The Structure of the ProtonA.M.Cooper-SarkarFeb 6th 2003RSEParton ModelQCD as the theory of strong interactionsParton Distribution FunctionsExtending QCD calculations across the kinematic plane understanding small-x, high density, non-perturbative regions

dF ~2LmnWmnEeEtEpq = k k, Q2 = -q2Px = p + q , W2 = (p + q)2s= (p + k)2x = Q2 / (2p.q) y = (p.q)/(p.k)W2 = Q2 (1/x 1)Q2 = s x ys = 4 Ee EpQ2 = 4 Ee E sin2e/2y = (1 E/Ee cos2e/2)x = Q2/syThe kinematic variables are measurableLeptonic tensor - calculableHadronic tensor- constrained by Lorentz invariance

d2s(eN) = [ Y+ F2(x,Q2) - y2 FL(x,Q2) K Y_xF3(x,Q2)], YK = 1 K (1-y)2dxdyF2, FL and xF3 are structure functions The Quark Parton Model interprets themds = 2pa2 ei2 xs [ 1 + (1-y)2] , for elastic eqQ4 dy d2s = 2pa2 s [ 1 + (1-y)2] Gi ei2(xq(x) + xq(x))

dxdy Q4 for eNisotropicnon-isotropicNow compare the general equation to the QPM prediction F2(x,Q2) = Gi ei2(xq(x) + xq(x)) Bjorken scaling FL(x,Q2) = 0 - spin quarks xF3(x,Q2) = 0 - only ( exchange(xP+q)2=x2p2+q2+2xp.q ~ 0for massless quarks and p2~0 so x = Q2/(2p.q)The FRACTIONAL momentum of the incoming nucleon taken by the struck quark is the MEASURABLE quantity xfor charged lepton hadron scattering

Compare to the general form of the cross-section for n/n scattering via W+/-FL (x,Q2) = 0xF3(x,Q2) = 2Gix(qi(x) - qi(x)) Valence F2(x,Q2) = 2Gix(qi(x) + qi(x)) Valence and Sea And there will be a relationship between F2eN and F2nNNOTE n,n scattering is FLAVOUR sensitivenm-duW+W+ can only hit quarks of charge -e/3 or antiquarks -2e/3 s(np) ~ (d + s) + (1- y)2 (u + c)s(np) ~ (u + c) (1- y)2 + (d + s)

Consider n,n scattering: neutrinos are handedds(n)= GF2 x s ds(n) = GF2 x s (1-y)2

dy dy ppFor n q (left-left)For n q (left-right)d2s(n) = GF2 s Gi [xqi(x) +(1-y)2xqi(x)]dxdypFor nNd2s(n) = GF2 s Gi [xqi(x) +(1-y)2xqi(x)]

dxdy p For nNClearly there are antiquarks in the nucleon3 Valence quarks plus a flavourless qq Sea q = qvalence +qsea q = qsea qsea= qsea

So in n,n scattering the sums over q, q ONLY contain the appropriate flavours BUT-high statistics n,n data are taken on isoscalar targets e.g. Fe Y (p + n)/2=Nd in proton = u in neutronu in proton = d in neutronA TRIUMPH(and 20 years of understanding the c c contribution)GLS sum ruleTotal momentum of quarks

QCD improves the Quark Parton ModelWhat iforBefore the quark is struck?I F2( xi > xi+1.

pt2 of quark relative to proton increases from target to probept2i-1 < pt2i < pt2 i+1Dominant diagrams have STRONG pt orderings s(Q2)The DGLAP equationsxi+1xixi-1xxyyy > x, z = x/y

Terrific expansion in measured range across the x, Q2 plane throughout the 90sHERA dataPre HERA fixed target mp,mD NMC,BDCMS, E665 and n,n Fe CCFRBjorken scaling is broken ln(Q2)Note strong rise at small x

Valence distributions evolve slowlySea/Gluon distributions evolve fastParton Distribution Functions PDFs are extracted by MRST, CTEQ, ZEUS, H1Parametrise the PDFs at Q20 (low-scale)xuv(x) =Auxau (1-x)bu (1+ eu x + gu x)xdv(x) =Adxad (1-x)bd (1+ ed x + gd x) xS(x) =Asx-ls (1-x)bs (1+ es x + gs x)xg(x) =Agx-lg(1-x)bg (1+ eg x + gg x)Some parameters are fixed through sum rules-others by model choices- typically ~15 parametersUse QCD to evolve these PDFs to Q2 > Q20Construct the measurable structure functions in terms of PDFs for ~1500 data points across the x,Q2 planePerform c2 fitNote scale of xg, xSNote error bands on PDFs

The fact that so few parameters allows us to fit so many data points established QCD as the THEORY OF THE STRONG INTERACTION and provided the first measurements of s (as one of the fit parameters)These days we assume the validity of the picture to measure PDFs which are transportable to other hadronic processesBut where is the information coming from?F2(e/mp)~ 4/9 x(u +u) +1/9x(d+d)F2(e/mD)~5/18 x(u+u+d+d) u is dominant , valence dv, uv only accessible at high x(d and u in the sea are NOT equal, dv/uv Y 0 as x Y 1)Valence information at small x only from xF3(nFe) xF3(nN) = x(uv + dv) - BUT Beware Fe target!

HERA data is just ep: xS, xg at small x xS directly from F2xg indirectly from scaling violations dF2 /dlnQ2

Fixed target :p/D data- Valence and Sea

HERA at high Q2 Y Z0 and W+/- become as important asexchange Y NC and CC cross-sections comparablefor NC processes F2 = 3i Ai(Q2) [xqi(x,Q2) + xqi(x,Q2)]xF3= 3i Bi(Q2) [xqi(x,Q2) - xqi(x,Q2)]Ai(Q2) = ei2 2 ei vi ve PZ + (ve2+ae2)(vi2+ai2) PZ2Bi(Q2) = 2 ei ai ae PZ + 4ai ae vi ve PZ2PZ2 = Q2/(Q2 + M2Z) 1/sin2W

a new valence structure function xF3 measurable from low to high x- on a pure proton targetY sensitivity to sin2W, MZ and electroweak couplings vi, ai for u and d type quarks- (with electron beam polarization)

CC processes give flavour informationd2s(e-p) = GF2 M4W [x (u+c) + (1-y)2x (d+s)] dxdy2px(Q2+M2W)2d2s(e+p) = GF2 M4W [x (u+c) + (1-y)2x (d+s)]

dxdy2px(Q2+M2W)2

MW informationuv at high xdv at high xMeasurement of high x dv on a pure proton target (even Deuterium needs corrections, does dv/uv Y 0, as x Y 1? )

Valence PDFs from ZEUS data alone-NC and CC e+ and e- beams Y high xvalence dv from CC e+, uv from CC e- and NC e+/-Valence PDFs from a GLOBAL fit to all DIS data Y high x valence from CCFR xF3(n,nFe) data and NMC F2(mp)/F2(mD) ratio

Parton distributions are transportable to other processesAccurate knowledge of them is essential for calculations of cross-sections of any process involving hadrons. Conversely, some processes have been used to get further information on the PDFs E.G DRELL YAN p N Y m+m- X, via q q Y (* Y m+m-, gives information on the SeaAsymmetry between pp Y m+m- X and pn Y m+m- X gives more information on d - u differenceW PRODUCTION- p p Y W+(W-) X, via u d Y W+, d u Y W- gives more information on u, d differencesPROMPT g - p N Y g X, via g q Y g q gives more information on the gluon (but there are current problems concerning intrinsic pt of initial partons)HIGH ET INCLUSIVE JET PRODUCTION p p Y jet + X, via g g, g q, g q subprocesses gives more information on the gluon for ET > 200 GeV an excess of jets in CDF data appeared to indicate new physics beyond the Standard Model BUT a modification of the u PDF which still gave a reasonable fit to other data could explain itCannot search for physics within (Higgs) or beyond (Supersymmetry) the Standard Model without knowing EXACTLY what the Standard Model predicts Need estimates of the PDF uncertainties

c2 = 3i [ FiQCD Fi MEAS]2

(siSTAT)2+(siSYS)2 = si2Errors on the fit parameters evaluated from )c2 = 1, can be propagated back to the PDF shapes to give uncertainty bands on the predictions for structure functions and cross-sectionsTHIS IS NOT GOOD ENOUGHExperimental errors can be correlated between data points- e.g. Normalisations BUT there are more subtle cases- e.g. Calorimeter energy scale/angular resolutions can move events between x,Q2 bins and thus change the shape of experimental distributions c2 = 3i 3j [ FiQCD Fi MEAS] Vij-1 [ FjQCD FjMEAS] Vij = dij(siSTAT)2 + 3l DilSYS DjlSYSWhere )i8SYS is the correlated error on point i due to systematic error source 8c2 = 3i [ FiQCD 38 slDilSYS- Fi MEAS]2 + 3 sl2(siSTAT) 2s8 are fit parameters of zero mean and unit variance Ymodify the measurement/prediction by each source of systematic uncertaintyHOW to APPLY this ?

OFFSET methodPerform fit without correlated errorsShift measurement to upper limit of one of its systematic uncertainties (sl = +1)Redo fit, record differences of parameters from those of step 1Go back to 2, shift measurement to lower limit (sl = -1)Go back to 2, repeat 2-4 for next source of systematic uncertaintyAdd all deviations from central fit in quadratureHESSIAN methodAllow fit to determine the optimal values of slIf we believe the theory why not let it calibrate the detector?In a global fit the systematic uncertainties of one experiment will correlate to those of another through the fitWe must be very confident of the theory/modelxg(x)Offset methodHessian methodQ2=2.5Q2=7Q2=20

Q2=200

Model Assumptions theoretical assumptions later!Value of Q20, form of the parametrizationKinematic cuts on Q2, W2, xData sets included .Changing model assumptions changes parametersY model errorsMODEL n sEXPERIMENTAL - OFFSET sMODEL o sEXPERIMENTAL - HESSIANYou win some and you lose some!The change in parameters under model changes is frequently outside the )c2=1 criterion of the central fite.g. the effect of using different data sets on the value of asAre some data sets incompatible? Y PDF fitting is a compromise, CTEQ suggest )c2=50 may be a more reasonable criterion for error estimation size of errors determined by the Hessian method rises to the size of errors determined by the Offset method

Comparison of ZEUS (offset) and H1(Hessian) gluon distributions Yellow band (total error) of H1 comparable to red band (total error) of ZEUSComparison of ZEUS and H1 valence distributions. ZEUS plus fixed target data H1 data alone- Experimental errors alone by OFFSET and HESSIAN methods resp.

Value of s and shape of gluon are correlateds increases harder gluon dF2 = s(Q2) [ Pqq q F2 + 2 3i ei2 Pqg q xg]dlnQ22pNLOQCD fit resultsDeterminations of s

Pqq(z) = P0qq(z) + s P1qq(z) +s2 P2qq(z) LO NLO NNLOTheoretical Assumptions- Need to extend the formalism?

What ifOptical theorem2ImThe handbag diagram- QPMQCD at LL(Q2)Ordered gluon ladders (asn lnQ2 n)NLL(Q2) one rung disordered asn lnQ2 n-1Or higher twist diagrams? low Q2, high xEliminate with a W2 cutBUT what about completely disordered Ladders?

Ways to measure the gluon distributionKnowledge increased dramatically in the 90sScaling violations dF2/dlnQ2 in DISHigh ET jets in hadroproduction- TevatronBGF jets in DIS g* g Y q qPrompt gHERA charm production g* g Y c cFor small x scaling violation data from HERA are most accuratePre HERAPost HERA

xg(x,Q2) ~ x -lgAt small x,small z=x/yGluon splitting functions become singular t = ln Q2/72s ~ 1/ln Q2/72Gluon becomes very steep at small x AND F2 becomes gluon dominated F2(x,Q2) ~ x -ls, ls=lg - ,

Still it was a surprise to see F2 still steep at small x - even for Q2 ~ 1 GeV2should perturbative QCD work? s is becoming large - s at Q2 ~ 1 GeV2 is ~ 0.32

The steep behaviour of the gluon is deducedfrom the DGLAP QCD formalism BUT thesteep behaviour of the Sea is measured fromF2 ~ x -ls, ls = d ln F2MRST PDF fitxg(x) ~ x -lgxS(x) ~ x -lsat low xFor Q2 >~ 5 GeV2lg > ls lglsPerhaps one is only surprised that the onset of the QCD generated rise appears to happen at Q2 ~ 1 GeV2 not Q2 ~ 5 GeV2 d ln 1/x

Need to extend formalism at small x?The splitting functions Pn(x), n= 0,1,2for LO, NLO, NNLO etcHave contributionsPn(x) = 1/x [ an ln n (1/x) + bn ln n-1 (1/x) .These splitting functions are used indq/dlnQ2 ~ s I dy/y P(z) q(y,Q2)And thus give rise to contributions to the PDFs p (Q2) (ln Q2)q (ln 1/x) rConventionally we sump = q r 0 at Leading Log Q2 - (LL(Q2)) p = q+1 r 0 at Next to Leading log Q2 (NLLQ2) DGLAP summationsBut if ln(1/x) is large we should considerp = r q 1 at Leading Log 1/x (LL(1/x))p = r+1 q 1 at Next to Leading Log (NLL(1/x)) - BFKL summations

LL(Q2) is STRONGLY ordered in pt. At small x it is also STRONGLY ordered in x Double Leading Log Approximations p (Q2) (ln Q2)q(ln 1/x)r, p=q=rLL(1/x) is STRONGLY ordered in ln(1/x) and can be disordered in pt Ys p(Q2)(ln 1/x)r, p=rBFKL summation at LL(1/x) Yxg(x) ~ x -ll = s CA ln2 ~ 0.5, for s ~ 0.25

steep gluon even at moderate Q2

But this is considerable softened at NLL(1/x)(Y way beyond scope !) p

Furthermore if the gluon density becomes large there maybe non-linear effectsGluon recombination g g Y gs~ s2r2/Q2may compete with gluon evolution g Y g gs~ s rwhere D is the gluon densityr~ xg(x,Q2) no.of gluons per ln(1/x)

BR2nucleon sizeNon-linear evolution equations GLRd2xg(x,Q2) = 3s xg(x,Q2) s2 81 [xg(x,Q2)]2dlnQ2dln1/xp16Q2R2as r as2 r2/Q2The non-linear term slows down the evolution of xg and thus tames the rise at small xThe gluon density may even saturate(-respecting the Froissart bound)Extending the conventional DGLAP equations across the x, Q2 plane

Plenty of debate about the positions of these lines!Colour Glass Condensate, JIMWLK, BKHigher twist

Do the data NEED unconventional explanations ?In practice the NLO DGLAP formalism works well down to Q2 ~ 1 GeV2 BUT below Q2 ~ 5 GeV2 the gluon is no longer steep at small x in fact its becoming NEGATIVE!We only measure F2 ~ xq dF2/dlnQ2 ~ Pqg xgUnusual behaviour of dF2/dlnQ2 may come fromunusual gluon or from unusual Pqg- alternative evolution?We need other gluon sensitive measurements at low xLike FL, or F2charm`Valence-like gluon shape

Current measurements of FL and F2charm at small x are not yet accurate enough to distinguish different approachesFLF2 charm

xg(x)Q2 = 2GeV2The negative gluon predicted at low x, low Q2 from NLO DGLAP remains at NNLO (worse)

The corresponding FL is NOT negative at Q2 ~ 2 GeV2 but has peculiar shapeIncluding ln(1/x) resummation in the calculation of the splitting functions (BFKL `inspired) can improve the shape - and the c2 of the global fit improvesAre there more defnitive signals for `BFKL behaviour?In principle yes, in the hadron final state, from the lack of pt orderingHowever, there have been many suggestions and no definitive observations- We need to improve the conventional calculations of jet production

The use of non-linear evolution equations also improves the shape of the gluon at low x, Q2The gluon becomes steeper (high density) and the sea quarks less steepNon-linear effects gg Y g involve the summation of FAN diagrams Q2 = 1.4 GeV2Q2 = 2Q2=10Q2=100 GeV2Non linearDGLAPxgxuxsxuvxdxcSuch diagrams form part of possible higher twist contributions at low x Y there maybe further clues from lower Q2 data?xg

Linear DGLAP evolution doesnt work for Q2 < 1 GeV2, WHAT does? REGGE ideas? Regge regionpQCD regionSmall x is high W2, x=Q2/2p.q . Q2/W2

s(g*p) ~ (W2) -1 Regge prediction for high energy cross-sections is the intercept of the Regge trajectory =1.08 for the SOFT POMERONSuch energy dependence is well established from the SLOW RISE of all hadron-hadron cross-sections - including s(gp) ~ (W2) 0.08 for real photon- proton scattering For virtual photons, at small x s(g*p) = 4p2 F2 Q2Y s~ (W2)-1 Y F2 ~ x 1- = x -l so a SOFT POMERON would imply l = 0.08 Y only a very gentle rise of F2 at small xFor Q2 > 1 GeV2 we have observed a much stronger rise Ypx2 = W2qp

The slope of F2 at small x , F2 ~x -l , is equivalent to a rise of s(g*p) ~ (W2)l which is only gentle for Q2 < 1 GeV2F((*p)gentle risemuch steeper riseGBW dipoleQCD improved dipoleRegge regionpQCD generated slopeSo is there a HARD POMERON corresponding to this steep rise?A QCD POMERON, (Q2) 1 = l(Q2)A BFKL POMERON, 1 = l = 0.5A mixture of HARD and SOFT Pomerons to explain the transition Q2 = 0 to high Q2?What about the Froissart bound ? the rise MUST be tamed non-linear effects?

Dipole models provide a way to model the transition Q2=0 to high Q2At low x, g* Y qq and the LONG LIVED (qq) dipole scatters from the protonThe dipole-proton cross section depends on the relative size of the dipole r~1/Q to the separation of gluons in the target R0s =s0(1 exp( r2/2R0(x)2)), R0(x)2 ~(x/x0)l~1/xg(x)r/R0 small Y large Q2, x F ~ r2~ 1/Q2 r/R0 large Y small Q2, x s ~ s0 Y saturation of the dipole cross-section GBW dipole modelF((*p) But s(g*p) = 4pa2 F2 is generalQ2 s(gp) is finite for real photons , Q2=0. At high Q2, F2 ~flat (weak lnQ2 breaking) and s(g*p) ~ 1/Q2(at small x)Now there is HERA data right across the transition region

t is a new scaling variable, applicable at small xIt can be used to define a `saturation scale , Q2s = 1/R02(x) . x -l ~ x g(x), gluon density- such that saturation extends to higher Q2 as x decreasesSome understanding of this scaling, of saturation and of dipole models is coming from work on non-linear evolution equations applicable at high density Colour Glass Condensate, JIMWLK, Balitsky-Kovchegov. There can be very significant consequences for high energy cross-sections e.g. neutrino cross-sections also predictions for heavy ions- RHIC, diffractive interactions Tevatron and HERA, even some understanding of soft hadronic physicsF = F0 (1 exp(-1/t))Involves only t =Q2R02(x) Y t = Q2/Q02 (x/x0)lAnd INDEED, for x 0.01Q2 < Q2sQ2 > Q2s

SummaryMeasurements of Nucleon Structure Functions are interesting in their own right- telling us about the behaviour of the partons which must eventually be calculated by non-perturbative techniques- lattice gauge theory etc.They are also vital for the calculation of all hadronic processes- and thus accurate knowledge of them and their uncertainties is vital to investigate all NEW PHYSICSHistorically these data established the Quark-Parton Model and the Theory of QCD, providing measurements of the value of as(MZ2) and evidence for the running of as(Q2)There is a wealth of data available now over 6 orders of magnitude in x and Q2 such that conventional calculations must be extended as we move into new kinematic regimes at small x, at high density and into the non-perturbative region. The HERA data has stimulated new theoretical approaches in all these areas.