000)