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DEEP INELASTIC SCATTERING J. Drees Fachbereich Physik

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DEEP INELASTIC SCATTERING J. Drees Fachbereich Physik DEEP INELASTIC SCATTERING J. Drees Fachbereich Physik, Universität Wuppertal, Wuppertal, Fed. Rep. Germany INTRODUCTION These notes summarize the main results on deep inelastic lepton nucleón scattering, a field which has greatly influenced the progress in physics over a period of the last 12 years. In parallel to the original electron scattering experiments at SLAC the quark parton model was developed. Neutrino experiments confirmed basic parton model predictions and greatly extended our knowledge of the structure of hadronic matter. Later muon and neutrino experiments established vital predictions of the new theory of the strong interactions, the quantum chromodynamics. The notes are divided in three parts. After a brief introduction of the kinematics in part 1 I will summarize all results from electron, muon, and neutrino experiments which have to do with our belief that the quark parton model has some relation to truth. Part 2 discusses the current theoretical interest in "scaling violations" of the structure functions and the experimental situation. Part 3 contains some aspects of the hadronic final state produced in the deep inelastic collision. 1. TESTS OF THE QUARK PARTON MODEL 1.1 Fundamentals The processes we are concerned with in these lectures are represented by the Feynman diagrams Fig. 1: a) for e,y scattering, b) for v scattering with weak charge changing current, c) for v scattering with weak neutral current. a) Electromagnetic b) Charged current c) Neutral current reaction reaction reaction Fig. 1 k and k' are the four momenta of the initial and final lepton, E and E' their laboratory energies, and e the laboratory scattering angle, q and p are the four momenta carried by the exchanged virtual boson and the target nucleón. Presently most knowledge about the structure of the nucleón comes from the study of processes a) and b) - 2 - which are therefore the main subject of these lectures. A detailed discussion of the kinematics of lepton scattering can be found in Refs. 1) >i) »3) >•») _ por reasons 0f consistency I will summarize the main features for the case that only the scattered lepton is detected. From the four momenta p and q one can form two Lorentz scalar variables Q2 = - q2 = 4 EE' sin2 | v = n-E^LM - = E - E' . proton The mass of the hadronic final state is given by W2 = p2, = M2 - Q2 + 2Mv = M2 + 2MK. We shall also use the ratios y - !• For evaluating the cross sections we are interested in the square of the matrix elements corresponding to the diagrams of Fig. 1 summed over all undetected hadronic final states. These are of the form: a) For e,y scattering 92 J v I = L £j w" uv Q4 b) For charged current v scattering |M|2 = L g4 , , yv (Q2 + ^2 where is the mass of the exchanged boson. The first factor arises from the coupling to the lepton current and is completely known in all cases. The second factor contains the contributions from the boson propagator and the couplings to the lepton and hadron currents. For neutrino scattering at available 2 2 energies MjJ » Q and 4 4 2 (Q2 + M2,)2 Mj 2 where G is the Fermi constant. The tensor W^v arises from the interaction of the hadron current and is a priori unknown. However, for unpolarized target nucléons, Wuv can be expressed in the most 2 general case in terms of three independent scalar structure functions W^(Q ,v) with i = 1,2,3. The number 3 corresponds to the three helicity states of the exchanged vector boson. In principle the structure functions are different for each scattering process, - 3 - but they are connected via the quark parton model as we will discuss in the following. In the case of the electromagnetic interaction there are only two independent functions W.p since parity conservation relates the amplitudes for the exchange of virtual photons with helicity +1 and -1. 1.1.1 Cross Section for Charged Lepton Scattering Here the tensors L and Wyv are of the form L = 1 (k k1 + k k' - g ¿ and 2 w = - (g^ + Ä Wl(Q ,v) + Cp * 2? qVKpv + 2lä vjw (Q2jV)/M2 . To derive this decomposition of Wyv one needs only Lorentz invariance, current conservation (i.e. q^ Wyv = Wuv q^ = 0) and parity conservation. The double differential cross section is 2 2 2 r d a 4irq 1 T Myv _ 4ira E r~.7 rrl -, „-„2 0 ,_2 -, , „2 0n f1 1~1 d?dJ "Tf ? uv ~o?~X[2WlW 'v) sin 1 +W2« 'v) C0S 2L <1'1> There is also the possibility to consider inelastic electron nucleón scattering as a collision between the exchanged virtual photon and the nucleón. One can then define virtual photon nucleón cross sections for the absorption of transversely (helicity of the 2 1 photon + 1) and of longitudinally polarized photons, Op(Q ,v) = (cr+ + a ) and CT^(Q^,V). In terms of these cross sections 2 2 "A" = t i C^HaT(Q ,v) • e aL(Q ,v)) , (1.2) dQ dv Q E where 2 _^ e = (1 + 2(1 + \) tan2 i) is the virtual photon polarization parameter, 0 - e - 1. The 2 ^ CTt L are related by = - W1 TI CT 4w a (1.3) 2 (tJ + W2 " TT" 7X^-2 T alP • 4TT a Q + v 2 In the limit Q 0, Op approaches the total photoabsorption cross section of real photons on nucléons while •*• 0. It is worthwhile to mention that for the special case of scattering off a pointlike target Op = 0 for spin 0 for spin 1 aL - 4 - 1.1.2 Higher Order QED Effects The aim of the experimentalist measuring inelastic eN or yN cross sections is to determine the structure functions. Therefore, he has to separate the cross section due to single photon exchange (Feynman diagram, Fig. 1a) from the measured cross section containing the following additional electromagnetic contributions: 2 Re 6 " 7 ' 8 Fig. 2 Second order contributions to e,y scattering. Here only contributions in lowest order of are shown, in principle one has to add all higher order corrections. Diagrams 2 (vertex correction) and 6,7 (internal bremsstrahlung) contain the corrections to the lepton current. Their contribution together with the contribution due to diagram 3 (vacuum polarization) is usually included in the "radiative correction" procedure5-' traditionally performed by the experimentalists. Fig. 3 shows the size of the corrections to ep scattering at 20 GeV and yp scattering at 200 GeV. Fig. 3 Size of radiative corrections. - 5 - Plotted are contour lines of constant values of the ratio: d a [dQ2dv Bom measured in the Q ,v plane for an incident electron energy of 20 GeV and a muon energy of 200 GeV. Large corrections occur at v close to E, i.e. at y > 0.9, a region normally left out when analyzing deep inelastic e and y scattering experiments. The corrections to the hadron current, diagrams 4 and 8 can shown to be small and are often neglected. The contributions due to two photon exchange diagram 5 can be determined by measuring e+/e~ (or y+/iO cross section ratios ReA7 1 + 4 where = single photon exchange amplitude and A2 = two photon exchange amplitude. The single photon exchange amplitude is real, thus the interference term picks out the real part of the two photon exchange amplitude. Fig. 4 shows the ratios of experimental yields measured in SLAC electron experiments6-''7-' and a Fermilab muon experiment8-'. 1.20 ° FANCHER et al IU.S CBJ-f|.4 0 • ROCHESTER et aL (SLAC! A E.26 (NAL) r :1.30 1.10 1.20 1.10 »4U 1.0 :0.90 0.90- -0.80 -0.70 0.80 0.60 10 20 30 10 50 + 2 Fig. 4 Ratios of e /e yields as a function of Q . The ratios are consistent with unity, and we may conclude that the two photon exchange part of the measured double differential cross section is small. After having convinced ourselves that we are able to determine the single photon exchange cross section from measurements of deep inelastic eN and pN scattering by performing corrections in the order of a few times 10%, let us proceed to a discussion of the main features of the structure functions. - 6 - 1.2 Scaling and the Quark Parton Model 1.2.1 The Gross Features of Deep Inelastic ep Scattering The first experiments at energies high enough to observe the characteristic features of deep inelastic scattering were performed at the Stanford Linear Accelerator Center (SLAC) at a primary electron energy of typically 20 GeV. Fig. 5 gives an impression of the experimental arrangement9-'. END STATION A Fig. 5 Spectrometers at SLAC. Electrons scattered off a short hydrogen target are analyzed with an optical spectrometer of high angular and momentum resolution. At fixed spectrometer angle, double differential cross sections for the full range of final electron energies are measured by changing the field of the spectrometer magnets. Several spectrometers optimized for the various ranges of scattering angles enable a complete mapping of the accessible part of the Q ,v plane. Obviously, at small scattering angles e (with e near 1) one measures essentially vW2 while at large 0 (e small) one measures . A collection of data measured at SLAC is presented in Fig. ö10-'. - 7 - 10 ..;r j , ! , , , 1 I, i (b) 1 ^ 0.1 S s, E M \ % - e<0.5 W>2GeV n Fig. 6 vW2 and 2MW^ as determined 0.01 Q2>1 Gé/2 R = 0 1 from the double differential cross 0.001 sections assuming a^/a^ = 0.
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