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Emmanuelle Perez (CERN-PH)

School on QCD, Low x physics, Saturation and Diffraction Calabria, Italy, July 1-14, 2007. Determination of parton distribution functions, impact of HERA data & consequences for LHC. Emmanuelle Perez (CERN-PH). Plot shown by A. Martin yesterday. In these lectures we see how we

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Emmanuelle Perez (CERN-PH)

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  1. School on QCD, Low x physics, Saturation and Diffraction Calabria, Italy, July 1-14, 2007 Determination of parton distribution functions, impact of HERA data & consequences for LHC Emmanuelle Perez (CERN-PH) Plot shown by A. Martin yesterday. In these lectures we see how we get there : - experimental data - QCD fit techniques Emmanuelle.Perez@cern.ch

  2. Lecture 1 : Deep-inelastic scattering and HERA • Deep-inelastic scattering, formalism & basics • The HERA collider and experiments • DIS measurements at HERA Lecture 2 : QCD fits and low x physics • QCD fits : generalities • fits using HERA data alone • “global” fits • - Back to the rise of F2 at low x Lecture 3 : Looking at the future… • Determination of uncertainties related to pdfs • Pdf uncertainties for LHC processes • A better knowledge of pdfs from LHC experiments ? • and beyond LHC… Low x School, July 07

  3. ^ •  is the cross-section of the partonic subprocess, calculable in perturbative • QCD at a given order in S. • Beyond leading order, both the pdfs and the hard cross-section depend on • - the factorisation scale F (separates long & short distance parts of the • scattering process) • - the renormalisation scheme (usually MS , sometimes DIS ) and the • renormalisation scale R Parton distribution functions • Parton distribution functions are the non-perturbative inputs necessary to • calculate a process in a collision involving hadron(s). F2) F2) F2, • Ga/A (xa, F2) represents the probability to find a parton a in hadron A, • carrying a fraction xa of the hadron longitudinal momentum. • They cannot be calculated from first principles. Have to be measured. • Universality of pdfs : the above formula holds for any process. • i.e. measure pdfs from e.g. deep-inelastic scattering, apply them to predict • cross-sections at the LHC. Low x School, July 07

  4. nucleus nucleon quark NC+CC+gluon Low x, diff Probing the nucleon… Scattering of a punctual (e.g. lepton) probe on a target : used for long to underpin the target contents. GE, GM(Q2)  W1, W2() Fi(x,Q2) Form factors, Bjorken scaling, structure functions. Deep inelastic scattering of a lepton off a nucleon (nuclei) is the golden process to study the parton distribution functions. Other processes also bring important constraints, as will be seen later. Low x School, July 07

  5. W2 = squared mass of the hadronic system = ( P + q )2 e PT * e PT2 = ( 1 – y ) Q2 q q Deep-inelastic scattering q = k – k’, Q2 = -q2squared momentum transfer S = (p + k)2 square of center of mass energy Q2 = x y S • V can be : • a  or Z : Neutral Current DIS • a W : Charged Current DIS “inelasticity variable” In nucleon rest-frame, y = (E – E’ ) / E Cross-section depends on 2 variables, generally choose (x, Q2). Partonic interpretation: xBjorken is the fraction of the nucleon longitudinal momentum taken by the “struck” quark, in the frame of infinite momentum for the nucleon (light cone variables : p+(quark) = x P+(N) ) Low x School, July 07

  6.    A  2x P + q = 0 P k ’ eout pin q X ein k +z PX q = (0, 0, 0, Q) P = (Q, 0, 0, -Q) / (2x) (P.k) = (1/2x) (EeQ + Q2/2) = Sep/2 = Q2 / (2xy) Ee = Q(2-y) / (2y)  “Breit” frame : P = (Ep, 0, 0, - Ep) q = (E, 0, 0, qZ) Ep = qZ / 2x A02 = (2xP + q)2 = - Q2 + 4x(P.q) = Q2 (x = Q2 / 2(P.q) ) Hence Q2 = ( 2x Ep + E )2 = (qZ + E )2 Q2 = -q2 = qZ2 - E2 = (qZ - E) (qZ + E) Lepton side : E(ein) = E(eout) since E = 0 same pT  pZ(ein) = - pZ(eout) pZ(ein) = pZ(eout) + Q  pZ(ein) = Q/2 Low x School, July 07

  7. - + 2 2 2 a a ( 1 y ) ( ) 1 æ ö æ ö 1 • For transverse bosons : + = + a = 2 1 cos 2 2 ç ÷ ç ÷ cos sin - 2 ( 2 y ) è ø è ø 2 2 4 a a 2 æ ö 2 = - Contribution of L vanishes at y = 1 2 cos sin ( 1 y ) ç ÷ x ( ) - 2 è ø 2 2 2 y • For longitudinal bosons : Example : amplitudes for right-handed leptons : eout  q  ein z Projection of ein spin on z axis is ½ with d1/21/2, 1/2= cos(/2) and -½ with sin(/2). This gives the following terms in the differential cross-section : • Parity violating term : (R + L) Jz=+1 – (R + L) Jz=-1 goes like : Low x School, July 07

  8. L : 2 (1-y) T : 1 + (1-y)2 PV : y (2-y) Putting everything together : Contain the dependence on the nucleon structure The sum of the first two terms can be rewritten as : [ (1-y)2 + 1 ] [ FT + FL ] - y2FL FT + FL  F2 0  FL  F2 With Y = 1  (1-y)2 the Born-level cross-sections read : Effect of FL large only at large y. Low x School, July 07

  9. _ _ F2 =  e2qi[ qi(x) + qi(x) ] d2 / dydx =  (d / dy )qi [ qi(x) + qi(x) ] un = dp  d dn = up  u _ xF3 : given by the difference (e+ N) - (e- N) : xF3 ~  ci [ qi(x) - qi(x) ] valence e+ e- - from Z interference : 5 coupling ~ ae aq - frompure Z exchange : aeveaqvq - _ q q Partonic interpretation of Neutral Current DIS (I) Cross-section for eq scattering : ( d / dy )q = 22/Q4 e2q [ 1 + (1-y)2 ] qi(x) = probability to find a quark qi with momentum fraction x _ _ In lepton-proton : ~ 4 (u + u) + (d + d) In lepton-neutron : ~ 4 (d + d) + (u + u ) _ _ _ _ In lepton-deuterium : ~ (u + d + u + d ) lp & ld allows to “separate” the flavors. QCD improved parton model, qi(x)  qi(x,Q2) : these two eq. hold at Leading Order. At NLO : F2 and xF3 are given by convolutions of pdfs with coeff. functions. Low x School, July 07

  10. - At NLO, FL is given by a convolution involving the “quark singlet”  (q + q) (contribution from q  qg) and the gluon density (g  qq). - FL ~ gluon density (away from the “valence” region). - FL sensitive to kT of quarks produced in g  qq.  _ qin    0  JZ qout  Partonic interpretation of Neutral Current DIS (II) Quarks have spin 1/2 Helicity conservation FL : back to the Breit frame : JZ,in = JZ,out   must be transverse FL = 0 at Leading Order Non-zero FL at Next-to-Leading Order : when qg in the final state, or when the incoming quark has a non-zero transverse momentum. Happens when the interacting quark comes from a gluon splitting g  qq. Hence : Low x School, July 07

  11. _ _ e+R R R e+R W _ _ uL dL dR uR e+R    e+R qR e+R   qR   _ e+R qL Hence the e+p cross-section goes as (1 –y)2 xD + xU _  qL the e-p cross-section goes as (1 –y)2 xD + xU Comparing with eq. on slide 9 gives : with U = u + c, D = d + s ( + b) _ W2+ = x ( U + D ) W2- = x ( U + D ) _ _ xW3+ = x ( D – U ) xW3- = x ( U – D ) _ Partonic interpretation of Charged Current DIS (charged incident lepton, lp) J = 1 W ( d11,1 )2 ~ ( 1 + cos *)2 ~ ( 1 – y)2 W couples to left-handed fermions and right-handed antifermions. J = 0 ( d00,0 )2 ~ 1 CC(e-p) >> CC(e+p) CC DIS brings important information to separate up / down quarks. Low x School, July 07

  12. _ R +R W _ _ dL, uR uL, dR _ _ _ _ _ _ L -L n : (1-y)2 D + U p : (1-y)2 U + D p : (1-y)2 U + D n : (1-y)2 D + U W _ _ un = dp  d dL, uR uL, dR dn = up  u _ _ W2 = x ( U + D + U + D ) _ _ xW3 = x ( U - U + D - D ) Neutrino Deep-inelastic scattering On proton : On neutron : Weak cross-sections experiments use a heavy target, e.g. Pb (CCFR), Fe (NuTeV), which are nearly isoscalar (i.e. #n ~ #p) _ (  +  ) Hence : _ (  -  ) The data need to be corrected : - for nuclear effects ( q in heavy nuclei differs from q in proton) - to isoscalar target (e.g. 5.7% excess of n over p in Fe) Corrections are not trivial... Low x School, July 07

  13. Via  N -  N _ DIS Fixed target results F2: 1< Q2 < 200 GeV2 xF3: 1< Q2 < 200 GeV2 scaling violations Q2 > 5 GeV2 : data down to 10-2 only approximate Bjorken scaling FL Measurements only at high x Low x School, July 07

  14. Large spread in the theoretical predictions. Pre-HERA status…  What happens towards low x ??  What happens at high Q2 ?? Low x School, July 07

  15. DESY, Hamburg H1 ZEUS The HERA electron-proton collider Equivalent to fixed target exp. with 50 TeV e± Two colliding experiments : H1 and ZEUS HERA I (1992-2000) : ~ 130 pb-1 - mainly e+ p data (only 20 pb-1 of e- p) - several analyses getting finalised on HERA I data HERA II (2003 – 30/06/07) : - ~ 500 pb-1 of data, ~ equally shared between e+ and e- - polarised leptons, P typically 30 -40 % Low x School, July 07

  16. The H1 detector Asymmetric detector : reflects the beam energies asymmetry. p Complete 4π detector Tracking: - central jet chamber - z drift chambers - forward track. detector - Silicon μ-Vtx (operate in a B field of 1.2 T) Calorimeters: - Liquid Argon cal. (em : 10%/E had: 50%/E) - Lead-Fiber cal. (SPACAL) (7% / E) Muon chambers Very forward detectors (e.g. “roman pots”) for diffractive physics. e Low x School, July 07

  17. The ZEUS detector Complete 4π detector Tracking: - central tracking detector - Silicon μ-Vtx (operate in a B field of 1.43 T) Calorimeters: - uranium-scintillator (CAL) σ(E)/E=0.18/√E [emc] σ(E)/E=0.35/√E [had] - instrumented-iron (BAC) Muon chambers Low x School, July 07

  18. HERA kinematic domain Q2 from 0.1 to 105 GeV2 x from 10-6 to ~ 0.8 Up to very high Q2 ~ 105 GeV2 Q2 = xyS with S = (320)2 ~ 1.2 105 GeV2 Huge extension of the kin. domain compared to fixed target expts. For Q2 above ~ 1 GeV2 (perturbative regime), x down to ~ 10-5 Very low Q2 accessible, allow to study the pert – non-pert transition region. Down to very low x, ~ 10-6 ! Low x School, July 07

  19. - From electron only : Q2e = 2 E0e E’e ( 1+cos e) ye = 1 – (E’e/E0e) sin2(e/2) Drawbacks : - resolution in y goes as 1 / ye i.e. bad at low ye - in case of initial state radiation, E0e Ebeam Kinematic Reconstruction • Charged Current DIS : measure only the hadronic final state. Bad resolution at high yh (1) • Neutral Current DIS : HERA experiments measure both the scattered lepton • and the hadronic final state  kinematics is over-constrained. • Combine e and had : replace 2E0e in (1) by 2E0e = (E-pZ)h + (E-pZ)eand use • p2T,e = Q2(1-y) to get Q2 • Kinematics can also be • reconstructed only from • the measured angles of e • and of the had. final state • (HFS). B. Heinemann, PhD Thesis Used for the calibration of the el. energy. Low x School, July 07

  20. Kinematics (I) “kinematic peak” : Ee = Ee0, x = Ee0 / Ep H1 data, Q2 < 150 GeV2 electron Hadronic final state Q2 below ~ 150 GeV2 : e “backward”, Ee limited Higher Ee when e is “central”. Highest Q2 for  < 90o. High x & y NC High x & y CC HFS goes more central as y increases. e p  backward Low x School, July 07

  21. Kinematics (II) – focus on low x and low Q2 Angular acceptance limits the measurements down to Q2 ~ 1.5 – 2 GeV2. Going lower in Q2 requires special techniques or dedicated apparatus. • ~ 180o : y = 1 – E’ / E i.e. high y  low E’ Going towards highest y (for max. sensitivity to FL ) requires : - dedicated triggers, with an energy threshold down to 2-3 GeV - a good understanding of experimental backgrounds (easy to “fake” a low E el.) Low x School, July 07

  22. The first HERA results (1993) Q2 = 15 GeV2 Strong rise of F2 towards low x ! HERA DIS measurements Low x School, July 07

  23. e.g. de Rujula et al., PRD 10 (1974) 1649 Why this was somehow a surprise Extrapolation from pre-HERA data indicated a “flattish” F2 at low x – that’s also what came out from Regge-like arguments. However, it was known (QCD) that the gluon should rise at low x, for Q2 high enough. What was not known is “where” (in x and Q2) the rise should start. dg(x,Q2) / dlnQ2 ~ x dy/y Pgg(x/y) g(x/y) with Pgg(z) ~ 1/z DGLAP equation : Approximate solution : g ~ exp ( -K(Q2) ln(1/x) ) with K(Q2) ~ (ln Q2) Starting from a ~ flat gluon at the “starting” scale, a rising gluon is obtained at higher scales. The “starting” scale, i.e. the scale down to which pQCD was taken to hold, was believed to be ~ a few GeV2. The evolution between Q02 and Q2 ~ 15-20 GeV2 is not long enough to generate a rising gluon from a flattish distribution. Such a rise could be obtained : - from a steep input gluon which could be expected due to large ln(1/x) terms (resummed in the BFKL evolution equation) - from DGLAP and a flattish starting gluon, but at a much lower Q02. Low x School, July 07

  24. (not the final word …) 96-97 data ~ 20 pb-1 stat. accuracy < 1 % systematics ~ 3 % Compared with where we are now… No “gap” between HERA & fixed target data. Low x School, July 07

  25. H1 Collab., EPJ C21 (2001) 33 What is measured is actually not F2, but rather F2 – y2 FL / Y+ (cf slide 9) FL The reduction of the cross-section as y  1 (lowest x), due to the disappearance of the contribution of longitudinal photons, is observed indeed. (more on FL later…) Also note that there is no “gap” between HERA & fixed target data. Low x School, July 07

  26. With increasing luminosity, important statistics over the full kinematics domain. • Good agreement between H1 and ZEUS • and with fixed target measurements • Strong scaling violations observed at • low x – sign of a large gluon density • ( g  qq ) • Negative scaling violations at high x • ( q  qg, a high x quark splits into a • gluon and a lower x quark) Overlaid curves are the results of QCD fits based on the DGLAP equations (see later). Within DGLAP : via F2/lnQ2, access to the gluon density. Excellent agreement with DGLAP, over 5 orders in magnitude in Q2 and 4 orders of magnitude in x. Low x School, July 07

  27. Very precise reconstruction of event’s characteristics and kinematic variables: Phase-space region: 6x10-5 < x < 0.65 2.7 < Q2 < 30000 GeV2 How did we get there… (example of ZEUS analysis) Count nr. of events in appropriately defined (x,Q2) bins Extract reduced cross section Uncertainties are systematics dominated for Q2 < 800 GeV2 Slide from E. Tassi, CTEQ 2003 Low x School, July 07

  28. More on high Q2 NC measurements Recall that due to Z exchange, NC(e-p)  NC(e+p). The difference gives access to xF3. High Q2 measurements done both in e-p and in e+p : - Z exchange contributes for Q2 above a few 103 GeV2 - Z interference is constructive in e-p, destructive in e+p Statistics is limited ! ~ 2uv + dv Neglect pure Z contribution (small) and correct for propagator terms  bring all data to Q2 = 1500 GeV2 Test the x dependence of valence quarks. Low x School, July 07

  29. (subset ot meas.) Pushing towards highest x… ZEUS Collab., EPJ C49 (2007) 523. Quark pdfs are not well known at highest x : - highest measured points at x=0.75 (BCDMS) (data at higher x exist but are in the resonance region, can not easily be interpreted in terms of pdfs) - HERA, standard techniques: xmax=0.65 increasing x Method : if no jet is found at  > 7o, don’t try to reconstruct x, but put the event in a bin [ xedge; 1 ] and measure the integrated . Should bring constraints on high x quarks, especially with the full HERA statistics. No jet found Low x School, July 07

  30. Measurements of CC DIS Recall (slide 12) : _ CC(e+p) goes as (1 –y)2 xD + xU _ CC(e-p)goes as (1 –y)2 xD + xU As an example, our e+p measurements are shown on the plot. HERA I measurements are statistically limited. However, for x ~ 0.1, precision of about 15% is reached (but at highest Q2 where the stat. is low) Brings constraints on “flavor separation”, which are missing from F2p (4U+D) alone. Low x School, July 07

  31. CC(e,SM) ~ (1  Pe) • e+ p : first publications • on HERAII data e- p • e- p : prelim. measurements with the • full available 05 stat. 69 pb-1 HERA I (syst.) typically 4%, (stat.) from 2% to 8% 30 pb-1 27 pb-1 21 pb-1 H1, PLB 634 (2006) 173. ZEUS, PLB 637 (2006) 210. e+ p Extrapolations to Pe =  1 consistent with no WR Low x School, July 07

  32. The charm and beauty contents of the proton • Exclusive measurements : • D* D0 slow  K  slow • and b  X, exploiting PT,rel() and impact parameter • Semi-inclusive measurements : • distributions of the significance of track • impact parameters are used to fit simultaneously • the light q, c and b contributions to F2. • Use silicon vertex devices around the interaction • point. H1 Collab., EPJ C45 (2006) 23 H1 Central Silicon Tracker 2 cylindrical layers, at radii of ~ 5 cm and ~ 10 cm. Impact parameter resolution: As F2, F2bb,cc shows large scaling violations at low x. Note the difference between the MRST and CTEQ predictions. Data now included in the most recent CTEQ analysis. Low x School, July 07

  33. Charm fraction fcc - roughly constant with Q2 - around 24% on average Beauty fraction fbb - increases rapidly with Q2 from ~0.3% to ~3% (i.e. by a factor of 10) Low x School, July 07

  34. e  Extending to lowest Q2 Q2 = 2 E0e E’e ( 1 + cos e) e p To go lower in Q2, needs to : - access larger angles - or lower the incoming energy E0e Larger angles : • dedicated apparatus, e.g. ZEUS Beam Bipe Tracker : silicon strip tracking • detector & EM calorimeter very close to the Beam Pipe. • Was present at HERA I. Allows to cover the range 0.045 < Q2 < 0.65 GeV2 Shifted vertex • shift the interaction vertex in the • forward direction. Two short runs • runs such a setting, with zvtx = 70 cm. • QED Compton events 1 Final electron (2) in the acceptance, 1 larger. 2 Q2 = xyS, S reduced i.e. access higher x at a given Q2. Lower the incoming energy : Exploit initial state radiation events. Low x School, July 07

  35. Example of measurements at lowest Q2 H1 Collab., PLB 598 (2004) 159 • Good agreement with fixed target exps. • F2 continues to rise at low x, even at the • lowest Q2… • as Q2 0, F2 ~ Q2 as required by the • conservation of the EM current. ZEUS Collab., PLB 487 (2000) 53 Low x School, July 07

  36. End of first lecture… _ _ • F2 (ep) measures 4 ( U + U ) + D + D. • Precise measurements from HERA over a huge kinematic range, in particular • extending towards very low x, and towards very high Q2. • Strong rise of F2 towards low x. Unitarity ? • Strong scaling violations at low x. • Withing DGLAP, the measurements of the scaling violations give access to • the gluon density. • The measurement of Charged Current DIS brings information on the flavor • separation. The other source of information (F2p vs F2d,  N vs  N) requires • non trivial corrections to the data. • Measuring xF3 brings information on the valence quark distributions – but • statistics is quite limited. _ Next time : • Extracting pdfs from these measurements • and from these measurements plus those from other experiments • Take a closer look at the low x behavior of HERA measurements. Low x School, July 07

  37. Parameterize a set of pdfs at a “starting scale” Q02 e.g. xg(x) = A x (1-x) P(x) and a set of quark pdfs, e.g. uval, dval, TotalSea =  q, d - u _ _ _ - quite some freedom in choosing what to parameterize - quite some freedom in choosing the form of the parameterization QCD fits to DIS data • Choose the “order of the fit”, LO or NLO. • Usually fits are performed at NLO. NNLO is coming – but so far not many • processes calculated to NNLO. • For NLO : choose the renormalization scheme, MS or DIS. • When pdfs are used to predict e.g. a pp cross-section, scheme should match ! • Most NLO calculations done in MS  most often, fits in MS. • Choose the scales used in the calculation, e.g. R = F = Q2. • Choose how to deal with heavy flavors. • Choose the datasets, and kinematic cuts to be applied to the data points. • e.g. cut away very low x (where DGLAP may break down), very low W2 • (higher twist effects), very high x (would need a resum. of ln(1-x) ) Low x School, July 07

  38. and do assumptions to supplement the lack of sensitivity of the fitted data. • e.g. - s – s not well constrained yet, often assumed that s – s = 0. • - If only lepton-p data are fitted, no information on d – u, set to zero • or to something consistent with other data. _ _ _ _ • Starting scale : not too high, to keep as much data as possible (mainly DIS) • not too low, to be in the perturbative domain. • Typical value Q02 ~ a few GeV2. Id. for c, b • Usually impose number sum rules : And momentum sum rule : • Helps fix the gluon normalisation • “connects” the low x and high x behaviors of g(x) • DGLAP equations give f(x,Q2) at any Q2, once f(x, Q02) is known. • Allows to calculate theo (DIS, DY, jet data,…) and fit theory to data. Low x School, July 07

  39. F2 can be written from a “singlet” and a “non-singlet” distribution. E.g. at LO, below the charm threshold : _ _ _ Non-singlet singlet F2 = 4/9 ( u + u ) + 1/9 ( d + d + s + s ) = 4/18  + 1/6 ( ud – s+) H1 in “H1Pdf2000” fit : choice as close as possible to what is actually measured : g, U = u + c, D = d + s ( + b) , U = u + c , D = d + s ( + b ) _ _ _ _ _ _ _ _ _ _ _ With  =  (q + q ) (singlet), ud = ( u + u ) – (d + d ) and s+ = s + s -  / 3 “Flavor decomposition” : choice of input combinations First, need to define the set of pdf’s combinations that will be evolved (one does not fit the 11 q, q and g distributions, since the data do not contain enough information). The gluon distribution is always one of those. _ Singlet & non-singlet evolve differently  need at least two quark distributions. Need more if interested in more than just the gluon density. Common examples : _ _ g, uVal, dVal, Total Sea, d – u Possibly add strange combinations if some of the included data are sensitive. Low x School, July 07

  40. Singlet part, i.e. gluon and sea quarks :  xg and xq ~ ( 1 / x) P(0) - 1 with P(0) = 1 +   1.08 i.e. xq, xg ~ “flatish” _ _ - Valence quarks : (qqq) ns = 2,  = 3 - Gluons : (qqqg) ns = 3,  = 5 - Antiquarks : (qqq qq ) ns = 4,  = 7 _ Choice of the parametrisation Early days : parameterize xg(x) and xq(x) as ~ x (1-x). Some “predictions” : • Low x behavior : suggested by Regge phenomenology Optical theorem relates F2p to tot(* p) at W2 = Q2 (1-x)/x Regge : tot(* p) ~ A Pomeron ( W2 )P(0) - 1 + A Reggeon ( W2 ) R(0) - 1 Pomeron has no contribution to the non-singlet part  xqval ~ ( 1 / x) R(0) - 1 with R(0) = 0.5 i.e. xqval ~ x 0.5 • High x behavior : simple “dimension arguments”,=2ns-1 where ns is the • number of “spectator quarks” : S. Brodsky et al., ~ 1973, e.g. PRL 31 (1973) 1153 and references therein Low x School, July 07

  41. Don’t forget that the parameterization includes assumptions, e.g. : _ _ d – u = A x (1-x) _ _ assumes that d – u  0 as x  0. Correct ? But : - at which Q2 should these naïve predictions hold ?? - current data show that x (1-x) is too simple ! • Current parameterizations : • e.g. xf(x) = A x (1-x) ( 1 + p0 x + p1 x + p2 x2..) • May use some more or less ad-hoc “2 saturation” procedure to decide whether • or not one adds a new term. • Param : needs to be flexible to allow for a good fit. But should still avoid • unstable fits, secondary minima… • The “best” parameterization depends on the starting scale. • E.g. one finds that, for a starting scale of a few GeV2, a param. • xg(x) = A x (1-x) ( Polynom ) works fine. • For a lower starting scale, ~ 1 GeV2, better fits can be obtained by giving • more flexibility to the gluon density, e.g. (MRST global fits) • xg(x) = A x (1-x) ( Polynom ) - x (1-x)Big Low x School, July 07

  42. Heavy flavor treatment Several “schemes” exist : • Zero-mass Variable Flavor Number scheme (ZM-VFNS) : • c = 0 until Q2 ~ 4 m2c. Above, charm is generated by gluon splitting and is • treated as massless. • drawback: ignores mc near the threshold… • Fixed Flavor Number scheme (FFNS) : • No pdf for c, b. Only 3 active flavors. • For W2 above the threshold (4m2c) • a cc pair can be produced by • photon-gluon fusion. • Drawback : at large Q2, large logs, ln(Q2/m2c). • General-mass Variable Flavor Number scheme • (GM-VFNS) : • the “state-of-the-art”. • somehow interpolates between the two • above approaches. • Not easy to implement esp. at NNLO. _  c c g _ Low x School, July 07

  43. Higher twists and target mass corrections “Higher twists” operators : generally related to correlations between partons in the nucleon. They contribute as additional terms in f(x)/Q2, g(x)/Q4,… to the structure functions, with f(x), g(x) large at high x only. • HT are important at low Q2 and high x • A cut W2 ( + Q2 (1-x)/x) > typically 10 GeV2 allows to be on the safe side. Important esp. for the data from fixed target experiments. • and/or one can correct the data for these HT effects, using some parameterized expressions for f(x) – but no general consensus. Target mass effects : take into account the finite mass of the nucleon (d or heavier). ( P + q)2 = 0 = -Q2 +  m2N + 2  (x.P) q And the measured F2 is related to the “massless” one by : P “higher twist” like, kinematic origin Effects largest at low Q2 and large x. Low x School, July 07

  44. x x x x x W2 > 10 GeV2 x Illustration of HT : MRST(HT) : fit with F2 = F2LT ( 1 + D(x) / Q2) Differences observed between MRST and MRST(HT) at low W2. A cut W2 > about 10 GeV2 allows to stay away from region where HT are important. Note that this cuts out many of the SLAC data points… Low x School, July 07

  45. Goal : determine the gluon density and S. Datasets : H1 low Q2, BCDMS p data. “Massive scheme”, assume symmetric sea : usea = u = d = dsea, s + s = ( u + d ) /2 Need only 3 input densities, the gluon & two quark combinations. _ _ _ _ _ Example 1 : QCD fit to low Q2 ep DIS data H1 Collab., EPJ C21 (2001) 33 Choice : Pure valence  sum rule ( V dx = 3 ) closest to what is measured at low x • Constraints on the gluon density only from • scaling violations. • Correlation between g and S, can be • alleviated only with other data. • Only ep data : no binding corrections needed. • More data needed to disentangle the quarks ! Low x School, July 07

  46. 0.7% 1.4% Model uncertainty : Scale uncertainties : > 5% ! (slide from M. Klein) Low x School, July 07

  47. _ _ Parameterize : xg(x), xU = xu + xc, xD = xd + xs ( + xb), xU , xD by : _ _ _ _ Relation between AU and AD such that d / u  1 as x  0 Example 2 : adding high Q2 DIS data H1 Collab., EPJ C30 (2003) 1 Low Q2 ep DIS data alone (~ 4U + D) give no information about the flavor separation. Would need data on deuterium, see fits presented later. Still with ep only : adding high Q2 CC, e+p and e-p, brings some information: - Charged Current in e+ p : mainly probes the d density - Charged Current in e- p : mainly probes the u density - Neutral Current e-p vs e+p : brings xF3 i.e. combination of uval and dval ( “2 saturation” procedure ) Assumptions : Such that the valence distributions vanish when x  0. Low x assumptions needed. Because no distinction of the rise at low x between xU and xD. xc constant fraction of xU, xs of xD, at starting scale Q02=4 GeV2 Results in 10 free parameters. Low x School, July 07

  48. Q20 = 4 GeV2 Chi2 / ndf = 0.88 for 621 data points (H1 only) Good determination with ep data alone ! Best precision for xU (1.5% at x=10-3, 6.5% at x=0.4). xD mainly from CC e+ (1.6% at x=10-3, 27% at x=0.4) Low x School, July 07

  49. _ _ _ Flavor decomposition : g, uval, dval, total sea =  (q + q ), d - u Assumptions : • p2 = 0.5 for uval and dval (little information on low x valence – cf Regge…) _ _ • d - u : the less constrained pdf with only DIS data… •  fix p2 = 0.5, fix the high-x parameter a la MRST, i.e. fit normalisation only. • ( Hence d = u is imposed by the param. as x  0, cf H1 assumptions) _ _ • xs = 20% of Total Sea (as suggested by dimuon data from CCFR) ZEUS Collab., PRD 67 (2003) 012007 Example 3 : fit (ZEUS) to DIS data Two fits performed : - to ZEUS DIS data only (low Q2 and high Q2, NC and CC) - to ZEUS NC DIS data and DIS data from fixed target experiments : - p and d data (BCDMS, NMC, E665) - NMC data for F2d / F2p  constraints d / u - CCFR data on xF3  constraints the high-x valence Starting scale Q02 = 7 GeV2, Q2min = 2.5 GeV2  11 parameter fit Low x School, July 07

  50. Reasonable agreement with e.g. the H1 and the CTEQ fit… ZEUS data + fixed target Differences however that are not embedded in the error bands, esp. for the valence distributions. • Sensitivity to those has a • different origin in the • H1 and ZEUS fits : • H1 : uses W & Z to do • the flavor separation • ZEUS : this comes mainly • from p vs. d and xF3 • measured in fixed target • experiments. Low x School, July 07

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