Low- x physics at HERA

The low-x Structure Function Data Introduction Brian Foster Bristol/DESY Corfu Summer School, 4.9.01 Low-x physics at HERA Other probes of QCD dynamics @ HERA Diffraction and its connection with low-x DIS Summary & Outlook Brian Foster - Corfu lectures

For low x, HERA ~ only game in town. Where are we now? Brian Foster - Corfu lectures

QCD as a description of low x Factorization - hard processes can be regarded as convolution of “sub-process” cross section with probability to find participating partons in target & probe - subsequent hadronisation ~ independent process For DIS can (normally) consider virtual photon as d-function => s ~ f  swhere s is sub-process cross section f is parton dist. function, satisfying f/ m ~ f P (m = renormalisation scale, Pis a splitting function) Brian Foster - Corfu lectures

where are AP splitting functions QCD evolution In general Ps are perturbative expansions to particular orders, keeping terms most important for particular regions: Leading lnQ2 terms come, in axial gauge, from evolution along parton chain strongly ordered in transverse momenta, LO DGLAP sums up terms - NLO sums terms which arise when two adjacent kts become comparable, losing factorlnQ2. Brian Foster - Corfu lectures

In small x region, leading terms in ln 1/x must be summed independent of Q2. This is done by the BFKL equation. LO terms arise from strong x ordering Generally, however, QCD coherence  angular ordering - work in unintegrated f(x,kt2,m2) - 2 hard scales  more complicated CCFM evolution equation. DGLAP/BFKL two limits of angular ordering. DGLAP, q  kt/kl, q grows since kt grows; in BFKL, q grows because kl  x falls. QCD evolution Brian Foster - Corfu lectures

Kinematics e(k) e'(k') 2 Q Q2 = xys g *(q) 2 W xP p(P) Low-x structure function data  s = k+P=energy in the ep c.m.s. Q2 = -(k-k')2 = -q2 =virtuality of the exchanged  x = Q2/(2P•q)=fraction of proton momentum carried by the struck quark y = (P•q)/(P•k) =fraction of beam lepton energy transferred to the photon W 2 = ys ~ Q2/xenergy in the *p c.m.s. Brian Foster - Corfu lectures

To reach lowest possibleQ2, some tricks needed! As well as exquisite understanding of detector - BPT F2(low Q2)from ZEUS Brian Foster - Corfu lectures

ZEUS BPT dataAt lowQ2 , F2falls likeQ2 F2(low Q2)from ZEUS & H1 Brian Foster - Corfu lectures

Since => sr =F sr ~F2 for small y, æ sr ~F2 - fory 1, so = F2fit - ç - F F F (x,Q2) ç L L L è (x,Q2) 2 FLfrom QCDfitfrom H1 Brian Foster - Corfu lectures

FLfrom QCDfitfrom H1 Brian Foster - Corfu lectures

III: Other probes of QCD dynamicsForward p0 @ H1 Brian Foster - Corfu lectures

Forward jets @ H1 Brian Foster - Corfu lectures

Forward Jets @ H1 Brian Foster - Corfu lectures

There are many parameterisations of the structure function data on the market - some more deeply based on physics others rather just convenient functional forms. e.g. DL fit ; wheree0 is “hard Pomeron” SinceW 2 ~ Q2/x , and , Regge theory, which governs high-energys s relevant for lowx IV. Interpretation & models Brian Foster - Corfu lectures

Another model exploits the “double-logarithmic” limit of QCD: Haidt, coming from a different direction, uses Ball-Forte fit Logarithms Brian Foster - Corfu lectures

Erdmann uses More Logarithms Brian Foster - Corfu lectures

Several on the market - MRST, CTEQ essentially global NLO QCD fits to all DIS data (HERA & fixed target) plus other relevant channels; GRV attempts to generate structure functions by evolution from “valence-like” gluon at very low Q . All give excellent fits to the data, with many free parameters. NLO QCD fits GRV’98 CTEQ Deviation of exp. data from CTEQ fit Brian Foster - Corfu lectures

Although using a subset (DIS) of the data, recently “home-grown” pdfs appeared which give errors on fitted pdfs as well as the correlations - e.g. Botje PDFs with errors Botje Brian Foster - Corfu lectures

NNLO estimates for the splitting fns. now becoming available - (Van Neerven&Vogt) MRS, CTEQ groups using them. Some strange effects! “Premature” (K.Ellis, DIS2000) NNLO PDF fitting Brian Foster - Corfu lectures

Thorne achieves interesting improvements by incorporating ln(1/x) terms in splitting fns after NNLO BFKL using running coupling BFKL eq. NNLO PDF fitting Brian Foster - Corfu lectures

ZEUS has very precise F2 data over 6 orders of magnitudein (x, Q2). What can it tell us? Look at the log. derivative since ~ LO gluon - most sensitive to low-xdynamics - fit x bins with form F2 = A+B(ln Q2)+C(ln Q2)2 Plot derivative as fn. of x& Q2in bins of constantW F2 & its derivatives Brian Foster - Corfu lectures

Errors on F2 syst. +stat. in quadrature (correlations ignored. F2 & its derivatives Brian Foster - Corfu lectures

There is no turn-over at constant Q2 One can look at 3-D surface of log. Slopes. The fundamental point is that the precision and kinematic range of the data is now opening up qualitatively new areas of study. The question is - what does it mean? F2 & its derivatives Brian Foster - Corfu lectures

As x falls, as we have seen, the gluon radiation drives a strong increase in parton density and hence increase inF2. At some point, the number of partons becomes so large that they cannot “fit” inside the proton and their wave-functions overlap - this is known as parton saturation. What is happening at low x? Brian Foster - Corfu lectures

Recently, great deal of interest in dipole models & saturation. L.T. Breit,  mom. prest In principle offers unification of inclusive DIS, diffraction + 1 g exchange 2 g octet exchange 2 g singlet exchange Diffraction Dipole Models Inclusive F2 Brian Foster - Corfu lectures

Example of this type of model: Golec-Biernat & Wuesthoff predicts -Q2s0 Dipole Models Brian Foster - Corfu lectures

The Golec-Biernat&Wusthoff model does a reasonable qualitative job - but so does QCD, and/or a variety of simple parameterisations. Fits to slopes Brian Foster - Corfu lectures

Although one can make QCD fit the logarithmic slopes, the resultant pdfs, as we saw earlier, are strange to say the least! ZEUS prel.. ZEUS prel.. QCD fit pdfs Brian Foster - Corfu lectures

But the predictedFLis even stranger! ZEUS prel.. QCD fit FL Brian Foster - Corfu lectures

The agreement of the data with dipole models and the saturation concept is intriguing - are we seeing the first departure from linear evolution in QCD? Clearly premature to draw this conclusion - NLO QCD can also reproduce the data to the same level - although at the cost of producing pdfs that are very difficult to interpret in a sensible way. The fundamental point is that the precision and kinematic range of the data is now opening up qualitatively new areas of study. Perhaps we are seeing a qualitatively new behaviour of QCD - but we can’t be certain. One of the problems is that the interesting “critical line” is down atQ2 ~ 1 GeV2 - we need to measure at low x but higher Q2. - needs a higher energy than HERA can achieve. F2 derivatives - summary Brian Foster - Corfu lectures

In diffraction, proton stays intact In great majority of DIS events, proton breaks up into hadrons + “remnant” in forward direction We saw that, in dipole models, there was an intimate connection between DIS & diffraction. Is this borne out by the data? Diffraction Brian Foster - Corfu lectures

The most basic measurement is the total cross section for diffraction. Does it agree with our expectations? } GB &W No. It has sameW2dependence asstot - W0.4. Contradicts optical theoremstot ~ Wa => sdiff ~ W2a; and ifstot ~ g, sdiff ~g2; and Regge, from Pomeron traj. stot~ W0.16 Diffraction Brian Foster - Corfu lectures

What about the structure functions? The analogue to F2 is F2D Diffraction Brian Foster - Corfu lectures

Note steep rise inW dependence ofs - indicative of hard processes becoming dominant. The intimate link between diffraction & non-diffractive DIS via dipole models & saturation also clearly applicable to vector meson production. Diffraction - vector mesons Brian Foster - Corfu lectures

Inset shows fit with Wd. For theJ/y, the charm mass seems to be large enough to provide a hard scale even atQ2 = 0. Diffraction - vector mesons Brian Foster - Corfu lectures

Fit to measured cross sections with Wd. For ther, theQ2provides a hard scale. Diffraction - vector mesons Brian Foster - Corfu lectures

Fit to measured cross sections with Wd. For all vector mesons , Q2 + M2seems to be a common hard scale. Diffraction - vector mesons Brian Foster - Corfu lectures

For Q2 > 5 GeV2, lVM ~ 2* lDIS Compare the W dependence for VM production and DIS. Diffraction - vector mesons Brian Foster - Corfu lectures

But the J/y wave function needs to be modelled so that model dependence enters extraction. What else can we learn from VMs? Since J/y seems to always be in the pQCD realm, we can in principle learn about proton gluon distribution. Diffraction - vector mesons Brian Foster - Corfu lectures

By fitting cross-section t dependence we can look at Pomeron trajectory. Diffraction - vector mesons Brian Foster - Corfu lectures

What is the appropriate hard scale in VM production? Diffraction - vector mesons Brian Foster - Corfu lectures

VM production at high t. Diffraction - vector mesons Brian Foster - Corfu lectures

J/y f/r f Production s ratios BFKL prediction (Forshaw et al.) y/r r 2g BFKL Diffraction - vector mesons Brian Foster - Corfu lectures

Simplest final state in diffraction Deeply virtual Compton scattering Measures Re part of a QCD amplitude Measures “skewed” parton distributions - generalisation on normal proton pdf’s. Brian Foster - Corfu lectures

Data cf. QEDC only QEDC & DVCS MC Deeply virtual Compton scattering DVCS process clearly necessary - extract cross-section Brian Foster - Corfu lectures

Now cross section measured, can go onto to look at interference etc. Deeply virtual Compton scattering Brian Foster - Corfu lectures

The agreement of the data with dipole models and the saturation concept is intriguing - are we seeing the first departure from linear evolution in QCD? Clearly premature to draw this conclusion - one of the problems is that the interesting “critical line” is down at Q2~ 1 GeV2 - we want to measure at low x but higher Q2. This I guess will have to wait for THERA, LHC ep option, ….? Is there something else we can do “now”? Yes, possibly. Running HERA with nuclei rather than p gives access to high-density of partons at low x. V Summary & Outlook Brian Foster - Corfu lectures

Many open questions - and of course this is not only important to those interested in QCD! If we want to useW,Z production at LHC as lumi. monitor, we had better understand smallxat HERA! h Summary & Outlook Brian Foster - Corfu lectures

The quality & precision of the HERA data are driving studies of low-x physics. Watching the Herculean labours of the F2 experts extracting the 96-97 result tells me that we are nearing the end of the road for improved precision in the standard inclusive F2 at low x - from now one attention will turn to semi-inclusive(particularly F2charm) and rare processes Much theoretical help required (as always) to tell us where/how to look The connection between diffraction and DIS is certainly a very interesting one that can throw much light on low-x physics. It may well justify a “HERA-III” programme - but all this will depend on TESLA! Summary & Outlook Brian Foster - Corfu lectures

Low- x physics at HERA