1. Deep Inelastic Scattering (DIS): Parton Distribution Functions and low-x physics Goa Sept 2008 A.M.Cooper-Sarkar Oxford What have we learnt from DIS in the last 30 years? QPM, QCD Parton Distribution Functions, α s Low-x physics

2. dσ~dσ~ 2 L μν W μν EeEe E’E’ 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 θ e /2 y = (1 – E’/E e cos 2 θ e /2) x = Q 2 /sy The kinematic variables are measurable Leptonic tensor - calculable Hadronic tensor- constrained by Lorentz invariance PDFs were first investigated in deep inelastic lepton-hadron scatterning -DIS

3. d 2  (e±N) = [ Y + F 2 (x,Q 2 ) - y 2 F L (x,Q 2 ) ± Y_xF 3 (x,Q 2 )], Y± = 1 ± (1-y) 2 dxdy F 2, F L and xF 3 are structure functions which express the dependence of the cross-section on the structure of the nucleon – The Quark-Parton model interprets these structure functions as related to the momentum distributions of quarks or partons within the nucleon AND the measurable kinematic variable x = Q2/(2p.q) is interpreted as the FRACTIONAL momentum of the incoming nucleon taken by the struck quark (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 Completely generally the double differential cross-section for e-N scattering Leptonic part hadronic part

4. d  =   e i 2 s [ 1 + (1-y) 2 ], so for elastic electron quark scattering, quark charge e i e Q4Q4 dy d 2  =   s [ 1 + (1-y) 2 ] Σ i e i 2 (xq(x) + xq(x)) dxdy Q 4 so for eN, where eq has c. of m. energy 2 equal to xs, and q(x) gives probability that such a quark is in the Nucleon isotropicnon-isotropic Now compare the general equation to the QPM prediction to obtain the results F 2 (x,Q2) = Σ i e i 2 (xq(x) + xq(x)) – Bjorken scaling – this depends only on x F L (x,Q2) = 0 - spin ½ quarks xF 3 (x,Q2) = 0 - only γ exchange Consider electron muon scattering d  =   s [ 1 + (1-y) 2 ], for elastic eμ Q4Q4 dy

5. Compare to the general form of the cross- section for / scattering via W +/- F L (x,Q2) = 0 xF 3 (x,Q2) = 2 Σ i x(q i (x) - q i (x)) Valence F 2 (x,Q2) = 2 Σ i x(q i (x) + q i (x)) Valence and Sea And there will be a relationship between F 2 eN and F 2  Also 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 Σ i [xq i (x) +(1-y) 2 xq i (x)] dxdy  For N d 2  ( ) = G F 2 s Σ 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

6. So in  scattering the sums over q, qbar ONLY contain the appropriate flavours BUT- high statistics  data are taken on isoscalar targets e.g. Fe ~ (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

7. BUT – Bjorken scaling is broken F2 does not depend only on x it also depends on Q2 – ln(Q 2 ) Particularly strongly at small x

8. QCD improves the Quark Parton Model What if or Before the quark is struck? PqqPgq PqgPgg The DGLAP parton evolution equations xx y y y > x, z = x/y So F 2 (x,Q 2 ) = Σ i e i 2 (xq(x,Q 2 ) + xq(x,Q 2 )) in LO QCD The theory predicts the rate at which the parton distributions (both quarks and gluons) evolve with Q 2 - (the energy scale of the probe) -BUT it does not predict their shape

9. What if higher orders are needed? Pqq(z) = P 0 qq(z) + α s P 1 qq(z) + α s 2 P 2 qq(z) LO NLO NNLO Note q(x,Q 2 ) ~ α s lnQ 2, but α s (Q 2 )~1/lnQ 2, so α s lnQ 2 is O(1), so we must sum all terms α s n lnQ 2n Leading Log Approximation x decreases from  s   s (Q 2 ) 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 F2 is no longer so simply expressed in terms of partons - convolution with coefficient functions is needed – but these are calculable in QCD FL is no longer zero.. And it depends on the gluon

10. Formalism NLO DGLAP MSbar factorisation Q 0 2 functional form @ Q 0 2 sea quark (a)symmetry etc. Data DIS (SLAC, BCDMS, NMC, E665, CCFR, H1, ZEUS, CCFR, NuTeV ) Drell-Yan (E605, E772, E866, …) High E T jets (CDF, D0) W rapidity asymmetry (CDF) etc. Who? Alekhin, CTEQ, MRST, H1, ZEUS http://durpdg.dur.ac.uk/hepdata/ http://projects.hepforge.org/lhapdf/ LHAPDFv5 f i (x,Q 2 )   f i (x,Q 2 ) α S (M Z ) How do we determine Parton Distribution Functions ? Parametrise the parton distribution functions (PDFs) at Q 2 0 (~1-7 GeV 2 )- Use NLO QCD DGLAP equations to evolve these PDFs to Q2 >Q20 Construct the measurable structure functions and cross-sections by convoluting PDFs with coefficient functions: make predictions for ~2000 data points across the x,Q2 plane- Perform χ2 fit to the data

11. 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, bar Fe CCFR We have to impose appropriate kinematic cuts on the data so as to remain in the region when the NLO DGLAP formalism is valid The DATA – the main contribution is DIS data 1. Q 2 cut : Q 2 > few GeV 2 so that perturbative QCD is applicable- α s (Q 2 ) small 2.W 2 cut: to avoid higher twist terms- usual formalism is leading twist 3.x cut: to avoid regions where ln(1/x) resummation (BFKL) and non-linear effects may be necessary….

12. Need to extend the formalism? Optical theorem 2 Im The handbag diagram- QPM QCD at LL(Q 2 ) Ordered gluon ladders ( α s n lnQ 2 n ) NLL(Q 2 ) one rung disordered α s n lnQ 2 n-1 ? And what about Higher twist diagrams ? Are they always subdominant? Important at high x, low Q 2 BUT what about completely disordered Ladders? at small x there may be a need for BFKL ln(1/x) resummation?

13. Non-linear fan diagrams form part of possible higher twist contributions at low x The strong rise in the gluon density at small-x leads to speculation that there may be a need for non-linear equations?- gluons recombining gg→g Strong coupling

14. In practice it has been amazing how low in Q 2 the standard formalism still works- down to Q 2 ~ 1 GeV 2 : cut Q 2 > 2 GeV 2 is typical It has also been surprising how low in x – down to x~ 10 -5 : no x cut is typical Nevertheless there are doubts as to the applicability of the formalism at such low-x.. (See later) there could be ln(1/x) corrections and/or non-linear high density corrections for x < 5 10 -3 The CUTS

15. Higher twist terms can be important at low-Q2 and high-x → this is the fixed target region (particularly SLAC- and now JLAB data). Kinematic target mass corrections and dynamic contributions ~ 1/Q 2 Fits establish that higher twist terms are not needed if W 2 > 15 GeV 2 – typical W 2 cut Fit with F2=F2 LT (1 +D 2 (x)/Q 2 ) X→ 2x/(1 + √(1+4m 2 x 2 /Q 2 ))

16. xu v (x) =A u x au (1-x) bu (1+ ε u √ x + γ u x) xd v (x) =A d x ad (1-x) bd (1+ ε d √ x + γ d x) xS(x) =A s x - λ s (1-x) bs (1+ ε s √ x + γ s x) xg(x) =A g x - λ g (1-x) bg (1+ ε g √ x + γ g x) 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 parameters control the low-x shape Parameters Ag, Au, Ad are fixed through momentum and number sum rules  – other parameters may be fixed by model choices- Model choices  Form of parametrization at Q 2 0, value of Q 2 0,, flavour structure of sea, cuts applied, heavy flavour scheme → typically ~15-22 parameters Use QCD to evolve these PDFs to Q 2 >Q 2 0 Construct the measurable structure functions by convoluting PDFs with coefficient functions: make predictions for ~2000 data points across the x,Q2 plane Perform χ2 fit to the data These parameters control the high-x shape These parameters control the middling-x shape The form of the parametrisation Parametrise the parton distribution functions (PDFs) at Q 2 0 (~1-7 GeV 2 ) Alternative form for CTEQ xf(x) = A 0 x A1 (1-x) A2 e A3x (1+e A4 x) A5

17. The form of the parametrisation at Q 2 0 Ultimately we may get this from lattice QCD, or other models- the statistical model is quite successful (Soffer et al). But we can make some guesses at the basic form: x a (1-x) b ….. at one time (20 years ago?) we thought we understood it! --------the high x power from counting rules ----(1-x) 2ns-1 - ns spectators valence (1-x) 3, sea (1-x) 7, gluon (1-x) 5 --------the low-x power from Regge – low-x corresponds to high centre of mass energy for the virtual boson proton collision (x = Q2 / (2p.q)) -----Regge theory gives high energy cross-sections as s (α-1) -----------which gives x dependence x (1-α), where α is the intercept of the Regge trajectory- different for singlet (no overall flavour) F2 ~x 0 and non-singlet (flavour- valence-like) xF3~x 0.5 The shapes of F2 and xF3 even looked as if they followed these shapes – pre HERA

18. – Valence distributions evolve slowly Sea/Gluon distributions evolve fast But at what Q2 would these be true? – Valence distributions evolve slowly but sea and gluon distributions evolve fast. We input non-perturbative ideas about shapes at a low scale, but we are just parametrising our ignorance. It turns out that we need the arbitrary polynomial In any case the further you evolve in Q2 the less the parton distributions look like the low Q2 inputs and the more they are determined by QCD evolution (Some people don’t use a starting parametrization at all- but let neural nets learn the shape of the data- NNPDF)

19. Assuming u in proton = d in neutron – strong- isospin Where is the information coming from? Originally- pre HERA Fixed target e/μ p/D data from NMC, BCDMS, E665, SLAC F 2 (e/  p)~ 4/9 x(u +ubar) +1/9x(d+dbar) + 4/9 x(c +cbar) +1/9x(s+sbar) F 2 (e/  D)~5/18 x(u+ubar+d+dbar) + 4/9 x(c +cbar) +1/9x(s+sbar) Also use ν, νbar fixed target data from CCFR( now also NuTeV/Chorus) (Beware Fe target needs nuclear corrections) F2( ν,νbar N) = x(u +ubar + d + dbar + s +sbar + c + cbar) xF 3 ( ν,νbar N) = x(u v + d v ) (provided s = sbar) Valence information for 0< x < 1 Can get ~4 distributions from this: e.g. u, d, ubar, dbar – but need assumptions like q=qbar for all flavours, sbar = 1/4 (ubar+dbar) and heavy quark treatment. Note gluon enters only indirectly via DGLAP equations for evolution

20. Flavour structure Historically an SU(3) symmetric sea was assumed u=u v +u sea, d=d v +d sea u sea = ubar = d sea = dbar = s = sbar =K and c=cbar=0 Measurements of F 2 μn = u v + 4d v +4/3K F 2 μp 4u v + d v +4/3K Establish no valence quarks at small-x F 2 μn /F 2 μp →1 But F 2 μn /F 2 μp →1/4 as x → 1 Not to 2/3 as it would for d v /u v =1/2, hence it look s as if d v /u v →0 as x →1 i.e the d v momentum distribution is softer than that of u v - Why? Non-perturbative physics --diquark structures? How accurate is this? Could d v /u v →1/4 (Farrar and Jackson)?

21. Flavour structure in the sea dbar ≠ubar in the sea Consider the Gottfried sum-rule (at LO) ∫ dx (F2p-F2n) = 1/3 ∫dx (uv-dv) +2/3∫dx(ubar-dbar) If ubar=dbar then the sum should be 0.33 the measured value from NMC = 0.235 ± 0.026 Clearly dbar > ubar…why? low Q2 non-perturbative effects, Pauli blocking, p →nπ +,pπ 0,Δ ++ π - -- W+W+ sbar≠(ubar+dbar)/2, in fact sbar ~ (ubar+dbar)/4 Why? The mass of the strange quark is larger than that of the light quarks Evidence – neutrino opposite sign dimuon production rates And even s≠sbar? Because of p→ΛK+ s c→s μ + ν

22. Low-x – within conventional NLO DGLAP Before the HERA measurements most of the predictions for low-x behaviour of the structure functions and the gluon PDF were wrong HERA ep neutral current (γ-exchange) data give much more information on the sea and gluon at small x….. xSea directly from F 2, F 2 ~ xq xGluon from scaling violations dF 2 /dlnQ 2 – the relationship to the gluon is much more direct at small-x, dF 2 /dlnQ 2 ~ Pqg xg So what did HERA bring?

23. xg(x,Q 2 ) ~ x - λ g At small x, small z=x/y Gluon splitting functions become singular t = ln Q 2 /  2 α s ~ 1/ln Q 2 /  2 A flat gluon at low Q 2 becomes very steep AFTER Q 2 evolution AND F 2 becomes gluon dominated F 2 (x,Q 2 ) ~ x - λ s, λ s= λ g - ε Low-x And yet people didn’t expect this…. See later

24. HERA data have also provided information at high Q 2 → Z 0 and W +/- become as important as γ exchange → NC and CC cross-sections comparable For NC processes F 2 =  i A i (Q 2 ) [xq i (x,Q 2 ) + xq i (x,Q 2 )] xF 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 θ W  a new valence structure function xF 3 due to Z exchange is measurable from low to high x- on a pure proton target → no heavy target corrections- no assumptions about strong isospin High Q 2 HERA data-still to be fully exploited

25. 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 d is not well known because u couples more strongly to the photon. Historically information has come from deuterium targets –but even Deuterium needs binding corrections. Open questions: does u in proton = d in neutron?, does dv/uv  0, as x  1?

26. How has our knowledge evolved?

27. The u quark LO fits to early fixed-target DIS data To view small and large x in one plot To reveal the difference in both large and small x regions

28. Rise from HERA data

29. The story about the gluon is more complex Gluon

30. HERA steep rise of F2 at low x

31. Gluon Does gluon go negative at small x and low Q? see MRST/W PDFs More recent fits with HERA data- steep rise even for low Q 2 ~ 1 GeV 2 Tev jet data

32. PDF comparison 2008 The latest HERAPDF0.1 gives very small experimental errors and modest model errors

33. Modern analyses assess PDF uncertainties within the fit Clearly errors assigned to the data points translate into errors assigned to the fit parameters -- and these can be propagated to any quantity which depends on these parameters— the parton distributions or the structure functions and cross- sections which are calculated from them = Σ j Σ k ∂ F V jk ∂ F ∂ pj ∂ pk The errors assigned to the data are both statistical and systematic and for much of the kinematic plane the size of the point- to-point correlated systematic errors is ~3 times the statistical errors This must be treated carefully in the χ2 definition

34. Some data sets incompatible/only marginally compatible? One could restrict the data sets to those which are sufficiently consistent that these problems do not arise – (H1, GKK, Alekhin ) But one loses information since partons need constraints from many different data sets – no- one experiment has sufficient kinematic range / flavour information (though HERA comes close) To illustrate: the χ2 for the MRST global fit is plotted versus the variation of a particular parameter (α s ). The individual χ2 e for each experiment is also plotted versus this parameter in the neighbourhood of the global minimum. Each experiment favours a different value of. α s PDF fitting is a compromise. Can one evaluate acceptable ranges of the parameter value with respect to the individual experiments?

35. This leads them to suggest a modification of the χ2 tolerance, Δχ2 = 1, with which errors are evaluated such that Δχ2 = T 2, T = 10. Why? Pragmatism. The size of the tolerance T is set by considering the distances from the χ2 minima of individual data sets from the global minimum for all the eigenvector combinations of the parameters of the fit. All of the world’s data sets must be considered acceptable and compatible at some level, even if strict statistical criteria are not met, since the conditions for the application of strict statistical criteria, namely Gaussian error distributions are also not met. One does not wish to lose constraints on the PDFs by dropping data sets, but the level of inconsistency between data sets must be reflected in the uncertainties on the PDFs. CTEQ6 look at eigenvector combinations of their parameters rather than the parameters themselves. They determine the 90% C.L. bounds on the distance from the global minimum from ∫ P(χ e 2, N e ) dχ e 2 =0.9 for each experiment illustration for eigenvector-4 Distance

36. Not just statistical improvement. Each experiment can be used to calibrate the other since they have rather different sources of experimental systematics Before combination the systematic errors are ~3 times the statistical for Q2< 100 After combination systematic errors are < statistical → very consistent data input HERAPDFs use Δχ2=1 Recent development: Combining ZEUS and H1 data sets

37. CTEQ6.5 MSTW08 Diifferent uncertainty estimates on the gluon persist as Q 2 increases Q 2 =10 Q 2 =10000 Note CTEQ in general bigger uncertainties…but NOT for low-x gluon

38. The general trend of PDF uncertainties for conventional NLO DGLAP is that The u quark is much better known than the d quark The valence quarks are much better known than the gluon/sea at high-x The valence quarks are poorly known at small- x but they are not important for physics in this region The sea and the gluon are well known at low-x but only down to x~10 -4 The sea is poorly known at high-x, but the valence quarks are more important in this region The gluon is poorly known at high-x And it can still be very important for physics

39. End lecture 1 Next PDFs for the LHC Be-(a)ware of low-x physics

40. Knowledge from HERA →the LHC- transport PDFs to hadron-hadron cross-sections using QCD factorization theorem for short-distance inclusive processes where X=W, Z, D-Y, H, high-E T jets, prompt-γ and  is known to some fixed order in pQCD and EW in some leading logarithm approximation (LL, NLL, …) to all orders via resummation ^ pApA pBpB fafa fbfb x1x1 x2x2 X The central rapidity range for W/Z production AT LHC is at low-x (5 ×10 -4 to 5 ×10 -2 ) The Standard Model is not as well known as you might think

41. MRST PDF NNLO corrections small ~ few% NNLO residual scale dependence < 1% W/Z production have been considered as good standard candle processes with small theoretical uncertainty. PDF uncertainty is THE dominant contribution and most PDF groups quote uncertainties <~5% (but note HERAPDF ~1-2%) BUT the central values differ by more than some of the uncertainty estimates. PDF set σ W+ B W →lν (nb) σ W- B W →lν (nb) σ z B z →ll (nb) ZEUS-200511.87±0.458.74±0.311.97±0.06 MRST0111.61±0.238.62±0.161.95±0.04 HERAPDF12.13±0.139.13±0.152.01±0.025 CTEQ6512.47±0.479.14±0.362.03±0.07 CTEQ6111.61±0.568.54±0.431.89±0.09

42. Pre HERAPost HERA Look at predictions for W/Z rapidity distributions Pre- and Post-HERA Why such an improvement ? It’s due to the improvement in the low-x gluon At the LHC the q- qbar which make the boson are mostly sea-sea partons at low-x And at Q2~MZ2 the sea is driven by the gluon

43. This is post HERA but just one experiment This is post HERA using the new (2008) HERA combined PDF fit However there is still the possibility of trouble with the formalism at low-x

44. Moving on to BSM physics Example of how PDF uncertainties matter for BSM physics– Tevatron jet data were originally taken as evidence for new physics-- iThese figures show inclusive jet cross-sections compared to predictions in the form (data - theory)/ theory Something seemed to be going on at the highest E_T And special PDFs like CTEQ4/5HJ were tuned to describe it better- note the quality of the fits to the rest of the data deteriorated. But this was before uncertainties on the PDFs were seriously considered

45. Today Tevatron jet data are considered to lie within PDF uncertainties. (Example from CTEQ hep-ph/0303013) We can decompose the uncertainties into eigenvector combinations of the fit parameters-the largest uncertainty is along eigenvector 15 – which is dominated by the high x gluon uncertainty

46. 2XD 4XD 6XD SM Such PDF uncertainties on the jet cross sections compromise the potential for discovery of any physics effects which can be written as a contact interaction E.G. Dijet cross section potential sensitivity to compactification scale of extra dimensions (M c ) reduced from ~6 TeV to 2 TeV. M c = 2 TeV, no PDF error M c = 2 TeV, with PDF error M c = 6 TeV, no PDF error

47. LHC is a low-x machine (at least for the early years of running) Low-x information comes from evolving the HERA data Is NLO (or even NNLO) DGLAP good enough? The QCD formalism may need extending at small-x BFKL ln(1/x) resummation High density non-linear effects etc. (Devenish and Cooper-Sarkar, ‘Deep Inelastic Scattering’, OUP 2004, Section 6.6.6 and Chapter 9 for details!) BEWARE of different sort of ‘new physics’

48. Before the HERA measurements most of the predictions for low-x behaviour of the structure functions and the gluon PDF were wrong Now it seems that the conventional NLO DGLAP formalism works TOO WELL _ there should be ln(1/x) corrections and/or non-linear high density corrections for x < 5 10 -3

49. xg(x,Q 2 ) ~ x - λ g At small x, small z=x/y Gluon splitting functions become singular t = ln Q 2 /  2 α s ~ 1/ln Q 2 /  2 A flat gluon at low Q 2 becomes very steep AFTER Q 2 evolution AND F 2 becomes gluon dominated F 2 (x,Q 2 ) ~ x - λ s, λ s= λ g - ε Low-x

50. So it was a surprise to see F 2 steep at small x - for low Q 2, Q 2 ~ 1 GeV 2 Should perturbative QCD work? α s is becoming large - α s at Q 2 ~ 1 GeV 2 is ~ 0.4

51. 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 evolution dq/dlnQ 2 ~  s 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 DGLAP sums- LL(Q 2 ) and NLL(Q2) etc STRONGLY ordered in pt. But if ln(1/x) is large we should consider Leading Log 1/x (LL(1/x)) and Next to Leading Log (NLL(1/x)) - BFKL summations LL(1/x) is STRONGLY ordered in ln(1/x) and can be disordered in pt BFKL summation at LL(1/x)  xg(x) ~ x - λ λ = α s C A ln2 ~ 0.5 π  steep gluon even at moderate Q 2  Disordered gluon ladders But NLL(1/x) changes this somewhat (many experts here)

52. The steep behaviour of the gluon is deduced from the DGLAP QCD formalism – BUT the steep behaviour of the low-x Sea can be measured from F 2 ~ x - λ s, λ s = d ln F 2 Obviously the virtual-photon proton cross-section only obeys the Regge prediction for Q2 < 1. Does the steeper rise of б (γ*p) require a HARD POMERON? --The BFKL Pomeron with alpha=1.5 ? What about the Froissart bound? d ln 1/x Small x is high W 2, x=Q 2 /2p.q = Q 2 /W 2. At small x б(γ*p) = 4π 2 α F 2 /Q 2 F 2 ~ x –λs → б (γ*p) ~ (W 2 ) λs But  (  *p) ~ (W 2 ) α-1 – is the Regge prediction for high energy cross-sections where α is the intercept of the Regge trajectory α =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

53. Furthermore if the gluon density becomes large there maybe non-linear effects Gluon recombination g g  g  ~ α s 2  2 /Q 2 may compete with gluon evolution g  g g  ~ α s  where  is the gluon density ~ xg(x,Q2) –no.of gluons per ln(1/x) R2R2 nucleon size Non-linear evolution equations – GLR d 2 xg(x,Q2) = 3 α s xg(x,Q2) – α s 2 81 [xg(x,Q2)] 2 dlnQ2dln1/x π 16Q 2 R 2 αs αs  α s 2  2 /Q 2 The non-linear term slows down the evolution of xg(x,Q 2 ) 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, Q2 plane Plenty of debate about the positions of these lines! Colour Glass Condensate, JIMWLK, BK Strong coupling

54. 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! xS(x) ~ x –λs, xg(x) ~ x –λg λg < λs at low Q2, low x So far, we only used 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?. Non-linear effects? We need other gluon sensitive measurements at low x, like FL or F2charm…. `Valence-like’ gluon shape

55. But charm is not so simple to calculate Heavy quark treatments differ Massive quarks introduce another scale into the process, the approximation m q 2 ~0 cannot be used Zero Mass Variable Flavour Number Schemes (ZMVFNs) traditional c=0 until Q 2 ~4m c 2, then charm quark is generated by g→ c cbar splitting and treated as massless-- disadvantage incorrect to ignore m c near threshold Fixed Flavour Number Schemes (FFNs) If W 2 > 4m c 2 then c cbar can be produced by boson-gluon fusion and this can be properly calculated - disadvantage ln(Q 2 /m c 2 ) terms in the cross-section can become large- charm is never considered part of the proton however high the scale is. General Mass variable Flavour Schemes (GMVFNs) Combine correct threshold treatment with resummation of ln(Q 2 /m c 2 ) terms into the definition of a charm quark density at large Q 2 Arguments as to correct implementation but should look like FFN at low scale and like ZMVFN at high scale. Additional complications for W exchange s→c threshold.

56. We are learning more about heavy quark treatments than about the gluon, so far

57. And now we have actually measured FL! FL looks pretty conventional- can be described with usual NLO DGLAP formalism But see later (Thorne and White)

58. No smoking gun for something new at low-x…so let’s look more exclusively Let’s look at jet production: First let’s just see what jets can do for us in a regular NLO DGLAP fit There is a decrease in gluon PDF uncertainty from using jet data in PDF fits Direct* Measurement of the Gluon Distribution ZEUS-jets PDF fit

59. Nice measurement of α s (M Z ) = 0.1183 ± 0.0028 (exp) ± 0.0008 ( model) From simultaneous fit of α s (M Z ) & PDF parameters And correspondingly the contribution of the uncertainty on α s (M Z ) to the uncertainty on the PDFs is much reduced Before jetsAfter jets And use of jet data can help to tie down both alphas itself and alphas related uncertainties on the gluon PDF

61. Look at the hadron final states..lack of pt ordering has its consequences. Forward jets with x j » x and k tj 2 ~ Q 2 are suppressed for DGLAP evolution but not for k t disordered BFKL evolution But this has served to highlight the fact that the conventional calculations of jet production were not very well developed. There has been much progress on more sophisticated calculations e.g DISENT, NLOJET, rather than ad-hoc Monte-Carlo calculations (LEPTO-MEPS, ARIADNE CDM …) The data do not agree with DGLAP at LO or NLO, or with the Monte-Carlo LEPTO-MEPS..but agree with ARIADNE. ARIADNE is not k t ordered but it is not a rigorous BFKL calculation either……… No smoking gun for seomthing new at low-x…so let’s look more exclusively Now let’s look at forward jets

62. NLO below data, especially at small x Bj but theoretical uncertainty is large Forward Jets DISENT vs data Comparison to LO and NLO conventional calculations

63. Lepto doesn’t suffice Ariadne default overestimates high E t jet, overestimates high η jet (proton remnant) Ariadne tuned is good Forward Jets Comparison to ARIADNE and LEPTO

64. jet calculations which go up to O(α s 3 ) can describe the data- including detailed correlations between x and pt, or x and azimuthal angle SO we started looking at more complex jet production processes….

65. 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 Back to considering inclusive quantities New work in 2007 by White and Thorne

66. White and Thorne have an NLL BFKL calculation accounting for running coupling AND heavy quark effects – this has various attractive features…… Plus improved χ2 for global PDF fit Other groups working on NLL BFKL are: Ciafaloni, Colferai, Salam, Stasto Altarelli, Ball, Forte But there are no corresponding global fits

67. 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  g involve the summation of FAN diagrams – Q 2 = 1.4 GeV 2 Non linear DGLAP xg xu xs xu v xd xc There is some phenomenology which supports the use of non-linear evolution equations as well, but it is not so well developed Part of our problem is that as well move to low-x we are also moving to lower Q2 and the strongly coupled region..

68. 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 ) α-1 – Regge prediction for high energy cross-sections α is the intercept of the Regge trajectory α=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 α F 2 Q2Q2 →  ~ (W 2 ) α-1 → F 2 ~ x 1-α = x - so a SOFT POMERON would imply = 0.08 gives only a very gentle rise of F 2 at small x For Q 2 > 1 GeV 2 we have observed a much stronger rise….. p x 2 = W 2 q p

69. 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, α(Q 2 ) – 1 = (Q 2 ) A BFKL POMERON, α – 1 = = 0.5 A mixture of HARD and SOFT Pomerons to explain the transition Q 2 = 0 to high Q 2 ?

70. Recent work by Caldwell suggests that a double Pomeron model fits gamma* p best If this is the case in γ*-p, it could be so in p-p But we haven’t noticed yet because the hard Pomeron is not strongly coupled at low energies At the LHC we could notice this But what about the Froissart bound ? – the rise MUST be tamed eventually – non-linear effects/saturation may be necessary Predictions for the p-p cross-section at LHC energies are not so certain Do we understand the rise of hadron-hadron cross-sections at all? Could there always have been a hard Pomeron- is this why the effective Pomeron intercept is 1.08 rather than 1.00? Does the hard Pomeron mix in more strongly at higher energies? What about the at the LHC?

71. Dipole models provide another way to model the transition Q 2 =0 to high Q 2 At low x, γ* → qqbar and the LONG LIVED (qqbar) 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 σ =σ 0 (1 – exp( –r 2 /2R 0 (x) 2 )), R 0 (x) 2 ~(x/x 0 ) λ ~1/xg(x) r/R 0 small → large Q 2, x σ~ r 2 ~ 1/Q 2 r/R 0 large →small Q 2, x σ ~ σ 0 → saturation of the dipole cross-section GBW dipole model б(γ*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

72. τ 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 -λ ~ x g(x), gluon density- such that saturation extends to higher Q 2 as x decreases (Q s is low ~1-2 GeV 2 at HERA) 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 (LHC and neutrino), predictions for heavy ions- RHIC, diffractive interactions etc. σ = σ 0 (1 – exp(-1/τ)) Involves only τ=Q 2 R 0 2 (x), τ= Q 2 /Q 0 2 (x/x 0 ) λ And INDEED, for x<0.01, σ(γ*p) depends only on τ, not on x, Q 2 separately x < 0.01 x > 0.01 Q 2 < Q 2 s Q 2 > Q 2 s

73. The Pomeron also makes less indirect appearances in HERA data in diffractive events, which comprise ~10% of the total. The proton stays more or less intact, and a Pomeron, with fraction X P of the proton’s momentum, is hit by the exchanged boson. One can picture partons within the Pomeron, having fraction β of the Pomeron momentum One can define diffractive structure functions, which broadly factorize in to a Pomeron flux (function of x P, t) and a Pomeron structure function (function of β, Q 2 ). The Pomeron flux has been used to measure Pomeron Regge intercept – which come out marginally harder than that of the soft Pomeron The Pomeron structure functions indicate a large component of hard gluons in the Pomeron

74. ColourDipoleModel fits to ZEUS diffractive data But this is not the only view of difraction. These data have also been interpretted in terms of dipole models If the total cross-section is given by σ ~∫ d 2 r dz ׀(ψ(z,r)׀ 2 σ dipole (W) Then the diffractive cross-section can be written as σ ~ ∫ d 2 r dz ׀(ψ(z,r)׀ 2 σ dipole 2 (W) The fact that the ratio of σ diff /σ tot is observed to be constant implies a constant σ dipole which could indicate saturation

75. Ther is more evidence for hard Pomeron behaviour from diffractive Vector meson production and DVCS DVCS also seems to show a form of geometric scaling These processes are ~elastic γ*p → V p Hence if σ tot ~Im A elastic ~ W 2(α0)-1) Then σ elastic ~ ׀A elastic׀ 2 ~ W 4(α(0)-1) A soft Pomeron α(0)~1.08 would give σ elastic ~ W 0.32 A hard Pomeron α(0)~1.3 would give σ elastic ~ W 1.2 The experimental measurements vary from soft to hard depending on whether there is a hard scale in the process We can plot the effective Pomeron intercepts vs Q 2 +M V 2

76. Summary What have we learnt from DIS in the last 30 -35years? Verified the basic idea of QPM Established QCD as the theory of the strong interaction Measurement of essential parameters: Parton Distribution Functions, α s (M Z ) and the running of α s Low-x physics- QCD beyond DGLAP..BFKL..non-linear..CGC..dipole models..diffraction etc The LHC could discover a different kind of new physics