Low-x physics at HERA A M Cooper-Sarkar Copenhagen Jan 2009

1. Low-x physics at HERAA M Cooper-SarkarCopenhagen Jan 2009 BEWARE of different sort of ‘new physics’ What if low-x behaves very differently? LHC is a low-x machine (at least for the early years of running) Is NLO (or even NNLO) DGLAP good enough for x < 10 -2. The QCD formalism may need extending at small-x. What is SAFE x? BFKL ln(1/x) resummation or non-linear evolution equations would change the deduced shape of the gluon

2. The Standard Model is not as well known as you might think fa pA x1 • where X=W, Z, D-Y, H, high-ET 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 ^ pB x2 fb X In the QCD sector the PDFs limit our knowledge - transport PDFs to hadron-hadron cross-sections using QCD factorization theorem for short-distance inclusive processes The central rapidity range for W/Z production AT LHC is at low-x (6 ×10-4 to 6 ×10-2) at 14 TeV (1.2 ×10-3 to 1.2 ×10-1) at 7 TeV

3. WHAT DO WE KNOW WELL? 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 ~3-4% W Z cross-sections at 10 TeV Agreement between PDFs has improved in recent years –only consider those which include massive heavy quark treatment. Can be used as a luminosity monitor?

4. Impact of HERA data on the LHC- just one example W and Z cross-sections, rapidity distributions are now thought to be very well known we owe this to HERA No HERA data Separate H1 +ZEUS HERA Combined 2009 w Z WHY? -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 high scale, Q2~MZ2,the sea is driven by the gluonby g→q qbar splitting

5. What does HERA measure? Leptonic tensor - calculable 2 Lμν Wμν dσ~ Hadronic tensor- constrained by Lorentz invariance E q = k – k’, Q2 = -q2 momentum transfer s= (p + k)2 c.of m. energy 2 Px = p + q , W2 = (p + q)2 x = Q2 / (2p.q) y = (p.q)/(p.k) inelasticity W2 = Q2 (1/x – 1) hadronic c.of m energy2 Q2 = s x y scale of the process Ee Ep s = 4 Ee Ep Q2 = 4 Ee E’ sin2θe/2 y = (1 – E’/Ee cos2θe/2) x = Q2/sy The kinematic variables are measurable

6. Completely generally the double differential cross-section for e-N scattering d2(e±N) = [ Y+ F2(x,Q2) - y2 FL(x,Q2) ± Y_xF3(x,Q2)], Y± = 1 ± (1-y)2 dxdy Leptonic part hadronic part F2, FL and xF3 are measurable structure functionswhich express the dependence of the cross-section on the structure of the nucleon– F2(x,Q2) = Σi ei2(xq(x,Q2) + xq(x,Q2)) in LO QCD The Quark-Parton model interprets these structure functions as related to the momentum distributions of quarks or partons within the nucleon – the PDFs- and the measurable kinematic variablex = Q2/(2p.q)is interpreted as the FRACTIONAL momentum of the incoming nucleon taken by the struck quark QCD predicts the rate at which the parton distributions (both quarks and gluons) evolve with Q2- (the energy scale of the probe) -BUT it does not predict their shape … THIS is why we need the DIS measurements At NLO the relationship of F2 to the PDFs is more complex but still calculabe

7. So what did HERA bring? Firstly an enormous extension of the kinematic range: both to high Q2 and to low-x. • Extension of the kinematic range to low-x brought very interesting effects related to Scaling Violation and the gluon. The DGLAP equations tell us how the quarks and the gluon evolve and at low-x the system is becoming gluon driven… BUT the steep rise at low-x was not expected…..

8. Recent development: Combining ZEUS and H1 data sets • Not just statistical improvement. Each experiment can be used to calibrate the othersince 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 HERA data set can be used as sole input to PDF fits with Δχ2=1. HERAPDF1.0 • So far for HERA-I data. HERA-II data will further reduce high-x, Q2 uncertainty Increase in PDF precision from combination

9. Comparison of HERAPDF1.0 to MSTW08 evaluated at 68% CL Major differences where the new data is much more precise at small-x

10. Low-x xSea directly from F2, F2 ~ xq xGluon from scaling violations dF2 /dlnQ2 – the relationship to the gluon is much more direct at small-x, dF2/dlnQ2 ~ Pqgxg Before the HERA measurements most of the predictions for low-x behaviour of the structure functions and the gluon PDF were wrong Most people didn’t expect the sharp rise at low-x which was seen in F2, and deduced in the gluon, But this IS what pQCD in the DGLAP formalism predicts!

11. Low-x t = ln Q2/2 Gluon splitting functions become singular αs ~ 1/ln Q2/2 At small x, small z=x/y A flat gluon at low Q2 becomes very steep AFTER Q2 evolutionAND F2 becomes gluon dominated F2(x,Q2) ~ x -λs, λs=λg -ε xg(x,Q2) ~ x -λg

12. Nevertheless the first results were much steeper than had been anticipated • And it was even more of a surprise to see the second results: F2 steep at small x - for very low Q2, Q2 ~ 1 GeV2 • Should perturbative QCD work? αs is becoming large - αs at Q2 ~ 1 GeV2 is ~ 0.4 • There hasn’t been enough lever arm in Q2 for evolution, but even the starting distribution is steep- the HUGE rise at low-x makes us think • there should beln(1/x) resummation(BFKL) as well as the traditional ln(Q2) DGLAP resummation- BFKL predicted F2(x,Q2) ~ x –λs, with λs=0.5, even at low Q2 • and/or there should be non-linear high density corrections for x < 5 10 -3

13. Need to extend formalism at small x? • The splitting functions Pn(x), n= 0,1,2……for LO, NLO, NNLO etc • Have contributions Pn(x) = 1/x [ an ln n (1/x) + bn ln n-1 (1/x) …. • These splitting functions are used in evolution dq/dlnQ2 ~ s dy/y P(z) q(y,Q2) • And thus give rise to contributions to the PDF sp (Q2) (ln Q2)q (ln 1/x) r • DGLAP sums- LL(Q2) 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 = αs CA ln2 ~ 0.5 • steep gluon even at moderate Q2  Disordered gluon ladders But NLL(1/x) softens this somewhat π

14. 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 And possible gluon saturation ? Colour Glass Condensate, JIMWLK, BK TW, ABF, CCSS Non-linear fan diagrams form part of possible higher twist contributions at low x But is there a ‘smoking gun’ for such new physics at low-x?

15. When you look at the sea and the gluon deduced from the DGLAP formalism at low Q2 there are odd features the gluon is no longer steep at small x – in fact its valence-like or even negative! The problem is that we are deducing this from limited information So far at low-x, we only use F2 ~ xq for the sea dF2/dlnQ2 ~ Pqgxg for the gluon Unusual behaviour of dF2/dlnQ2 may come from unusual gluon or from unusual Pqg- alternative BFKL evolution?. Non-linear effects? We need alternative ways to probe the gluon CTEQ6.5 MSTW08

16. We need other gluon sensitive measurementslike FL In NLO DGLAP FL is given by And at low-x this becomes gluon dominated LO approx, AMCS et al 1987!! Even last year I’d have said: FL looks pretty conventional- can be described with usual NLO DGLAP formalism BUT there are new lower Q2 FL measurements from HERA …. And predictions differ more at low Q2

17. Compared to various NLO DGLAP fits And compared to alternative theoretical predictions: White and Thorne (WT) which has NLL ln1/x resummation included Dipole Models which can accommodate non-linear effects/ saturation eg IIM colour glass condensate So do we have a smoking gun? Maybe not- but the circumstantial evidence is building up….

18. To search for something new at low-x let’s look at forward jets Look at the hadron final states..lack of pt ordering has its consequences.Forward jets with xj» x and ktj2 ~ Q2 are suppressed for DGLAP evolution but not for kt 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 …)

19. What about Forward Jets? SO we started looking at more complex jet production processes…. jet calculations which go up to O(αs3 ) can describe the 1f+2c data- but not 2f+1c – Note these tri-jet calculations have αS ln(1/x) terms- but its not quite enough for 2f+1c..more powers of ln(1/x) should be summed? Comparison to conventional LO and NLO calculations- NLO below data, especially at small xBj

20. MRST02 MRST03 What consequence would new low-x physics have at the LHC ? BFKL ln(1/x) resummation would change the deduced shape of the gluon –Thorne and White -And has the added benefit ofimproved χ2 for global PDF fit For other work on NLL BFKL see also Ciafaloni, Colferai, Salam, Stasto Altarelli, Ball, Forte MRST03 PDFs were a TOY PDF which distrusted all x < 10-3. This would affect the central region for W production. Thorne and White Central rapidity High density non-linear effects may induce gluon saturation this also affects the deduced shape of the gluon Far forward Drell-Yan M(ee) = 4GeV Eskola et al

21. But the TOY PDFs are unlikely to be realistic - a better way could be to look at pt spectra for W and Z production Lack of pt ordering at low-x is a further consequence BFKL resummation AND most non-linear treatments. This would affect the pt spectra for W and Z productionat the LHC (See hep-ph/0508215) Pt spectra are also used to measure MW -- dMW from PDF uncertainties, using pt(e), is ~20 MeV So we’d better be sure we’ve got the calculations for Pt spectra right Conventional Unconventional < pT(W)> Same pattern dMW(fit)

22. Another aspect of low-x Small x is high W2, x=Q2/2p.q Q2/W2 σ(γ*p) ~ (W2) α-1 – Regge prediction for high energy cross-sections whereαis 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 σ(γp) ~ (W2) 0.08 for real photon- proton scattering For virtual photons, at small x σ(γ*p) = 4π2α F2 q px2 = W2 p Q2 →σ~ (W2)α-1→ F2 ~ x 1-α = x -λ so we can re-interpret the rise in low-x F2 data as a rise of σ(γ*p) with W2 which is much stronger than predicted by a SOFT POMERON (λ= 0.08)

23. So is there a different Pomeron - a HARD POMERON - corresponding to this steep rise? A QCD POMERON, α(Q2) – 1 = λ(Q2) OR a BFKL POMERON, α – 1 = λ= 0.5 OR a mixture of HARD and SOFT Pomerons to explain the transition from Q2 = 0 to high Q2? Soft pomeron There can be very significant consequences for high energy cross-sections (LHC and neutrino), predictions for heavy ions- RHIC, diffractive interactions etc.

24. 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? 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

25. Summary • There is evidence that all is not well with conventional QCD calculations at low-x • Trouble sets in for x < 10-2 • Which means that SM calculations may not look as we expect even in the central rapidity region at the LHC • These effects can be explicitly studied in the forward detectors/ in heavy ion collisions/ in diffractive processes

26. Dipole models provide another way to model the transition Q2=0 to high Q2 At low x, γ* → qqbar and the LONG LIVED (qqbar) dipole scatters from the proton б(γ*p) Now there is HERA data right across the transition region The dipole-proton cross section depends on the relative size of the dipole r~1/Q to the separation of gluons in the target R0 σ =σ0(1 – exp( –r2/2R0(x)2)), R0(x)2 ~(x/x0)λ~1/xg(x) Buts(g*p) = 4pa2 F2 is general Q2 (at small x) r/R0 large →small Q2, x σ ~ σ0→ saturation of the dipole cross-section r/R0 small → large Q2, x σ~ r2~ 1/Q2 s(gp) is finite for real photons , Q2=0. At high Q2, F2 ~flat (weak lnQ2 breaking) and s(g*p) ~ 1/Q2 GBW dipole model

27. x < 0.01 σ= σ0 (1 – exp(-1/τ)) Involves only τ=Q2R02(x), τ= Q2/Q02 (x/x0)λ And INDEED, for x<0.01, σ(γ*p) depends only on τ, not on x, Q2 separately Q2 > Q2s Q2 < Q2s x > 0.01 τ is a new scaling variable, applicable at small x It can be used to define a `saturation scale’ , Q2s = 1/R02(x) . x -λ ~ x g(x), gluon density- such that saturation extends to higher Q2 as x decreases (Qs is low ~1-2 GeV2 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.

28. Further clues from diffractive data ? Diffractive Vector meson production γ*p → V p is almost elastic Hence if σtot ~Im Aelastic ~ W 2(α0)-1) Then σelastic ~ ׀Aelastic׀2 ~ W 4(α(0)-1) A soft Pomeron would give σelastic ~ W 0.32 A hard Pomeron would give σelastic ~ W1.2 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 Q2+MV2 If the total cross-section is given by σ ~∫ d2r dz ׀(ψ(z,r)׀2σdipole(W) Then the diffractive cross-section can be written as σ ~ ∫ d2r dz ׀(ψ(z,r)׀2σdipole2(W) The fact that the ratio of σdiff/σtotis observed to be constant implies a constant σdipole which could indicate saturation DVCS also seems to show geometric scaling

29. So how do we do the FL measurement at HERA? We measure the NC e+p differential cross-section at different beam energies Schematically y=Q2/sx And here are H1’s low Q2 results

30. Can we improve our knowledge of PDFs using LHC data itself? We actually measure the decay lepton spectra Generate pseudodata at 14TeV corresponding to 100pb-1- using CTEQ6.1MZEUS_S MRST2001 PDFs with full uncertainties At y=0 the total uncertainty is ~ ±6% from ZEUS ~ ±4% from MRST01E ~ ±8% from CTEQ6.1 To improve the situation we NEED to be more accurate than this:~4% Statistics are no problem there will be millions of W’s We need to control the systematic uncertainty generator level electron positron ATLFAST electron positron Hep-ex: 0509002

31. Can we improve the situation with early LHC data? Generate W+/W- data with 4% error using CTEQ6.1 PDF, pass through ATLFAST detector simulation and then include this pseudo-data in the global ZEUS PDF fit (actually use the decay lepton spectra)Central value of prediction shifts and uncertainty is reduced BEFORE including W data AFTER including W data e+ rapidity spectrum and gluon PDF BEFORE these data are included in the PDF fit e+ rapidity spectrum and gluon PDF AFTER these pseudodata are included in the PDF fit Gluon PDF uncertainties are reduced

32. Now let’s look at ratios: Z/W ratio is a golden benchmark measurement (10TeV) MSTW08 CTEQ6.6 CTEQ6.5 pre 2008 ZOOM in on Z/W ratio – there is fantastic agreement between PDF providers PDF uncertainty from the low-x gluon and flavour symmetric sea cancels out- and so do luminosity errors BUT there is somewhat more PDF uncertainty than we thought before 2008 (~1.5% rather than <1% in the central region) There is uncertainty in the strangeness sector that does not cancel out between Z and (W+ + W-)… it was always there we just didn’t account for it Z = uubar + ddbar + ssbar +ccbar +bbar W+ + W- ~ (udbar + csbar) + (dubar+scbar) YES this does translate to the Z/lepton ratio

33. But in the W asymmetry – there is NOT fantastic agreement (10 TeV) The biggest difference is MSTW08 to CTEQ66 The difference doesn’t look much until you home in on the discrepancy of the MSTW08 prediction to the CTEQ66 error band It exceeds my standard 10% scale Turns out to be~35% discrepancy! Where is it coming from?- compare W+ and W- distributions It looks like the W-agrees fairly well but the W+ is lower for MSTW We can trace this difference back to the different valence PDFs of CTEQ66 and MSTW08

34. Predictions for AW are different in the central region- because predictions for valence distributions at small-x are different Dominantly, at LO Aw= (u(x1) dbar(x2) – d(x1) ubar(x2)) (u(x1) dbar(x2) + d(x1) ubar(x2)) And at central rapidity x1= x2 and ubar ~ dbar ~ qbar at small x So Aw~ (u – d) = (uv – dv) (u + d) (uv + dv + 2 qbar ) Actually this LO approx. is pretty good even quantitatively The difference in valence PDFs you see here does explain the difference in AW between MRST and CTEQ As we move away from central rapidity: as x1 increases (decreases) the larger (smaller) difference is weighted by larger (smaller) sea distributions at smaller x2 x- range affecting W asymmetry in the measurable rapidity range at ATLAS (10TeV)