aa r X i v : . [ h e p - e x ] N ov The Physics of Deep-Inelastic Scattering atHERA
Cristinel Diaconu
Centre de Physique des Particules de Marseille andDeutsches Elektronen Synchrotron, Notkestrasse 85, 22607 Hamburg, Germany
Abstract.
In this paper an introduction to the physics of deep-inelastic scattering is given togetherwith an account of some of the most recent results on the proton structure obtained in electron– andpositron–proton collisions at the HERA collider.
Keywords:
Deep inelastic scattering, proton structure, HERA collider , QCD.
INTRODUCTION
The investigation of the matter structure using particle collisions started in early XXcentury with Geiger, Mardsen and Rutherford discovery of the atomic nucleus using a -particle scattering on a gold foil. The principle of the measurement is related to thespatial resolution obtained using high energy particles. The ‘resolving power’ can beexpressed as d = ∼
200 M eV / Q [ − m ] and is related to the uncertainity principle:the higher the transfer momentum (denoted by Q ), the smaller the details that can be"flashed" and imprinted in the distribution of the scattered (point-like) particle. Thesearch for further substructure levels continued with the scattering of leptons on lightnuclei ( H , D ) in order to investigate the structure of protons and neutrons, the maincomponents of the nuclear matter. The finite size of the proton was established inelastic scattering and its components, the quarks and the gluons, were discovered usingelectrons of higher and higher energies, which were ultimately able to break the nucleonin the so called deep inelastics scattering (DIS). In this paper, an introduction [1] to thephysics of DIS is given using as examples recent measurements of the proton structureperformed at HERA electron-proton collider . THE OBSERVABLES OF LEPTON–HADRON SCATTERING
The lepton–hadron scattering is described in the framework shown in figure 1. Thescattering can occur via the exchange of g or Z bosons (neutral currents NC) or via W bosons (charged currents CC). In the later case, a neutrino is expected in the finalstate. The incoming electron (with a four-momentum k ) scatters off the proton ( P ) to afinal state electron with four–momentum k ′ via a virtual photon g ∗ or a weak boson The paper is based on an introductory lecture presented at the “Carpathian Summer School of Physicson Exotic Nuclei and Nuclear/Particle Astrophysics” , August 20-31, 2007, Sinaia, Romania. (k) e’ / n ’ (k’) g ,Z / W ± (q)q f (xP)spectator jet p (P)scattered parton jet Q = − q = − ( k ′ − k ) y = P · qP · kx = Q P · q ( Bjorken ) n = P · qM p W = ( P ′ ) = ( P + q ) (= M X ) s = ( P + k ) ( fixed ) FIGURE 1.
Lepton–hadron scattering: an exchange of a boson in the t –channel. with a virtuality Q in the t -channel. The Bjorken variable x is associated with thefraction of the momentum of the proton carried by the struck parton. The total centre-of-mass energy is given by √ s and the energy of the g ∗ p system is given by W , whichis equivalent to the total mass of the hadronic system in the final state M X . In the caseof elastic scattering M X = M p and from M X expression follows that Q = Pq and x = n has a simple meaning in the proton rest frame, as the energy lostby the electron during the scattering n = E e − E ′ e , while y represents the fractional energyloss y = E e − E ′ e E e . Q can be expressed as a function of the electron energy and scatteredangle Q = E e E ′ e sin q . From these relations, it is obvious that the DIS kinematics canbe calculated from the measurement of the scattered electron only. The measurement ofthe hadrons in the final state, if available, can be exploited as an extra constraint or usedin case of CC reactions, where the outgoing neutrino is not measured. THE HERA PROJECT
The idea for a large electron–proton collider to mark a new step in the studies for protonstructure, beyond the fixed target experiments, was promoted already in seventies [3].The HERA collider project started in 1985 and produced the first electron–proton colli-sions in 1992. It is composed of two accelerators designed to store and collide counterrotating electrons ( e − ) or positrons ( e + ) with an energy of 27.5 GeV and protons with anenergy of 920 GeV. The operations came to an end in june 2007 and the final analysesusing the collected data are in progress at present.HERA ring hosted two collider mode detectors H1 and ZEUS. They were build ashermetic (4 p ) multi-purpose detectors equipped with internal trackers able to measurecharged particle momenta and calorimeters completing the measurement of the energyflow. Two other experiments used e ± or p beams for fixed target studies: HERMES, ded-icated to the study of polarised e ± p ( N ) collisions and (until 2003) HERA-B dedicated tothe study of beauty production in hadronic collisions. Since 2003, the e ± beam were lon-itudinally polarised in collision mode with an average polarisation of P e ± = − − (200 pb − ) has been collected in e + p ( e − p ) by eachthe two collider mode experiments, H1 and ZEUS. DEEP-INELASTIC SCATTERING MEASUREMENTS AT HERA y = ( H E R A √ s = G e V ) x Q ( G e V ) E665, SLACCCFR, NMC, BCDMS,Fixed Target Experiments:ZEUSH1 -1 -6 -5 -4 -3 -2 -1 FIGURE 2.
The kinematic plane accessible at HERA compared to former fixed target experiments(left) and event displays of a neutral current scattering event measured by H1 (center) and charged currentscattering event measured by ZEUS (right).
The H1 and ZEUS experiments measured both neutral current (NC) and chargedcurrent (CC) processes. The kinematic ( x , Q ) plane accessible at HERA is shown infigure 2 (left) with a Q domain up to 50000 GeV and x down to 10 − . The NC eventscontain a prominent electron and a jet of particles measured in the calorimeter, while inCC events only the jet is visible since the outgoing neutrino in not detected. Examplesof such events are shown in figure 2.Since a large domain in x and Q is accessed, the NC cross section is sensitive to weakforce effects. The proton structure, as revealed by the photon and Z boson in DIS, canbe incorporated into the so-called generalised structure functions. The cross section isparameterised as following:d s ± NC d x d Q = pa xQ ( Y + ˜ F ∓ Y − x ˜ F − y ˜ F L ) , (1)The generalised structure functions ˜ F and x ˜ F can be further decomposed as [4]˜ F ≡ F − v e k Q ( Q + M Z ) F g Z + ( v e + a e ) (cid:18) k Q Q + M Z (cid:19) F Z , (2) x ˜ F ≡ − a e k Q ( Q + M Z ) xF g Z + ( v e a e ) (cid:18) k Q Q + M Z (cid:19) xF Z , (3)with k − = M W M Z ( − M W M Z ) in the on-mass-shell scheme [5]. The quantities v e and a e are the vector and axial-vector weak couplings of the electron or positron to the Z [5].The electromagnetic structure function F originates from photon exchange only anddominates over the vast majority of the measured phase space. The functions F Z and ( pb / G e V / d Q s d -7 -5 -3 -1 ) (GeV Q p CC 03-04 (prel.) + H1 ep CC 2005 (prel.) - H1 e p CC 2004 + ZEUS ep CC 04-05 (prel.) - ZEUS ep CC (CTEQ6M) + SM ep CC (CTEQ6M) - SM e p NC 03-04 (prel.) + H1 ep NC 2005 (prel.) - H1 e p NC 2004 + ZEUS ep NC 04-05 (prel.) - ZEUS ep NC (CTEQ6M) + SM ep NC (CTEQ6M) - SM e y < 0.9 = 0 e P HERA II -2 -1 -2 -1 -2 -1 HERA Charged Current Q = 280 GeV s ∼ H1 e - pZEUS e - p 98-99 H1 e + p 94-00ZEUS e + p 99-00 SM e - p (CTEQ6D)SM e + p (CTEQ6D) Q = 530 GeV Q = 950 GeV Q = 1700 GeV Q = 3000 GeV Q = 5300 GeV Q = 9500 GeV Q = 17000 GeV Q = 30000 GeV x · u(1-y) x · d x FIGURE 3.
Left: the charged current and neutral current cross section as a function of Q measuredin e ± p collisions at HERA. Right: the charged current reduced cross section ˜ s CC as a function of x forvarious Q values measured in electron– and positron–proton collisions. xF Z are the contributions to ˜ F and x ˜ F from Z exchange and the functions F g Z and xF g Z are the contributions from g Z interference. These contributions are significant onlyat high Q . The structure functions provide however direct information on the protoncomponents. Within the so-called quark-parton model (QPM), the proton is composedof spin one half partons, called quarks q = u , d with fractionary charges e q . This modelpredicts the structure functions as combinations of quark densities q ( x ) . For instance F = (cid:229) q e q q ( x ) and is independent of Q (Bjorken scaling). This model is improvedwithin the Quantum Chromodynamics (QCD) predicting quark interactions via gluons,the careers of the strong force, and leading to F dependence on Q (scaling violation).The charged current (CC) interactions, e ± p → n ( ) e X , are mediated by the exchange ofa W boson in the t channel. The cross section is parameterised as: d s CC ( e ± p ) dxdQ = G F p x (cid:20) M W M W + Q (cid:21) ˜ s ± CC ( x , Q ) , (4)with ˜ s ± CC ( x , Q ) = (cid:2) Y + W ± ( x , Q ) ∓ Y − xW ± ( x , Q ) − y W ± L ( x , Q ) (cid:3) . ˜ s is the reducedcross section, G F is the Fermi constant, M W , the mass of the W boson, and W , xW and W L , CC structure functions. In the quark parton model, the structure functions W ± and xW ± may be interpreted as lepton charge dependent sums and differencesof quark and anti-quark distributions: W + = x ( U + D ) , xW + = x ( D − U ) , W − = x ( U + D ) , xW − = x ( U − D ) , whereas W ± L =
0. The terms xU , xD , xU and xD aredefined as the sum of up-type, of down-type and of their anti-quark-type distributions.The differential NC and CC cross sections as a function of Q are shown in fig-ure 3 (left) for e ± p collisions. At low Q the NC cross section, driven by the elec-tromagnetic interaction, is two orders of magnitude larger than the CC cross section ERA F F e m - l og ( x ) Q (GeV ) ZEUS NLO QCD fitH1 PDF 2000 fitH1 94-00H1 (prel.) 99/00ZEUS 96/97BCDMSE665NMCx=6.32E-5 x=0.000102x=0.000161x=0.000253x=0.0004x=0.0005x=0.000632x=0.0008x=0.0013x=0.0021x=0.0032x=0.005x=0.008x=0.013x=0.021x=0.032x=0.05x=0.08x=0.13x=0.18x=0.25x=0.4x=0.65 H C o ll abo r a t i on HERA -2 -1 x x F g Z Q =1500 GeV H1+ZEUS Combined (prel.)
H1 2000 PDFZEUS-JETS PDF
F2 xF3FL
FIGURE 4.
The determination of the structure functions F , F L and xF from HERA measurements. which correspond to a pure weak interaction. At large Q ∼ M W , Z the two cross sectionsare similar, which can be interpreted as a hint for electroweak unification. The largest Q measurement corresponds to a resolution d ≃ − m, i.e. 1/1000 the proton size.The agreement between the measurement and the prediction based on improved partonmodel constitutes a spectacular confirmation of QCD and suggests that no evidence forquark substructure is observed at present.The double differential reduced cross section ˜ s CC ( x , Q ) is shown in figure 3 (right).The CC processes are sensitive to individual quark flavours, especially visible at large Q : the e + p collisions probe the d quark distribution, while e − p are more sensitive tothe u distribution. This is a very useful feature of the CC processes compared to the NC,where the quark flavour separation is weaker. STRUCTURE FUNCTIONS MEASUREMENTS: F , F L AND xF The NC cross section is dominated over a large domain by the F contributions, definedin equation 1. The measurement of the NC cross section at HERA can therefore betranslated into an F measurement, which is shown in figure 4 together with the previousmeasurements performed at fixed target experiments. One can observe the Bjorkenscaling in the region at high x , but obvious scaling violation at lower x . This can beunderstood in terms of DGLAP equations [2] as a contribution driven by the gluon ¶ F ( x , Q ) / ¶ ln ( Q ) ≈ ( a s ( Q ) / p ) xg ( x , Q ) . while at high x the quark pdf’s arethe major contributors to the evolution in DGLAP equations.From the F measurements at fixed Q one can observe a steep increase of F towardslow x , as shown in figure 5 (left). The region at low x is populated by quarks whichhave undergone a hard or multiple gluon radiation and carry a low fraction of the protonmomentum at the time of the interaction. The observation of such large fluctuations to -4 -3 -2 -1 Q = 15 GeV x F e m H1 96/97ZEUS 96/97NMC, BCDMS, E665CTEQ6DMRST (2001)
FIGURE 5.
Left: The measurement of F as a function of x for Q =
15 GeV. Right: a sketch of thecorrespondence between F shape as a function of x and the quark-parton model. very high parton density is driven by the uncertainty principle, which requires that theinteraction time be very short and therefore at high Q . In this regime, it is expectedthat the structure function grows at low x and shrinks at large x , confirmed by theexperimental observation. The rise of the structure functions at low x is one of themost surprising observations at HERA. It is predicted in the double leading log limitof QCD [6]. It can be intuitively understood in terms of gluon driven parton productionat low x , as depicted in figure 5 (right).The longitudinal structure function F L is usually a small correction, only visible atlarge y . The F L measurement from the cross section has to proceed in such a way that F contribution is separated. Indirect methods assume some parameterisation of F toextract F L . Using this method, an F L determination can be performed and is shown infigure 4 at fixed W (the g ∗ p centre-of-mass energy). In the naive QPM the longitudinalstructure function F L = F − xF ≡ F L contains by definition the devi-ations from the Callan-Gross relation. It can be shown that F L is directly related to thegluon density in the proton [7, 8] xg ( x ) = . [ p a s F L ( . x ) − F ( . x ] ≃ . a s F L meaningthat at low x , to a good approximation F L is a direct measure for the gluon distribution.A direct measurement of F L can be performed if the cross section s ∼ F ( x , Q ) + f ( y ) F L ( x , Q ) is measured at fixed x and Q but variable y . This can only be performedif the centre-of-mass energy is varied, for instance by reducing the proton beam energy.Eliminating F ( x , Q ) , F L can be directly measured with reduced uncertainties fromthe difference of cross sections: F L ∼ C ( y ) ∗ ( s ( E p ) − s ( E p )) . Measurements of DIS atHERA at lower proton energies of 460 GeV and 575 GeV has been performed at the endof the run in 2007 in order to perform the first direct measurement of F L in the low x andhigh Q regime.The structure function x ˜ F can be obtained from the cross section difference betweenelectron and positron unpolarised data x ˜ F = Y + Y − (cid:2) ˜ s − ( x , Q ) − ˜ s + ( x , Q ) (cid:3) The domi-nant contribution to xF arises from the g Z interference. In leading order QCD the inter- -4 -3 -2 -1 x x f( x , Q ) H1 PDF 2000H1ZEUS-S PDFZEUS-S PDFCTEQ6.1CTEQ6.1 Q =10 GeV xu V xd V xg( × × -1 -0.5 0 0.5 1-1-0.500.51 -PDF (prel.) u -v u ZEUS-pol-a total uncert. uncorr. uncert. H1 prel. (HERA I+II 95-05) SM CDF LEP -1 -0.5 0 0.5 1-1-0.500.51 u a u v
68% CL
H1 and ZEUS
FIGURE 6.
Left: The parton distribution functions extracted from HERA data. Right: Axial and vectorcouplings of the u –quark measured from the combined electroweak–QCD fit at HERA and comparedwith measurements from LEP (using light quarks production at Z pôle e + e − → q ¯ q ) and Tevatron (fromDrell-Yan electron pair production q ¯ q → e + e − ). ference structure function xF g Z can be written as xF g Z = x [ e u a u ( U − U ) + e d a d ( D − D )] , with U = u + c and D = d + s thus provides information about the light quark axialvector couplings ( a u , a d ) and the sign of the electric quark charges ( e u , e d ). The averaged xF g Z , determined by H1 and ZEUS for a Q value of 1500 GeV , is shown in figure 4. PARTON DISTRIBUTION FUNCTIONS AND ELECTROWEAKEFFECTS
The NC and CC cross section measurements are used in a global fit in order to extractthe parton distribution functions (pdf’s) [9, 10]. The shapes for the quarks q ( x , Q ) andgluon g ( x , Q ) distributions are parametrised as a function of x at a given scale Q andevolved using DGLAP equations [2] to each ( x , Q ) point where the cross section hasbeen measured. The theoretical cross section can therefore be accurately calculated asa function of the pdf’s parameters. A c is then built using the measurements and thepredictions for all measurements points and minimised to extract the non-perturbativepdf’s parameters. Since the number of parameters (typically 10) is much lower thanthe number of measurements (several hundred) the fit also consitutes a very powerfultest of QCD. The structure functions from the fit are compared with data in figure 4.The parton distribution functions are extracted using the decomposition of the structurefunction described above. As an example, the pdf’s obtained for Q =
10 GeV areshown in figure 6. The valence distributions peak at 1 / u V twice as large as d V . Gluon distribution is enhanced at low x . Theknowledge of the proton structure deduced from inclusive CC/NC measurements canbe used to calculate the cross section of exclusive processes leading to a specific final P -1 -0.5 0 0.5 1 ( pb ) CC s p Scattering – Charged Current e X n fi p - e X n fi p + e > 400 GeV Q y < 0.9 MRST 2004 CTEQ6D H1 2005 (prel.)H1 98-99ZEUS 04-05 (prel.)ZEUS 98-99H1 99-04ZEUS 06-07 (prel.)ZEUS 99-00 e P -1 -0.5 0 0.5 1 ( pb ) CC s HERA -1-0.8-0.6-0.4-0.200.20.40.60.81 10 Q (GeV ) A H1+ZEUS Combined (prel.) A + A - H1 2000 PDFZEUS-JETS PDF
FIGURE 7.
Left: The dependence of the charged current cross section on the electron or positron beampolarisation at HERA. Right: The polarisation asymmetry of the NC cross section at HERA. state FS as a convolution of the parton level cross section with pdf’s, for instance: s ep → FS = s eq − > FS ⊗ q ( x , Q ) . This factorisation can also be used to calculate the crosssection of processes produced in proton–proton collisions using the pdf’s measured inDIS.Recently, a new approach has been adopted by the H1 and ZEUS collaborations[11],performing a combined QCD–electroweak fit. The strategy is to leave free in the fit theEW parameters together with the parameterisation of the parton distribution functions.Due to the t -channel electron-quark scattering via Z bosons, the DIS cross sectionsat high Q are sensitive to light quark axial (a q ) and vector (v q ) coupling to the Z .This dependence includes linear terms with significant weight in the cross sectionwhich allow to determine not only the value but also the sign of the couplings. Themeasurements of the u –quark couplings obtained at HERA, LEP and Tevatron are shownin figure 6. e ± p COLLISION WITH A POLARISED LEPTON BEAM
The polarisation of the electron beam at HERA II allows a test of the parity non-conservation effects typical of the electroweak sector. The most prominent effect ispredicted in the CC process, for which the cross section depends linearly on the e ± –beam polarisation: s e ± p ( P ) = ( ± P ) s e ± pP = . The results[13] obtained for the first time in e ± p collisions are shown in figure 7. The expected linear dependence is confirmed andprovides supporting evidence for the V-A structure of charged currents in the StandardModel.Due to parity violating couplings of the Z boson, the e ± beam polarisation effectscan also be measured in NC processes at high Q . The charge dependent longitudinalpolarisation asymmetries of the neutral current cross sections, defined as A ± = P R − P L · s ± ( P R ) − s ± ( P L ) s ± ( P R ) + s ± ( P L ) ≃ ∓ ka e F g Z F , (5)easure to a very good approximation the structure function ratio. These asymmetriesare proportional to combinations a e v q and thus provide a direct measure of parityviolation. In the Standard Model A + is expected to be positive and about equal to − A − .At large x the asymmetries measure the d / u ratio of the valence quark distributionsaccording to A ± ≃ ± k + d v / u v + d v / u v . The measurement from ZEUS and H1 [14], shown infigure 7, are in agreement with the theoretical predictions.
OUTLOOK
The study of deep-inelastic scattering is a fundamental branch of high energy physics.The structure of matter was last time resolved to new components, the quarks, in thefirst break-up of the proton at SLAC in 1968. Since then, the Standard Model of particlephysics has become a well established theory with the last quark, the top, discoveredin 1994. The knowledge of the structure of the baryonic matter, dominating the visibleuniverse, has made huge progress in the last decades, thanks to an impressive effort tounravel the nucleon structure in fixed target experiments and at the HERA ep collider.The knowledge acquired at HERA is invaluable also for the physics of the Large HadronCollider, foreseen to start pp collisions at 14 TeV in 2008. By enabling even moreambitious DIS experiments [15] beyond HERA, the matter structure investigations maygain a new momentum. REFERENCES
1. More details about DIS physics can be found in the following monographies: R.K. Ellis, W.J. Sterlingand B.R. Webber, “QCD and Collider Physics”, Cambridge, ISBN 521 58189 3, (1996);R. Devenish and A. Cooper-Sarkar, “Deep Inelastic Scattering”, Oxford University Press, ISBN13:978-0-19-850671-3, (2005) ;2. V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438, 675; L.N. Lipatov, Sov. J. Nucl.Phys. 20 (1975) 94; Yu. L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641; G. Altarelli and G. Parisi,Nucl. Phys. B126 (1977) 298.3. A. Febel, H. Gerke, M. Tigner, H. Wiedemann and B. H. Wiik, IEEE Trans. Nucl. Sci. (1973) 782.4. M. Klein and T. Riemann, Z. Phys. C (1984) 151.5. S. Eidelman et al. [Particle Data Group Collaboration], Phys. Lett. B592 (2004) 1.6. A. De Rújula et al. , Phys. Rev. D (1974) 1649; R. D. Ball and S. Forte, Phys. Lett. B (1994) 77[arXiv:hep-ph/9406385]; R. D. Ball and S. Forte, Phys. Lett. B (1994) 77 [arXiv:hep-ph/9405320].7. G. Altarelli and G. Martinelli, Phys. Lett. B (1978) 89.8. A. M. Cooper-Sarkar et al. , Z. Phys. C (1988) 281.9. C. Adloff et al. [H1 Collaboration], Eur. Phys. J. C , 1 (2003) [hep-ex/0304003].10. S. Chekanov et al. [ZEUS Collaboration], Phys. Rev. D , 012007 (2003) [hep-ex/0208023].11. H1 and ZEUS Collaboration, contribution to EPS Conference, Manchester, UK, july 2007 [ H1prelim-07-041, ZEUS-prel-06-003].12. D. Acosta et al. [CDF Collaboration], Phys. Rev. D , 052002 (2005) [hep-ex/0411059].13. H1 and ZEUS Collaborations, contributions to EPS Conference, Manchester, UK, july 2007[H1prelim-06-041, ZEUS-prel-06-003];14. H1 and ZEUS Collaborations, contribution to EPS Conference, Manchester, UK, july 2007[H1prelim-06-142, ZEUS-prel-06-022].15. J. B. Dainton, M. Klein, P. Newman, E. Perez and F. Willeke, “Deep inelastic electron nucleonscattering at the LHC,” JINST1