Hadronic Structur Functions 1 arxiv:hp-x/0009009v1 5 Sp 2000 Martin Erdmann Institut für Exprimntll Krnphysik, Univrsität Karlsruh, Engssrstr. 7, D-76128 Karlsruh, Martin.Erdmann@dsy.d Abstract Exprimntal rsults on hadronic structurs ar discussd in viw of our physics undrstanding. Achivmnts and challngs ar notd. 1 Invitd opning plnary talk at th 8th Intrnational Workshop on Dp Inlastic Scattring and QCD, Livrpool, UK (2000)
1 Motivation Today s motivation of masuring lpton hadron scattring procsss is at last four-fold. Fig.1 shows basic diagrams at lctron positron, hadron hadron, and lpton hadron collidrs: Only in lpton hadron collisions is th fusion diagram forbiddn within th Standard Modl, which strongly motivats sarchs for nw physics,.g. lptoquarks. Th xchang of bosons allows diffrnt hadronic structurs to b probd: Th prototyp for xisting hadronic structurs is th proton which currntly is th most prcisly studid hadronic objct. Gnsis of hadronic structurs is analysd using th structur dvloping in quantum fluctuations of th photon. Colour singlt xchang constituts a procss byond singl boson xchang. It s succssful dscription provids a prim challng for QCD. It is th purpos of this contribution to undrlin ths diffrnt aspcts of lpton-hadron scattring physics and thir prspctivs using as much as possibl th masurmnts thmslvs. LEP/TESLA TEV/LHC HERA LEP/TESLA q, g q, g? q, g q, g q q Figur 1: Basic diagrams at lctron positron, hadron hadron, and lpton hadron collidrs. Only th last diagram is forbiddn within th Standard Modl. 2 Lpton Quark Scattring at Attomtr Distanc Th larg cntr of mass nrgy at HERA of s p = 318 GV allows lpton quark scattring to b analysd at distancs down to almost 1 Attomtr = 10 18 m. Both nutral and chargd currnt intractions (Fig.2) ar usd to tst th Standard Modl prdictions. Th doubl diffrntial cross sctions in trms of th rsolution scal Q 2, which dnots th ngativ squard 1
Nutral and chargd currnt intractions obsrvd with th ZEUS and H1 xpri- Figur 2: mnts. four-momntum transfr carrid by th boson, and th quark fractional momntum x rlativ to th proton ar givn by: d 2 σ NC dq 2 dx d 2 σ CC dq 2 dx α 2 1 1 Q 4 x Φ NC(x, Q 2 ) (1) ( ) M G 2 2 2 W 1 F MW 2 + Q2 x Φ CC (x, Q 2 ). (2) Hr α and G F dnot th coupling strngth of th lctromagntic and wak intraction procsss. M W is th W -boson mass. Th Φ trms dnot th spin charactristics of th scattring togthr with th probabilitis xf(x, Q 2 ) of finding th diffrnt quark flavours in th proton. In addition Φ NC contains trms for Z xchang and γ-z intrfrnc. At high Q 2 it is ( ( )) ( Φ p θ NC 1 + cos 4 4 2 9 (xu + xū) + 1 ) 9 (xd + x d ) ± add. trms with γ, Z (3) ( ) Φ p θ CC xu + cos 4 x 2 d (4) ( ) Φ + p θ CC xū + cos 4 xd. (5) 2 θ dnots th scattring angl in th lpton-quark cntr of mass systm and can b calculatd from cos 4 (θ /2) = (1 Q 2 /s p /x) 2. Th two componnts in th angular distribution rsult from two spin configurations of th colliding lpton and quark: if th spins add up to zro, any scattring angl is allowd. If th spins add up to 1, backward scattring is forbiddn for masslss quarks and th angular distribution is wightd by cos 4 (θ /2). In chargd currnt intractions also th quark typ can b analysd:.g., positrons coupl only to ngativly chargd quarks. In addition, right-handd positrons coupl only to right-handd antiquarks, or lft-handd quarks. This offrs a uniqu handl to diffrntiat btwn quark flavours in th proton. 2
Elctron Proton Collisions dσ dq 2 p γ,z p W Q 2 [GV 2 ] Figur 3: Diffrntial cross sction for nutral and chargd currnt intractions as a function of th rsolution scal Q 2 from H1 and ZEUS data. Intgrating th doubl diffrntial cross sctions ovr x givs th singl diffrntial cross sction which is shown in Fig.3 as a function of Q 2 from H1 [1] and ZEUS [2] data. Around Q 2 10 4 GV 2 th cross sctions ar found to b of qual magnitud. Sinc in both nutral currnt and chargd currnt lctron-proton scattring at high Q 2 primarily th u-valnc quarks ar probd, qs. (3, 4), ths data stablish dirct obsrvation of th unification of th nutral currnt and chargd currnt intractions at a rsolution scal corrsponding to about 10 Attomtr. In Fig.4, th spin charactristics of chargd currnt positron-proton scattring is tstd in th masurmnt of th wightd cross sctions Φ, q. (2), as a function of cos 4 (θ /2) [3]. Within th prcision of th masurmnt, th data ar in ach x-bin compatibl with a linar ris as xpctd from q. (5). Th xtrapolation of th linar bhavior to th backward scattring rgion (cos 4 (θ /2) = 0) rvals a non-vanishing contribution of th ngativly chargd antiquarks, mainly ū. Thir rlativ contribution dcrass as x incrass. Th rising componnt rflcts th contribution of th d-valnc quarks. Th d-quark dnsity can b rad off th forward scattring cross sction (cos 4 (θ /2) = 1). Also in nutral currnt intractions, th data ar ovr a wid rang compatibl with a linar ris: thy rflct two qually larg componnts, xplaind by th two spin configurations of th lctromagntic procsss, q. (3). Th forward scattring rgion (cos 4 (θ /2) = 1) shows 3
φ CC 0.5 Chargd Currnt x=0.08 H1 + p 94-97 Standard Modl (NLO QCD Fit) 0 xd xū φ NC 0.5 0 Nutral Currnt x=0.08 H1 + p 94-97 Standard Modl (NLO QCD Fit) γ-exchang Fit 4 9 xu 4 9 xu x=0.13 x=0.13 0.5 0.5 0 0 0.5 x=0.25 0.5 x=0.25 0 0 0.25 0.5 0.75 1 (1-y) 2 cos 4 θ 2 0 0 0.25 0.5 0.75 1 (1-y) 2 cos 4 θ 2 Figur 4: H1 masurmnts of th doubl diffrntial chargd and nutral currnt positronproton cross sctions as a function of cos 4 (θ /2) in diffrnt bins of th parton fractional momntum x. θ dnots th scattring angl in th lpton-quark cntr of mass systm. approximatly th u-quark dnsity in th proton: 2 4/9 xu xu. In th rgion of backward scattring procsss (cos 4 (θ /2) = 0), th cross sction masurmnts dviat from th linar ris and dmonstrat th onst of a nw intraction: th lowr cross sction rsults from th ngativ intrfrnc btwn th photon and th Z-boson. Th comparison of th positron-proton nutral currnt and chargd currnt data in th forward scattring rgion of Fig.4 dmonstrats dirctly from th data that th u-quark dnsity is twic that of th d-quark. Thrfor th proton consists of th uud quark configuration also at th small distanc scals probd at HERA. Th HERA luminosity upgrad program, starting to tak data in 2001, is agrly awaitd: much mor prcis data will challng th Standard Modl prdictions for p procsss in th Attomtr rgim. 4
3 Existing Hadronic Structur: Proton As discussd in th prvious sction, th uud valnc structur of th proton has bn rconfirmd in th high Q 2 nutral and chargd currnt masurmnts at HERA. In th following our physics undrstanding of th proton structur function F 2 is discussd. F 2 is dtrmind from masurmnts of th doubl diffrntial nutral currnt cross sction (compar with qs. (1, 3)) d 2 σ dq 2 dx 1 α2 Q 4 (1 + cos 4 ( θ 2 )) 1 x F 2(x, Q 2 ) (6) and contains th individual quark distributions, wightd by th quark squard chargs: F 2 (x, Q 2 ) 4 9 (xu + xū ) + 1 9 (xd + x d ) + 1 (xs + x s ) +... (7) 9 Th QCD volution quations prdict that masurmnts of hadronic structurs dpnd on th logarithm of th rsolution scal Q 2 at which th structur is probd. On this basis, th following ansatz to analys th x-dpndnc of structur function data is xplord [4]: F 2 (x, Q 2 ) = a(x) [ ln ( )] Q 2 κ(x). (8) Hr Λ is a scal paramtr, a rflcts th charg squard wightd quark distributions xtrapolatd to ln(q 2 /Λ 2 ) = 1, and κ dtrmins th positiv and ngativ scaling violations of F 2. In Fig.5, publishd ZEUS [5] low-x data of th proton structur function F 2 for Q 2 > 2 GV 2 ar shown. In ach x-bin, th rsult of a two-paramtr fit according to q. (8) is shown, using a fixd valu of Λ = 0.35 GV which rprsnts a typical valu of th strong intraction scal. Only th total xprimntal rrors hav bn usd, ignoring corrlations btwn individual data points. Th sam fitting procdur has bn applid to BCDMS data [6] which ar takn hr as a rfrnc sampl for th high-x rgion. Th rsulting paramtrs a and κ ar summarizd in Fig.6 as a function of x togthr with fits to th publishd H1 low-x data [7]. Also shown ar fits to th prliminary H1 data [8] which ar much mor prcis than th prvious masurmnts. For a, th data fits xhibit two distinct rgions: around x 0.3 thy rflct th valnc quark distributions, implying that ach valnc quark carris 1/3 of th proton momntum. At low x, a(x) is compatibl with convrging to a constant valu. A comparison of th lowst point, drivd from th H1 prliminary masurmnt, with th nw ZEUS prliminary data prsntd at this confrnc will b of intrst. Th rsulting scaling violation trm κ appars to ris as x dcrass, xhibiting th ngativ and positiv scaling violations of F 2 for x abov and blow 0.1 rspctivly. Th rrors in Fig.6 rprsnt th statistical rrors of th fits. Both paramtrs a and κ ar anti-corrlatd as can b sn from nighbouring points. No significant Q 2 -dpndnc of a and κ has bn found in th publishd data whn th fits wr rpatd for two intrvals in Q 2 (abov and blow 20 GV 2 ). Λ 2 5
Figur 5: ZEUS and BCDMS masurmnts of th proton structur function F 2 ar shown as a function of Q 2 in th rang 10 4 < x < 1. Thy ar compard to th 2-paramtr fits according to q. (8) in ach x-bin. With a bing approximatly constant blow x 10 2, changs of F 2 at low x rsult from th scaling violation trm κ alon, indicativ of th intraction dynamics that drivs F 2 and in support of th prdictions [9, 10]. Th paramtr a has alrady bn idntifid as th charg squard wightd quark distributions xtrapolatd to ln (Q 2 /Λ 2 ) = 1 which corrsponds hr to Q 2 = 0.3 GV 2. An undrstanding of th paramtrs Λ and κ can b achivd by comparison with th QCD volution quation which is writtn hr in th lading ordr DGLAP approximation: df i (x, Q 2 ) = α s(q 2 ) d lnq 2 2π j 1 x dy y P ij ( ) x f j (y, Q 2 ). (9) y 6
sa quarks valnc quarks Figur 6: Th quark distribution a(x) of th proton xtrapolatd to Q 2 = 0.3 GV 2 and th scaling violations κ(x) from th fits to th publishd H1, ZEUS, BCDMS, and to th H1 prliminary data according to q. (8). Th dottd lins srv to guid th y. Hr f i, f j dnot th parton dnsitis, P ij ar th splitting functions, and is th strong coupling constant. α s = b ln (Q 2 /Λ 2 QCD ) (10) Th drivativ of th ansatz chosn hr, q. (8), with rspct to ln Q 2 givs df 2 (x, Q 2 ) d lnq 2 = 1 ln (Q 2 /Λ 2 ) κ(x) F 2(x, Q 2 ), (11) whr rlating 1/ ln(q 2 /Λ 2 ) with α s in q. (9) implis association of th scal paramtr Λ in q. (8) with th QCD paramtr Λ QCD. Th trm κ rlats to th sum ovr th diffrnt parton radiation trms in q. (9) dividd by F 2. κ incrass towards small x, consistnt with largr phas spac availabl for parton radiation. 7
Figur 7: Masurmnts of th photon structur function ar shown as a function of Q 2. Thy ar compard to th 2-paramtr fits according to q. (8) in ach x-bin. To match th dscription chosn hr, q. (8), with th doubl asymptotic approximation xpctd from QCD for th gluon-dominatd rgion at small x, F 2 xp ln x ln (ln (Q 2 /Λ 2 )) [9, 11], th scaling violation trm κ is rquird to hav a dpndnc lik ln x κ ln (ln (Q 2 /Λ 2 )). (12) Th Q 2 -dpndnc of κ is thrfor xpctd to b vry small which is in agrmnt with th xprimntal obsrvation statd abov. Mor prcis data and data raching smallr valus of x will dtrmin whthr or not th scaling violations furthr incras towards low-x and thrfor giv valuabl information on th parton dnsitis in th proton as x approachs 0. 4 Gnsis of Hadronic Structur: Photon Th photon structur rsults from fluctuations of a photon into a colour nutral and flavour nutral hadronic stat. For comparison with th proton data, th sam fits according to q. (8) hav bn applid to rcnt masurmnts of th photon structur function F γ 2 which hav bn prformd at + collidrs [12] (Fig.7). Th valus of th paramtrs a and κ ar shown in Fig.8 as th opn circls. Both paramtrs ar distinct from thos of a hadronic bound stat lik th proton: in a(x) th photon data xhibit no valnc quark structur. Instad, in th low-x rgion around x 0.1 th photon data prfr similar valus of a to th proton data for x 0.01. Th scaling violations κ ar positiv at all valus of x and κ is approximatly 1. This is as xpctd from QCD calculations which prdict F γ 2 for 0.1 < x < 1 [13]. 8
Figur 8: Th hadronic structurs a(x), xtrapolatd to Q 2 = 0.3 GV 2, and th scaling violations κ(x) from fits to structur function data according to q. (8) ar compard btwn th proton, photon, and colour singlt xchang. Th diagrams of splitting functions indicat rgions thy contribut to th QCD volution. Th lins srv to guid th y. Judgmnt on a univrsal low-x bhaviour of hadronic structurs will rsult from mor prcis masurmnts and lowr-x data of th photon structur function. If th photon data show a constant quark dnsity at small x similar to th low-x proton data, scaling violations of F γ 2, which dviat from thos rsulting from th photon splitting into quark-antiquark pairs and approach thos obsrvd for th proton, could bcom visibl blow or slightly abov x = 10 2 whr also for th proton data it is κ 1. Intrsting information on th qustion of univrsality coms alrady from masurmnts of th gluon in th photon probd in strong intraction procsss in photon-proton collisions. Th production of two-jt vnts is snsitiv to th gluons dvloping in photon fluctuations. In Fig.9, a rcnt masurmnt of xg(x) is shown [14]. Th gluons appar as th low-x companions of th nwly built hadronic structur: at larg x th gluon dnsity is small; it riss towards small valus of x. 9
p p Figur 9: Comparison of th H1 photon and proton gluon distributions as a function of x. In th sam figur, this gluon distribution is compard to th gluon distribution of th proton, dtrmind from masurmnts of th proton structur function [8]. Although th rror bars of th photon masurmnt ar larg and Q 2 and p 2 t may not rprsnt th vry sam rsolution scal, th similarity of th nwly built and th alrady xisting gluon distribution is striking. This obsrvation may b a first xprimntal indication of a univrsal gluon distribution dvloping in hadronic structurs. 5 Colour Singlt Exchang Furthr information on gluons in hadronic structurs rsults from structur function masurmnts of colour singlt xchang. In Fig.10, H1 F D(3) 2 data [15] ar compard to th sam two-paramtr fits as usd abov, q. (8). Hr x (frquntly calld β) dnots th fractional momntum of th scattrd parton rlativ to th colour nutral objct, which itslf carris a fractional momntum ξ = 0.003 rlativ to th proton and thrfor blongs to th low-x companions of th proton. Also ths data xhibit scaling violations κ that ar diffrnt from th proton masurmnts at th sam valus of x (Fig.8). Instad, for x < 0.5 thy hav th tndncy of bing largr than th photon data and ar similar to th low-x proton data. Th larg rat of vnts with colour singlt xchang togthr with th larg scaling violations of F D(3) 2 is suggstiv of a gluon dominatd xchang. 10
p Figur 10: H1 masurmnts of th structur function of colour singlt xchang ar shown as a function of Q 2. Thy ar compard to th 2-paramtr fits according to q. (8) in ach x-bin. Th valus of th normalization a ris towards x = 1 to about a = 10 (in Fig. 8, th paramtr a has bn scald by 1/25). Ths valus hav larg uncrtaintis of th ordr of 100%. If mor prcis data support such a singular parton dnsity for x 1 at low Q 2, thn ths colour nutral fluctuations consist of on gluon carrying ssntially all th colour singlt momntum and (at last) on furthr gluon with vry low momntum nutralizing th colour. 6 Prdictiv Powr for Proton Intraction Procsss Th proton structur rvals amazing simplicity: at low rsolution scal Q 2, th thr valnc quarks uud ach carry fractional momntum x = 1/3 (s sctions 2, 3). Th sa quark contribution is at low valus of x indpndnt of x (s sction 3). Gluons accompany th proton at low x with a possibly univrsal momntum distribution (s sction 4). Gluons initiat colour nutral configurations togthr with othr vry low momntum gluons (s sction 5). Howvr, to prdict intractions with protons, full information on all individual parton distributions of th proton ar rquird. Whil such parton distribution functions xf i hav bn availabl from global fits for many yars, rcnt pionring work has succdd in dtrmining th prcision of ths distribution functions taking into account th prcision of th masurmnts and corrlations btwn th diffrnt functions xf i [16] (Fig.11). This analysis shows a good knowldg of th functions xf i ovr a wid rang in x. Howvr, th knowldg for x 1 is not satisfactory: larg valus of x corrspond to high rsolution powr at hadron collidrs (.g. LHC) and point at th potntial discovry rgion for nw physics. An improvd dtrmination of th proton parton distribution as x approachs 1 by dp inlastic scattring xprimnts is thrfor mandatory and currntly is undr discussion [17]. 11
xq i d u x Figur 11: Parton distribution functions and d/u ratio as a function of x at Q 2 = 10 GV 2 from a global fit which taks into account xprimntal rrors and corrlations btwn th individual parton distribution functions. x Furthr qustions on th prdictiv powr of QCD calculations for proton-proton intractions rsult from th masurmnts of forward jt and forward π cross sctions in p collisions,.g. [18]. Ths masurmnts xplicitly tst QCD volution ovr som rapidity distanc and may signal limitations of th currnt approximations of QCD volution to simpl procss configurations at small distancs. Hr thortical work is ndd and ongoing. 7 Achivmnts and Challngs W currntly clbrat th 30 yars knowldg of valnc quarks in th proton. Th nw contribution of th HERA collidr xprimnts to th undrstanding of th proton is th low-x structur which appars as a consqunc of QCD dynamics. Opn qustions ar: is th parton dnsity of th proton finit as x 0? What is th parton dnsity at x 1? Is th QCD volution approximatd corrctly? Masurmnts on th gnsis procss of hadronic structurs us quantum fluctuations of th photon: sinc ovr 20 yars w know th momntum distributions of quarks rsulting from th photon splitting into quark-antiquark pairs. For th first tim, th HERA and LEP xprimnts hav masurd th gluon distribution of nwly built hadronic configurations, which is found to b vry similar to th gluon distribution masurd in protons. Th opn qustion to th photon data is: is hadronic structur at low x univrsal, i.., do th low-x partons know about th partons in th high-x rgion? Masurmnts of th partonic structur of colour singlt xchang at HERA and th TEVA- TRON [19] for th first tim show that such objcts dominantly consist of gluons. Will ths 12
masurmnts srv as a rfrnc procss for a gluon drivn rgim and offr nw insight into QCD dynamics? Major contributions of lpton-hadron scattring in th past 10 yars dpn our undrstanding of hadronic structurs. Burning opn qustions nsur that this fild of rsarch will rmain vry activ also in th coming dcad. Acknowldgmnts I wish to thank vry much th Livrpool tam for a wondrful confrnc! For carful rading and commnts to th manuscript I am gratful to E. Elsn and B. Fostr. I wish to thank Th. Müllr and th IEKP group of th Univrsity Karlsruh for thir hospitality, and th Dutsch Forschungsgminschaft for th Hisnbrg Fllowship. Rfrncs [1] H1 Collab., Masurmnt of th Chargd and Nutral Currnt Cross Sctions at HERA, contrib. papr 157b, Intrnational Europhysics Confrnc on High Enrgy Physics (HEP99), Tampr, Finland, 1999 [2] ZEUS Collab., Masurmnt of High Q 2 Nutral, Chargd Currnt Dp Inlastic Scattring Cross Sctions in p scattring at HERA, contrib. paprs 549, 558, Intrnational Europhysics Confrnc on High Enrgy Physics (HEP99), Tampr, Finland, 1999 [3] H1 Collab., C. Adloff t al., Eur. J. Phys. C 13, 609 (2000) [4] M. Erdmann, hp-x/0007058, Phys. Ltt. B 488, 131 (2000) [5] ZEUS Collab., M. Drrick t al., Z. Phys. C 72, 399 (1996) [6] BCDMS Collab., A.C. Bnvnuti t al., Phys. Ltt. B 223, 485 (1989) [7] H1 Collab., S. Aid t al., Nucl. Phys. B 470, 3 (1996) [8] M. Klin for th H1 Collab., hp-x/0001059, Proc. XIX Int. Symp. on Lpton and Photon Intractions at High Enrgis, Stanford, USA (1999) [9] A. D Rujula t al., Phys. Rv. D 10, 1649 (1974) [10] M. Glück, E. Rya, and A. Vogt, Z. Phys. C 53, 127 (1992) [11] R.D. Ball and S. Fort, Phys. Ltt. B 335, 77 (1994) [12] Photon structur function data from rviw by R. Nisius, hp-x/9912049, Phys. Rpt. 332, 165 (2000) [13] E. Wittn, Nucl. Phys. B 120, 189 (1977) 13
[14] H1 Collab., C. Adloff t al., Phys. Ltt. B 483, 36 (2000) [15] H1 Collab., C. Adloff t al., Z. Phys. C 76, 613 (1997) [16] M. Botj, hp-ph/9912439, Eur. J. Phys. C 14, 285 (2000) [17] Workshop on th Nuclon Structur in High x-bjorkn Rgion, HiX2000, Tmpl Univrsity, Jffrson Lab, Philadlphia, USA (2000) [18] H1 Collab., C. Adloff t al., Phys. Ltt. B 462, 440 (1999) [19] CDF Collab., T. Affoldr t al., Phys. Rv. Ltt. 84, 232 (2000) 14