Hadron interactions and hadron structure

+ q 10.1098/rsta.2000.0725 Hadron interactions and hadron structure By W. Ja es Stirling Departents of Matheatical Sciences and Physics, University of Durha, Durha DH1 3LE, UK While deep-inelastic scattering experients are the priary source of inforation on the parton structure of hadrons, hard-scattering processes in hadron{hadron collisions also provide useful inforation. We describe the theoretical fraework that relates hard-scattering cross-sections to parton distribution functions, and review soe of the recent phenoenology. Keywords: hadron structure; parton distributions; pheno enology 1. Introduction It was rst pointed out by Drell & Yan (1970) that parton odel ideas developed for deep-inelastic scattering could be extended to certain processes in hadron{hadron collisions. The paradig process, proposed by Drell & Yan, was the production of a assive lepton pair by quark{antiquark annihilation: the Drell{Yan process (illustrated in gure 1). For the production of a lepton pair of invariant ass M, the hadronic cross-section d¼ = dm 2 was to be obtained by weighting the subprocess cross-section d^ ¼ = dm 2 for q q! d^ ¼ dm = 4º 2 2 3M 2 e2 q (^s M 2 ); (1.1) with the parton distribution functions f q (x) extracted fro deep-inelastic scattering. Labelling the quark (antiquark) oentu fractions by x 1 (x 2 ), and noting that ^s = x 1 x 2 s, the cross-section is d 2 ¼ dm = 4º 2 2 3M 4 0 1 dx 1 dx 2 (x 1 x 2 ½ ) e 2 q f q (x 1 )f q (x 2 ) = 4º 2 3M 4 F(½ ) (scaling variable ½ = M 2 =s): (1.2) In the original derivation of this result by Drell & Yan, there was no additional `1=3 quark colour averaging factor, and in order to quantify the cross-section it was assued that f q = f q = f, hence F ¹ F 2 «F 2, where F 2 is the deep-inelastic structure function. The doain of validity of (1.2) is the asyptotic `scaling liit M 2, s! 1, ½ = M 2 =s xed, in which liit the cross-section M 4 d¼ = dm 2 depends only on the diensionless ratio ½. Scaling was subsequently con red experientally ( gure 2 shows an exaple), thus validating the parton-odel treatent of largeass lepton pair production. 359, 257{269 257 c 2001 The Royal Society

+ + 258 W. J. Stirling A q l + q l - B Figure 1. The Drell{Yan process. The signi cance of the Drell{Yan process was that it showed for the rst tie that a rigorous, quantitative analysis of certain types of hadronic cross-section was possible. Studies were subsequently extended to other `hard-scattering processes, for exaple the production of hadrons with large transverse oentu and of heavy quarks, with equally successful results. Drell & Yan (1970) realized the iportance of hard-scattering processes for obtaining inforation on the parton structure of the colliding partons: The full range of processes of the type A + B! + with incident p, p, p, K, g, etc., a ords the interesting possibility of coparing their parton and antiparton structures. Indeed, nowadays data fro hadron{hadron scattering processes are routinely used in `global analyses to deterine the parton distribution functions of the proton (see, for exaple, the Martin{Roberts{Stirling{Thorne (MRST) work of Martin et al. (1998) or the Coordinated Theoretical{Experiental Project on QCD (CTEQ) work of Lai et al. 2000). As we shall see in the following sections, the Drell{Yan process provides essential inforation on the light quark sea in the proton and on the valence (quark and antiquark) distributions in the pion. The production of large transverse oentu jets and `propt photons, on the other hand, is a vital source of inforation on the ediu- and high-x gluon distribution in the proton. In this brief review we will only have tie to discuss the inforation on quark structure fro hadron{hadron collisions. The extraction of the gluon distribution fro large transverse oentu jet production at the Tevatron p p collider is described in Montgoery (this issue). Before discussing the phenoenology in ore detail, we ention an iportant theoretical developent of the 1970s that elevated the Drell{ Yan cross-section fro a byproduct of the original `naive parton odel to a rigorous and central prediction of quantu chroodynaics (QCD). 2. QCD and factorization In a quantu eld theory such as QCD, in which quarks interact with assless (gluon) gauge bosons, gure 1 can be regarded as the leading-order (in the quark{ gluon coupling) contribution to the inclusive process q q! +X. Probles arise, however, when perturbative corrections fro real and virtual gluon eission are calculated. Large logariths fro gluons eitted collinear with the incoing quarks appear to spoil the convergence of the perturbative expansion. Writing the scaled

y t Hadron interactions and hadron structure 259 10-30 10-31 sd 2 s /dö dy y = 0 (GeV 2 c 2 ) 10-32 10-33 p lab = 200 GeV/c, E325 p lab = 300 GeV/c, E325 p lab = 400 GeV/c, E325 p lab = 800 GeV/c, E605 10-34 0.20 0.30 0.40 0.50 Figure 2. A coparison of Drell{Yan cross-sections at four di erent collision energies fro two Ferilab experients, E325 and E605 (p + Cu! + + X). The cross-sections are seen to depend only on the variable ½, con ring the `scaling property. parton-odel Drell{Yan subprocess cross-section of equation (1.1) as ^F q q = (1 ½ ), then the O( s ) correction, calculated fro the diagras of gure 3, isy Ö t ^F q q = (1 ½ ) + s ( 2 ) 2º 2 ln Q2 µ 2 P (0) q q (½ ) + C(½ ) + O( 2 s ); (2.1) where µ 2 is a diensionful paraeter introduced to regulate collinear divergences. The key point, however, is that the coe cient of the logarith in equation (2.1) is the sae as that which arises in the analogous deep-inelastic scattering structure function calculation. It is therefore absorbed in the rede nition of the parton distributions, giving rise to logarithic violations of scaling (see, for exaple, the discussion in Ellis et al. (1996)). In fact, all logariths appearing in the Drell{Yan perturbative corrections can be factored into renoralized parton distributions in this way, and factorization theores, which show that this is a general feature of hard-scattering processes, can be derived. Unlike the logarithic corrections, the nite corrections such as C(½ ) in equation (2.1) are process dependent, and odify the parton-odel-like cross-section. Only the q q corrections are exhibited here.

F 260 W. J. Stirling Figure 3. Next-to-leading-order diagras for the Drell{Yan process. The size of the perturbative corrections depends on the lepton-pair ass and on the overall centre-of-ass energy. At xed-target energies and asses the correction is generally large and positive, of the order of 50% or ore. In this regie of relatively large ½, the (negative) contribution fro the quark{gluon scattering diagras in gure 3 is quite sall. However, at p p collider energies, where ½ is uch saller, the quark{gluon contribution is ore iportant and the overall correction is saller. The general structure, valid for any hard-scattering process, is ¼ AB = a;b 2 2 dx a dx b f a=a (x a ; F )f b =B (x b ; )[^ ¼ 0 + 2 s ( R )^ ¼ 1 + ] ab! X: (2.2) Here F is the factorization scale and R is the renoralization scale for the QCD running coupling. Forally, the perturbation series is invariant under changes in these paraeters, the F and R dependence of the coe cients, e.g. ¼^1, exactly copensating the explicit dependence of the parton distributions and the coupling constant. This copensation becoes ore exact as ore ters are included in the perturbation series. To avoid unnaturally large logariths reappearing in the perturbation series it is sensible to choose F and R values of the order of the typical oentu scales of the hard-scattering process, for exaple F = R = M for the Drell{Yan cross-section. All the iportant hard-scattering hadronic processes have now been calculated to next-to-leading-order (NLO), i.e. up to and including the ¼^1 ters. One process, the Drell{Yan process, is even calculated to one order higher (see below). This allows a very high degree of precision in a wide variety of processes. In any cases, the residual renoralization and factorization scale dependence is weak, and the precision of the theoretical prediction is liited only by uncertainties in the knowledge of the parton distributions. 3. Hadron structure fro the Drell{Yan process (a) Pion structure Hadron{hadron interactions are the only way to obtain direct inforation on the parton structure of esons. There is a signi cant aount of data on hard-scattering

+ Hadron interactions and hadron structure 261 0.5 Q 2 = 5 GeV 2 0.4 x f (x,q 2 ) 0.3 0.2 q V 0.1 g/10 q S 0 0.2 0.4 0.6 0.8 1.0 x Figure 4. Parton distributions in the pion, fro the analysis of Martin et al. (1992). Table 1. Moents of parton distributions in the pion, fro the lattice calculation of Best et al. (1997) and the global t of Martin et al. (1992) (PDF, parton distribution function.) q V oent (quenched) lattice MRSS PDF t hxi 0:273 0:012 0:228 hx 2 i 0:107 0:035 0:099 hx 3 i 0:048 0:020 0:055 processes like p N! ( ; g ; J=Á; : : : ) + X. With the nucleon parton distributions deterined fro deep-inelastic scattering and other processes, the pion distributions can be extracted (Owens 1984; Aurenche et al. 1989; Martin et al. 1992; Gl uck et al. 1992). In this case Drell{Yan production is doinantly a valence{valence annihilation process. For the pion s valence quark distributions, therefore, the precision is essentially that of the input experiental data. It is uch ore di cult to extract precise inforation on the gluon and sea-quark distributions. Figure 4 shows the results of the global analysis of Martin et al. (1992). The sea-quark distribution is largely an educated (theoretical) guess. In recent years there has been signi cant progress in calculating hadronic parton structure `fro rst principles using lattice QCD. Results exist for the rst few oents of the polarized and unpolarized quark distributions in the pion in the socalled quenched approxiation. Table 1 copares the lattice calculations of Best et al. (1997) with oents calculated fro the valence quark distribution of gure 4. The agreeent is reasonable, especially for the higher oents, which presuably are less a ected by the quenched approxiation (the `issing sea-quark distributions are expected to be signi cant only at sall x).

y s + Ö t 262 W. J. Stirling x F = - 0.125 10 10 x F = - 0.075 M 3 d 2 /dx F dm (nb GeV - 2 nucleon - 1 ) 10 8 10 6 10 4 x F = - 0.025 x F = 0.025 x F = 0.075 x F = 0.125 x F = 0.175 x F = 0.225 10 2 x F = 0.275 1 0.2 0.3 0.4 0.5 Figure 5. Cross-section sd 2 ¼ = d p ½ dy for pcu! + X at p la b = 800 GeV=c ( p s = 38:8 GeV) easured by the E605 collaboration (Moreno et al. 1991), with theoretical predictions fro Martin et al. (1998) calculated at NLO. (b) The light quark sea in the proton In pp or pn collisions the cross-section is proportional to the sea-quark distribution, q(x; 2 ). This provides copleentary inforation to deep-inelastic scattering, and in fact Drell{Yan data can be used to constrain the sea-quark distributions in global parton distribution ts. Most of the data are fro xed-target experients using isoscalar targets, for which, to a good approxiation, ¼ p N ¹ u 1 ( u + d)2. It is therefore the su of the light quark distributions that is probed. Figure 5 shows an exaple of the use of high-precision Drell{Yan data in a global t (Martin et al. 1998). Data fro the E605 collaboration (Moreno et al. 1991) on the cross-section sd 2 ¼ = d p ½ dy for pcu! X at p lab = 800 GeV=c ( p s = 38:8 GeV) are copared with theoretical predictions calculated at NLO. Here the u and d seaquark distributions are adjusted to optiize the t. The resulting agreeent between theory and experient is excellent, and suggests that the (unknown) higher-order corrections in this case are sall.y An additional overall noralization factor is included in the t, to allow for unknown higher-order corrections. The value of this factor was found by Martin et al. (1998) to be 1.07.

Hadron interactions and hadron structure 263 1.2 1.0 0.8 MRST MRS (R2) d = u 0 0.1 0.2 0.3 x 2 Figure 6. The MRST description (continuous curve) of the E866 data for the ratio of the proton and deuteriu target Drell{Yan cross-sections versus x2, the fractional oentu of the parton p in the target (¼ d p =2¼ p ). The other curves are for coparison only. (c) The u, d asyetry in the quark sea It is di cult to deterine u and d separately fro deep-inelastic and isoscalar target Drell{Yan data. Indeed, before 1992, global parton analyses assued that the non-strange sea was avour syetric, that is u = d. The rst strong evidence that u 6= d was the Gottfried su easureent by NMC (Arneodo et al. 1994): which suggests 0 1 dx x (F p 2 F n 2 ) = 1 3 2 3 0 1 0 1 dx ( d u) = 0:235 0:026; (3.1) dx ( d u) º 0:15: Such a easureent, however, gives little inforation on the x dependence of the d u di erence. A powerful and independent ethod for obtaining further inforation on d u is to copare Drell{Yan lepton-pair production of pp and pd origin (Ellis & Stirling

264 W. J. Stirling 1991). De ning the cross-section ratio R d p ² ¼ p d 2¼ p p ; (3.2) it is straightforward to show that R d p º 1 if d = u. More generally, ¼ p d 2¼ p p 1 + (4u 1 d 1 )( d 2 u 2 ) 4u 1 u 2 + d 1 d2 ; (3.3) where the 1, 2 subscripts indicate that the partons are to be evaluated at x 1 ; x 2 = 1 2 ( x F + x 2 F + 4½ ); (3.4) with ½ = M 2 =s and x F the Feynan x of the produced lepton pair. Thus, by varying M and x F, the x dependence of d u can be easured. The rst experient of this type was perfored by the NA51 collaboration (Baldit et al. 1994) and yielded u= d = 0:51 0:06 at x = 0:18, 2 ¹ 30 GeV 2. More recently, the E866 collaboration (Hawker et al. 1998) have easured R d p over a uch wider range of M and x F, which enables a study of the x dependence of d u over the range 0:04 < x < 0:3. The continuous curve in gure 6 shows the MRST t (Martin et al. 1998) to these data. The dotted curve shows the d = u prediction for the ratio. The iplications for d= u are shown in gure 7. Interestingly, the structure of d= u shows that, at the axiu value of x that is easured, the ratio has decreased to give d u. Moreover, we see that the NA51 easureent occurs at a value of x for which d= u is essentially at a axiu. Nevertheless, the new data indicate a soewhat saller value of d= u at this point, x = 0:18. Despite uch theoretical work, largely inspired by the recent Drell{Yan data, the origin of u 6= d is not yet entirely clear. The Pauli exclusion principle would suggest that the creation of u u pairs is indeed suppressed relative to d d pairs because of the excess of u over d valence quarks. However, it is di cult to translate this observation into a prediction for the shape of d= u. A ore likely explanation is that there is a non-negligible contribution to the cross-section fro scattering o a eson cloud surrounding the nucleon target. In particular, if the aplitude for p! p + + + is suppressed relative to that for n! p + p, which appears to be the case, then e ectively u < d in the target, in agreeent with the Drell{Yan data. For a detailed discussion of the phenoenology of this see, for exaple, the work of Melnitchouk et al. (1999), and references cited therein. While odels of this type appear to account satisfactorily for the observed increase of d= u fro 1 as x increases fro 0 to about 0:15, they have di culty in accoodating the apparent turnover of the ratio at higher x, see gure 7. One r conclusion that can be drawn fro gure 7 is that ore-precise data at higher x are badly needed! 4. Precision phenoenology at hadron colliders: W and Z production We end this review with an exaple of how hadron{hadron interactions can achieve a very high level of precision in testing the Standard Model. In this case, parton distributions easured very accurately in deep-inelastic scattering are inputs into the hard-scattering cross-section.

q Hadron interactions and hadron structure 265 2.0 1.5 p N total d/u 1.0 0.5 0 0.1 0.2 0.3 x Figure 7. A coparison of the predictions of a `eson cloud odel for the parton ratio d=¹u ¹ with data fro the E866 collaboration (fro Melnitchouk et al. 1999). The separate contributions fro the p N and p states are shown (dashed lines), together with the total (solid line). In xed-target Drell{Yan production, the asses of the lepton pairs are typically M Á. M l + l. O(20 GeV). If the hadron collision energy is large enough, for exaple at the high-energy p p colliders, the annihilation of quarks and antiquarks can produce real W and Z bosons. Indeed, the discovery in 1983 of the W and Z gauge bosons in this way at the CERN p p collider (Arnison et al. 1983; Banner et al. 1983) provided draatic con ration of the Glashow{Sala{Weinberg electroweak odel. The decay widths of the W and Z are O(2 GeV), and so, instead of the di erential distribution in the resulting lepton pair (ln or l + l ) invariant ass, it is ore appropriate to consider the cross-sections for the production of approxiately stable on-shell particles with asses M W and M Z. These can then be ultiplied by branching ratios for the various hadronic and leptonic nal states. In analogy with the Drell{Yan cross-section derived above, the subprocess cross-sections for W and Z production are readily calculated to be q ¼^ p D 0! W = 1 3 º p 2GF MW 2 jv q q 0 j 2 (^s MW 2 ); q q! ¼^ Z = 1 3 º p 2GF MZ 2(v2 q + a 2 q ) (^s MZ 2); (4.1) where V q q 0 is the appropriate Cabibbo{Kobayashi{Maskawa atrix eleent, and v q (a q ) is the vector (axial vector) coupling of the Z to the quarks. The theoretical technology for calculating the total W and Z cross-sections is relatively robust. The O( s ) perturbative QCD correction to the W and Z cross-sections is the sae as the Drell{Yan correction (for a photon of the sae ass) discussed in the previous section: the gluon is ` avour blind and couples in the sae way to the annihilating quark and antiquark. Indeed, as already noted, these cross-sections are now known to next-to-next-to-leading-order (NNLO), i.e. O( 2 s ) (Haberg et al. 1991a; b; van Neerven & Zijlstra 1992). For Z production, the coplete set of O( ) electroweak corrections are also known (Baur et al. 1998) the analysis irrors that of Z production at LEP1 whereas for W production the QED (i.e. photon eission) part of the full O( ) correction has been calculated (Baur et al. 1999). The residual theoretical uncertainty fro unknown electroweak corrections is estiated to be below O(1%).

y s s a 266 W. J. Stirling 2.8 MRST99,00 partons 2.6 NNLO QCD ± 5% CDF(e) D0( ) Z B l + l - (nb) W B lv (nb) 2.4 2.2 2.0 0.27 0.25 0.23 99 g S q 00 00 NNLO (a) ± 5% D0(e) CDF(e) CDF( ) D0(e) 0.21 0.19 (b) 99 g a S q 00 00 NNLO Figure 8. Theoretical predictions based on MRST partons (Martin et al. 2000a) for (a) W and (b) Z cross-sections at the Tevatron, copared with data fro CDF (Abe et al. 1999a; b; A older et al. 2000) and D0 (Abbott et al. 1999). Predictions labelled `00 correspond to new preliinary NLO and NNLO MRST ts (see Martin et al. 2000b). The ain theoretical uncertainty, therefore, originates in the input parton distribution functions and, to a lesser extent, fro s.y For the hadro-production of a heavy object like a W boson, with ass M and at central rapidity, leading-order kineatics give x ¹ M= p s and Q 2 ¹ M 2. Notice that u,d quarks with x values typical of Tevatron and LHC production are already ore or less directly `easured in deep-inelastic scattering (in xed-target experients and at HERA, respectively), but at uch lower Q 2. Therefore, the two iportant sources of uncertainty in the parton distribution functions relevant to W, Z production are: The two are of course correlated.

y Hadron interactions and hadron structure 267 4.0 D0 CDF MRST parton range EW rad. corr. R = s (W) / s (Z) 3.5 3.0 run 0 run 1a run 1b cobined 2.5 Figure 9. The ratio of W and Z total cross-sections (i.e. without branching ratios) at the Tevatron. The data are obtained assuing B(W! en )=B(Z! ee) = 3:2155 0:0060, and the theoretical prediction is fro Martin et al. (2000a). (i) the uncertainty in the DGLAP evolution, which, except at high x, coes ainly fro the gluon and s, and (ii) the uncertainty in the quark distributions fro easureent errors on the structure function data used in the t. In order to investigate these e ects, variants of the standard MRST99 distributions have been constructed (Martin et al. 2000a), allowing the e ect of approxiate 1¼ variations in the gluon, s, the overall quark noralization, and the s and c parton distributions on the W and Z cross-sections to be quanti ed. A coparison of the resulting predictions with the latest CDF (Abe et al. 1999a; b; A older et al. 2000) and D0 (Abbott et al. 1999) easureents at the Tevatron p p collider is shown in gure 8. Evidently, the largest variation in the predictions (ca. 3%) coes fro the e ect of the noralization uncertainty in the input structure function data. Overall, 5% would appear to be a conservative estiate of the theoretical uncertainty.y Note that an iportant coponent of the experiental error is due to the luinosity easureent and uncertainty. The latter is quoted as 3:6% for CDF and 5:4% for D0. In addition, the value assued for the total p p cross-section is slightly di erent (by 6.2%) for the two experients, and this ay account in part for the systeatically saller D0 cross-sections displayed in gure 8. Predictions for the LHC pp collider have siilar theoretical errors (Martin et al. 2000a), which raises the interesting possibility of using W and Z cross-sections as `luinosity onitors at this achine. It is interesting to consider the ratios of the W and Z cross-sections in gure 8. These are experientally uch ore precisely easured, since the luinosity cancels Note that the ost recent `standard MRST and CTEQ distributions, MRST99(1) the central prediction in gure 8 and CTEQ5M1 (Lai et al. 2000), give alost identical cross-sections.

268 W. J. Stirling in the ratio. Many of the theoretical uncertainties, for exaple the overall structure function noralization error, also cancel. The predictions in gure 8 use Standard Model values for the partial and total decay widths to calculate the branching ratios. For the Z, these widths are very precisely pinned down by LEP1 data. However, the total W decay width is still rather poorly deterined experientally, and so the W=Z cross-section ratio provides an interesting indirect easureent: R = N(W! en ) N (Z! ee) B(W! en ) ¼ (W) = B(Z! ee) ¼ (Z) = (W! en ) Z (Z! ee) W ¼ (W) ¼ (Z) : (4.2) Figure 9 shows data on the cross-section ratio ¼ (W)=¼ (Z) fro CDF and D0, obtained by dividing the ratio of observed leptonic events by the Standard Model prediction B(W! en )=B(Z! ee) = 3:2155 0:0060. The band on the theoretical prediction is an estiate of the uncertainty due to parton distributions (coing ainly fro the d=u ratio), and the lines correspond to an estiated 1% uncertainty fro electroweak radiative corrections. The coparison between theory and experient is intriguing, as there is soe slight hint of a disagreeent. When this is translated into a easureent of W, one nds (Lancaster 1999) W(CDF + D0; indirect) = 2:171 0:027(stat.) 0:056(sys.) GeV; (4.3) to be copared with the Standard Model prediction of W = 2:0927 0:0025 GeV. I a grateful to y collaborators Alan Martin, Richard Roberts and Robert Thorne for providing uch of the aterial used in this paper. References Abbott, B. et al. 1999 Measureent of W and Z boson production cross sections. Phys. Rev. D 60, 052003. Abe, F. et al. 1999a Measureent of Z 0 and Drell{Yan cross sections using diuons in ¹pp collisions at p s = 1:8 TeV. Phys. Rev. D 59, 052002. Abe, F. et al. 1999b Measureent of ¼ B(Z 0! e + e ) in p¹p collisions at p s = 1:8 TeV. Phys. Rev. Lett. 76, 3070. A older, T. et al. 2000 The transverse oentu and total cross section of e + e pairs in the Z-boson region fro p¹p collisions at p s = 1:8 TeV. Phys. Rev. Lett. 84, 845. Arneodo, M. et al. 1994 A re-evaluation of the Gottfried su. Phys. Rev. D 50, 1. Arnison, G. et al. 1983 Experiental observation of isolated large transverse oentu energy electrons with associated issing energy at p s = 630 GeV. Phys. Lett. B 122, 103. Aurenche, P. et al. 1989 The gluon content of the pion fro high p T direct photon production. Phys. Lett. B 233, 517. Baldit, A. et al. 1994 Study of the isospin breaking in the light quark sea of the nucleon fro the Drell{Yan process. Phys. Lett. B 332, 244. Banner, G. et al. 1983 Observation of single isolated electrons of high transverse oentu in events with issing transverse energy at the CERN ¹pp collider. Phys. Lett. B 122, 476. Baur, U., Keller, S. & Sakuoto, W. K. 1998 QED radiative corrections to Z boson production and the forward backward asyetry at hadron colliders. Phys. Rev. D 57, 199. Baur, U., Keller, S. & Wackeroth, D. 1999 Electroweak radiative corrections to W boson production in hadronic collisions. Phys. Rev. D 59, 013002. Best, C. et al. 1997 Pion and rho structure functions fro lattice QCD. Phys. Rev. D 56, 2743.

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