1. The Structure of the Proton A.M.Cooper-Sarkar Feb 6 th 2003 RSE Parton Model QCD as the theory of strong interactions Parton Distribution Functions Extending QCD calculations across the kinematic plane – understanding small-x, high density, non- perturbative regions

2. dF~dF~ 2 LWLW EeEe EtEt EpEp q = k – k , Q 2 = -q 2 P x = p + q, W 2 = (p + q) 2 s= (p + k) 2 x = Q 2 / (2p.q) y = (p.q)/(p.k) W 2 = Q 2 (1/x – 1) Q 2 = s x y s = 4 E e E p Q 2 = 4 E e E  sin 2  e /2 y = (1 – E  /E e cos 2  e /2) x = Q 2 /sy The kinematic variables are measurable Leptonic tensor - calculable Hadronic tensor- constrained by Lorentz invariance

3. d 2  (e±N) = [ Y + F 2 (x,Q2) - y 2 F L (x,Q2) K Y_xF 3 (x,Q2)], Y K = 1 K (1- y) 2 dxdy F 2, F L and xF 3 are structure functions – The Quark Parton Model interprets them d  =   e i 2 xs [ 1 + (1-y) 2 ], for elastic eq Q4Q4 dy d 2  =   s [ 1 + (1-y) 2 ] G i e i 2 (xq(x) + xq(x)) dxdy Q 4 for eN isotropic non-isotropic Now compare the general equation to the QPM prediction F 2 (x,Q2) = G i e i 2 (xq(x) + xq(x)) – Bjorken scaling F L (x,Q2) = 0 - spin ½ quarks xF 3 (x,Q2) = 0 - only ( exchange (xP+q) 2 =x 2 p 2 +q 2 +2xp.q ~ 0 for massless quarks and p 2 ~0 so x = Q 2 /(2p.q) The FRACTIONAL momentum of the incoming nucleon taken by the struck quark is the MEASURABLE quantity x for charged lepton hadron scattering

4. Compare to the general form of the cross- section for / scattering via W +/- F L (x,Q2) = 0 xF 3 (x,Q2) = 2 G i x(q i (x) - q i (x)) Valence F 2 (x,Q2) = 2 G i x(q i (x) + q i (x)) Valence and Sea And there will be a relationship between F 2 eN and F 2  NOTE  scattering is FLAVOUR sensitive -- d u W+W+ W+ can only hit quarks of charge -e/3 or antiquarks -2e/3  p)  ~ (d + s) + (1- y) 2 (u + c)  p ) ~ (u + c) (1- y) 2 + (d + s) Consider, scattering: neutrinos are handed d  ( )= G F 2 x s d  ( ) = G F 2 x s (1-y) 2 dy  For q (left-left)For q (left-right) d 2  ( ) = G F 2 s G i [xq i (x) +(1-y) 2 xq i (x)] dxdy  For N d 2  ( ) = G F 2 s G i [xq i (x) +(1-y) 2 xq i (x)] dxdy  For N Clearly there are antiquarks in the nucleon 3 Valence quarks plus a flavourless qq Sea q = q valence +q sea q = q sea q sea = q sea

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

6. QCD improves the Quark Parton Model What if or Before the quark is struck? I F2( < N) dx ~ 0.5 where did the momentum go? PqqPgq PqgPgg Note q(x,Q 2 ) ~  s lnQ 2, but  s (Q 2 )~1/lnQ 2, so  s lnQ 2 is O(1), so we must sum all terms  s n lnQ 2n Leading Log Approximation x decreases from target to probe x i-1 > x i > x i+1 …. p t 2 of quark relative to proton increases from target to probe p t 2 i-1 < p t 2 i < p t 2 i+1 Dominant diagrams have STRONG p t ordering  s   s (Q 2 ) The DGLAP equations x i+1 xixi x i-1 xx y y y > x, z = x/y

7. Terrific expansion in measured range across the x, Q 2 plane throughout the 90’s HERA data Pre HERA fixed target  p,  D NMC,BDCMS, E665 and, Fe CCFR Bjorken scaling is broken – ln(Q 2 ) Note strong rise at small x

8. Valence distributions evolve slowly Sea/Gluon distributions evolve fast Parton Distribution Functions PDFs are extracted by MRST, CTEQ, ZEUS, H1 Parametrise the PDFs at Q 2 0 (low-scale) xu v (x) =A u x au (1-x) bu (1+  u √x +  u x) xd v (x) =A d x ad (1-x) bd (1+  d √x +  d x) xS(x) =A s x - s (1-x) bs (1+  s √x +  s x) xg(x) =A g x - g (1-x) bg (1+  g √x +  g x) Some parameters are fixed through sum rules- others by model choices- typically ~15 parameters Use QCD to evolve these PDFs to Q 2 > Q 2 0 Construct the measurable structure functions in terms of PDFs for ~1500 data points across the x,Q 2 plane Perform  2 fit Note scale of xg, xS Note error bands on PDFs

9. 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 processes But where is the information coming from? F 2 (e/  p)~ 4/9 x(u +u) +1/9x(d+d) F 2 (e/  D)~5/18 x(u+u+d+d) u is dominant, valence d v, u v only accessible at high x (d and u in the sea are NOT equal, d v /u v Y 0 as x Y 1) Valence information at small x only from xF 3 ( Fe) xF 3 ( N) = x(uv + dv) - BUT Beware Fe target! HERA data is just ep: xS, xg at small x xS directly from F 2 xg indirectly from scaling violations dF 2 /dlnQ 2 Fixed target : p/D data- Valence and Sea

10. HERA at high Q 2 Y Z 0 and W +/- become as important as  exchange Y NC and CC cross-sections comparable for NC processes F 2 = 3 i A i (Q 2 ) [xq i (x,Q 2 ) + xq i (x,Q 2 )] xF 3 = 3 i B i (Q 2 ) [xq i (x,Q 2 ) - xq i (x,Q 2 )] A i (Q 2 ) = e i 2 – 2 e i v i v e P Z + (v e 2 +a e 2 )(v i 2 +a i 2 ) P Z 2 B i (Q 2 ) = – 2 e i a i a e P Z + 4a i a e v i v e P Z 2 P Z 2 = Q 2 /(Q 2 + M 2 Z ) 1/sin 2  W Ya new valence structure function xF 3 measurable from low to high x- on a pure proton target Y sensitivity to sin 2  W, M Z and electroweak couplings v i, a i for u and d type quarks- (with electron beam polarization)

11. CC processes give flavour information d 2  (e - p) = G F 2 M 4 W [x (u+c) + (1-y) 2 x (d+s)] dxdy 2  x(Q 2 +M 2 W ) 2 d 2  (e + p) = G F 2 M 4 W [x (u+c) + (1-y) 2 x (d+s)] dxdy 2  x(Q 2 +M 2 W ) 2 M W information u v at high x d v at high x Measurement of high x d v on a pure proton target (even Deuterium needs corrections, does d v /u v Y 0, as x Y 1? )

12. Valence PDFs from ZEUS data alone- NC and CC e+ and e- beams Y high x valence d v from CC e+, u v from CC e- and NC e+/- Valence PDFs from a GLOBAL fit to all DIS data Y high x valence from CCFR xF3 (, Fe) data and NMC F2(  p)/F2(  D) ratio

13. Parton distributions are transportable to other processes Accurate 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  +  - X, via q q Y ( * Y  +  -, gives information on the Sea Asymmetry between pp Y  +  - X and pn Y  +  - X gives more information on d - u difference W PRODUCTION- p p Y W + (W - ) X, via u d Y W +, d u Y W - gives more information on u, d differences PROMPT  - p N Y  X, via g q Y  q gives more information on the gluon (but there are current problems concerning intrinsic pt of initial partons) HIGH E T 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 it Cannot 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

14.  2 = 3 i [ F i QCD – F i MEAS ] 2 (  i STAT ) 2 +(  i SYS ) 2 =  i 2 Errors on the fit parameters evaluated from )  2 = 1, can be propagated back to the PDF shapes to give uncertainty bands on the predictions for structure functions and cross-sections THIS IS NOT GOOD ENOUGH Experimental 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,Q 2 bins and thus change the shape of experimental distributions  2 = 3 i 3 j [ F i QCD – F i MEAS ] V ij - 1 [ F j QCD – F j MEAS ] V ij =  ij (  i STAT ) 2 + 3   SYS  j SYS Where ) i 8 SYS is the correlated error on point i due to systematic error source 8  2 = 3 i [ F i QCD – 3 8 s  i SYS - F i MEAS ] 2 + 3 s 2 (  i STAT ) 2 s 8 are fit parameters of zero mean and unit variance Y modify the measurement/prediction by each source of systematic uncertainty HOW to APPLY this ?

15. OFFSET method 1.Perform fit without correlated errors 2.Shift measurement to upper limit of one of its systematic uncertainties (s = +1) 3.Redo fit, record differences of parameters from those of step 1 4.Go back to 2, shift measurement to lower limit (s = -1) 5.Go back to 2, repeat 2-4 for next source of systematic uncertainty 6.Add all deviations from central fit in quadrature HESSIAN method Allow fit to determine the optimal values of s If 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 fit We must be very confident of the theory/model xg(x) Offset method Hessian method Q 2 =2.5 Q 2 =7 Q 2 =20 Q 2 =200

16. Model Assumptions – theoretical assumptions later! Value of Q 2 0, form of the parametrization Kinematic cuts on Q 2, W 2, x Data sets included ……. Changing model assumptions changes parameters Y model error  MODEL n  EXPERIMENTAL - OFFSET  MODEL o  EXPERIMENTAL - HESSIAN You win some and you lose some! The change in parameters under model changes is frequently outside the )  2 =1 criterion of the central fit e.g. the effect of using different data sets on the value of  s Are some data sets incompatible? Y PDF fitting is a compromise, CTEQ suggest )  2 =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

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

18. Value of  s and shape of gluon are correlated  s increases  harder gluon dF 2 =  s (Q 2 ) [ Pqq q F 2 + 2 3 i e i 2 Pqg q xg] dlnQ 2 22 NLOQCD fit results Determinations of  s

19. Pqq(z) = P 0 qq(z) +  s P 1 qq(z) +  s 2 P 2 qq(z) LO NLO NNLO Theoretical Assumptions- Need to extend the formalism? What if Optical theorem 2 Im The handbag diagram- QPM QCD at LL(Q 2 ) Ordered gluon ladders (  s n lnQ 2 n ) NLL(Q 2 ) one rung disordered  s n lnQ 2 n-1 Or higher twist diagrams? low Q 2, high x Eliminate with a W 2 cut BUT what about completely disordered Ladders?

20. Ways to measure the gluon distributionKnowledge increased dramatically in the 90’s Scaling violations dF2/dlnQ2 in DIS High ET jets in hadroproduction- Tevatron BGF jets in DIS  * g Y q q Prompt  HERA charm production  * g Y c c For small x scaling violation data from HERA are most accurate Pre HERAPost HERA

21. xg(x,Q 2 ) ~ x - g At small x, small z=x/y Gluon splitting functions become singular t = ln Q 2 / 7 2  s ~ 1/ln Q 2 / 7 2 Gluon becomes very steep at small x AND F 2 becomes gluon dominated F 2 (x,Q 2 ) ~ x - s, s= g -,

22. Still it was a surprise to see F 2 still steep at small x - even for Q 2 ~ 1 GeV 2 should perturbative QCD work?  s is becoming large -  s at Q 2 ~ 1 GeV 2 is ~ 0.32

23. The steep behaviour of the gluon is deduced from the DGLAP QCD formalism – BUT the steep behaviour of the Sea is measured from F 2 ~ x - s, s = d ln F 2 MRST PDF fit xg(x) ~ x - g xS(x) ~ x - s at low x For Q 2 >~ 5 GeV 2 g > s g s Perhaps one is only surprised that the onset of the QCD generated rise appears to happen at Q 2 ~ 1 GeV 2 not Q 2 ~ 5 GeV 2 d ln 1/x

24. Need to extend formalism at small x? The splitting functions P n (x), n= 0,1,2……for LO, NLO, NNLO etc Have contributions P n (x) = 1/x [ a n ln n (1/x) + b n ln n-1 (1/x) …. These splitting functions are used in dq/dlnQ 2 ~  s I dy/y P(z) q(y,Q 2 ) And thus give rise to contributions to the PDF  s p (Q 2 ) (ln Q 2 ) q (ln 1/x) r Conventionally we sum p = q ≥ r ≥ 0 at Leading Log Q 2 - (LL(Q 2 )) p = q+1 ≥ r ≥ 0 at Next to Leading log Q 2 (NLLQ2) – DGLAP summations But if ln(1/x) is large we should consider p = 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(Q 2 ) is STRONGLY ordered in p t. At small x it is also STRONGLY ordered in x – Double Leading Log Approximation  s p (Q 2 ) (ln Q 2 ) q (ln 1/x) r, p=q=r LL(1/x) is STRONGLY ordered in ln(1/x) and can be disordered in p t Y  s p (Q 2 )(ln 1/x) r, p=r BFKL summation at LL(1/x) Y xg(x) ~ x - =  s C A ln2 ~ 0.5, for  s ~ 0.25 Ysteep gluon even at moderate Q2 But this is considerable softened at NLL(1/x) ( Y way beyond scope !) 

25. Furthermore if the gluon density becomes large there maybe non-linear effects Gluon recombination g g Y g  ~  s 2  2 /Q 2 may compete with gluon evolution g Y g g  ~  s  where D is the gluon density  ~ xg(x,Q2) –no.of gluons per ln(1/x) BR2BR2 nucleon size Non-linear evolution equations – GLR d 2 xg(x,Q2) = 3  s xg(x,Q2) –  s 2 81 [xg(x,Q2)] 2 dlnQ2dln1/x  16Q 2 R 2 s s   s 2  2 /Q 2 The non-linear term slows down the evolution of xg and thus tames the rise at small x The gluon density may even saturate (-respecting the Froissart bound) Extending the conventional DGLAP equations across the x, Q 2 plane Plenty of debate about the positions of these lines! Colour Glass Condensate, JIMWLK, BK Higher twist

26. Do the data NEED unconventional explanations ? In practice the NLO DGLAP formalism works well down to Q 2 ~ 1 GeV 2 BUT below Q 2 ~ 5 GeV 2 the gluon is no longer steep at small x – in fact its becoming NEGATIVE! We only measure F 2 ~ xq dF 2 /dlnQ 2 ~ Pqg xg Unusual behaviour of dF 2 /dlnQ 2 may come from unusual gluon or from unusual Pqg- alternative evolution? We need other gluon sensitive measurements at low x Like F L, or F 2 charm `Valence-like’ gluon shape

27. Current measurements of F L and F 2 charm at small x are not yet accurate enough to distinguish different approaches FLFL F 2 charm

28. xg(x) Q 2 = 2GeV 2 The negative gluon predicted at low x, low Q 2 from NLO DGLAP remains at NNLO (worse) The corresponding F L is NOT negative at Q 2 ~ 2 GeV 2 – but has peculiar shape Including ln(1/x) resummation in the calculation of the splitting functions (BFKL `inspired’) can improve the shape - and the  2 of the global fit improves Are there more defnitive signals for `BFKL’ behaviour? In principle yes, in the hadron final state, from the lack of p t ordering However, there have been many suggestions and no definitive observations- We need to improve the conventional calculations of jet production

29. The use of non-linear evolution equations also improves the shape of the gluon at low x, Q 2 The gluon becomes steeper (high density) and the sea quarks less steep Non-linear effects gg Y g involve the summation of FAN diagrams – Q 2 = 1.4 GeV 2 Q 2 = 2Q 2 =10Q 2 =100 GeV 2 Non linear DGLAP xg xu xs xu v xd xc Such diagrams form part of possible higher twist contributions at low x Y there maybe further clues from lower Q 2 data? xg

30. Linear DGLAP evolution doesn’t work for Q 2 < 1 GeV2, WHAT does? – REGGE ideas? Regge region pQCD region Small x is high W 2, x=Q 2 /2p.q. Q 2 /W 2  (  *p) ~ (W 2 )  -1 – Regge prediction for high energy cross-sections  is the intercept of the Regge trajectory  =1.08 for the SOFT POMERON Such energy dependence is well established from the SLOW RISE of all hadron-hadron cross-sections - including  (  p) ~ (W 2 ) 0.08 for real photon- proton scattering For virtual photons, at small x  (  *p) = 4  2  F 2 Q2Q2 Y  ~ (W 2 )  -1 Y F 2 ~ x 1-  = x - so a SOFT POMERON would imply = 0.08 Y only a very gentle rise of F 2 at small x For Q 2 > 1 GeV 2 we have observed a much stronger rise Y p x 2 = W 2 q p

31. The slope of F 2 at small x, F 2 ~x -, is equivalent to a rise of  (  *p) ~ (W 2 ) which is only gentle for Q 2 < 1 GeV 2 F ( ( *p) gentle rise much steeper rise GBW dipole QCD improved dipole Regge regionpQCD generated slope So is there a HARD POMERON corresponding to this steep rise? A QCD POMERON,  (Q 2 ) – 1 = (Q 2 ) A BFKL POMERON,  – 1 = = 0.5 A mixture of HARD and SOFT Pomerons to explain the transition Q 2 = 0 to high Q 2 ? What about the Froissart bound ? – the rise MUST be tamed – non-linear effects?

32. Dipole models provide a way to model the transition Q 2 =0 to high Q 2 At low x,  * Y qq and the LONG LIVED (qq) dipole scatters from the proton The dipole-proton cross section depends on the relative size of the dipole r~1/Q to the separation of gluons in the target R 0  =  0 (1 – exp( –r 2 /2R 0 (x) 2 )), R 0 (x) 2 ~(x/x 0 ) ~1/xg(x) r/R 0 small Y large Q 2, x F ~ r 2 ~ 1/Q 2 r/R 0 large Y small Q2, x  ~  0 Y saturation of the dipole cross-section GBW dipole model F ( ( * p) But  (  *p) = 4  2 F 2 is general Q2Q2  (  p) is finite for real photons, Q 2 =0. At high Q 2, F 2 ~flat (weak lnQ 2 breaking) and  (  *p) ~ 1/Q 2 (at small x) Now there is HERA data right across the transition region

33.  is a new scaling variable, applicable at small x It can be used to define a `saturation scale’, Q 2 s = 1/R 0 2 (x). x - ~ x g(x), gluon density - such that saturation extends to higher Q 2 as x decreases Some 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 physics F = F 0 (1 – exp(- 1/  )) Involves only  =Q 2 R 0 2 (x) Y  = Q 2 /Q 0 2 (x/x 0 ) And INDEED, for x<0.01,  (  *p) depends only on , not on x, Q 2 separately x < 0.01 x > 0.01 Q 2 < Q 2 s Q 2 > Q 2 s

34. Summary Measurements 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 PHYSICS Historically  these data established the Quark-Parton Model and the Theory of QCD, providing measurements of the value of  s (M Z 2 ) and evidence for the running of  s (Q 2 ) There is a wealth of data available now over 6 orders of magnitude in x and Q 2 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.