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Quark Distributions at Large X

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Quark Distributions at Large X Workshop on Intersections of Nuclear Physics with Neutrinos and Electrons JLab, May 4-5, 2006 Quark distributions at large x Wally Melnitchouk Jefferson Lab 3.1 The New Standard PDF Sets The standard set of parton distributions in the MS scheme, referred to as CTEQ6M, provides an excellent global fit to the data sets listed in Sec. 2.1. An overall view of these PDF’s is shown in Fig. 1, at two scales Q = 2 and 100 GeV. The overall χ2 for the CTEQ6M fit is 1954 for 1811 data points. The parameters for this fit and the individual χ2 values for the data sets are given in Appendix A. In the next two subsections, we discuss the comparison of this fit to the data sets, and then describe the new features of the parton distributions themselves. Quantitative comparison of data and fit is studied in more depth in Appendix B Fig. 1 : Overview of the CTEQ6M parton distribution functions at Q = 2 and 100 GeV. Lai et al., Eur. Phys. J. C12 (2000) 375 Significant advances in determination of quark and gluon 3.1.1 Comparison with Data distributions at small x in recent years The fact that correlated systematic errors are now fully included in the fitting procedure allows a more detailed study of the quality of fits than was possible in the past. We can take the correlated systematic errors into account explicitly when comparing data and theory, by using the procedure discussed in Sec. B.2 of Appendix B. In particular, based on the formula for the extended χ2 function expressed in the simple form Eq. (11), we obtain a precise graphical representation of the quality of the fit by superimposing the theory curves on the shifted data points D containing { i} the fitted systematic errors. The remaining errors are purely uncorrelated, hence are properly represented by error bars. We use this method to present the results of our fits when!ever possible. Figure 2 shows the comparison of the CTEQ6M fit to the latest data of the H1 experiment [14]. The extensive data set is divided into two plots: (a) for x < 0.01, and (b) for x > 0.01. In order to keep the various x bins separated, the values of F2 on the plot have been offset vertically 2 for the kth bin according to the formula: ordinate = F2(x, Q ) + 0.15 k. The excellent fit seen in the figure is supported by a χ2 value of 228 for 230 data points. Similarly, Fig. 3 shows the comparison to the latest data from ZEUS [15]. One again sees very good overall agreement. 8 0.6 2 2 Q = 25 GeV uv d v 0.4 !u d! s x f (x,Q) g/15 0.2 0 0 0.2 0.4 0.6 0.8 1 x sea quarks & gluons valence quarks q¯ = u¯, d¯, s¯... u g q = u, d, s... p d 0.6 Valence quarks Nucleon structure at intermediate & largeuv x d dominated by valence quarks v 0.4 !u Most direct connection between quark distributionsd! and models of the nucleon is through valences quarks x f (x,Q) g/15 0.2 0 0 0.2 0.4 0.6 0.8 1 x Valence quarks At large x, valence u and d distributions extracted from p and n structure functions 4 1 F p ≈ u + d 2 9 v 9 v 4 1 F n ≈ d + u 2 9 v 9 v u quark distribution well determined from p d quark distribution requires n structure function 4 − n p d ≈ F2 /F2 n p u 4F2 /F2 − 1 Higher twists (a) (b) (c) Valence quarks RatioHigher of d to twists u quark distributions particularly sensitive to quark dynamics in nucleon SU(6) spin-flavour symmetry τ = 2 τ > 2 (a) (b) (c) proton wave function single quark qq and qg ↑ 1 ↑ √2 ↓ scattering correlations p = d (uu)1 d (uu)1 −3 − 3 Higher twists √2 ↑ 1 ↓ 1 ↑ + u (ud)1 u (ud)1 + u (ud)0 τ = 2 6 τ >− 23 √2 (τ = 2) (b) (c) (a) diquark spin single quark interactingqq and qg corquarkrelationsspectator scattering diquark τ = 2 τ > 2 single quark qq and qg scattering correlations Valence quarks Ratio of d to u quark distributions particularly sensitive to quark dynamics in nucleon SU(6) spin-flavour symmetry proton wave function ↑ 1 ↑ √2 ↓ p = d (uu)1 d (uu)1 −3 − 3 √2 ↑ 1 ↓ 1 ↑ + u (ud)1 u (ud)1 + u (ud)0 6 − 3 √2 u(x) = 2 d(x) for all x n F2 2 p = F2 3 Valence quarks scalar diquark dominance M∆ > MN =⇒ (qq)1 has larger energy than (qq)0 . =⇒ scalar diquark dominant in x → 1 limit . =⇒ since only u quarks couple to scalar diquarks . d → 0 u n F2 1 p → F2 4 Feynman 1972, Close 1973, Close/Thomas 1988 Valence quarks hard gluon exchange at large x, helicity of struck quark = helicity of hadron . =⇒ x → 1 . q↑ ! q↓ . =⇒ . helicity-zero diquark dominant in limit d 1 → u 5 F n 3 2 → F p 2 7 Farrar, Jackson 1975 Valence quarks BUT no free neutron targets! (neutron half-life ~ 12 mins) use deuteron as ‘‘effective neutron target’’ d p n However: deuteron is a nucleus, and F2 != F2 + F2 nuclear effects (nuclear binding, Fermi motion, shadowing) obscure neutron structure information “nuclear EMC effect” Nuclear ‘‘EMC effect’’ anti-shadowing FA 2 pions? d F2 Fermi motion shadowing multiple ! Ca, SLAC scattering " Ca, NMC # Fe, SLAC $ Fe, BCDMS x A d 40 56 “EMC effect” Fig. 3.1. The structure function ratio F2 /F2 for Ca and Fe. The data are taken from NMC [71], SLACwhat[72], an aboutd BCDMS d[73 /]. N? binding, N off-shell A d Figure 3.1 presents a compilation of data for the structure function ratio F2 /F2 over A the range 0 x 1. Here F2 is the structure function per nucleon of a nucleus with ≤ ≤ d mass number A, and F2 refers to deuterium. In the absence of nuclear effects the ratios A d d F2 /F2 are thus normalized to one. Neglecting small nuclear effects in the deuteron, F2 can N approximately stand for the isospin averaged nucleon structure function, F2 . However, the more detailed analysis must include two-nucleon effects in the deuteron. Several distinct regions with characteristic nuclear effects can be identified: at x < 0.1 one observes a A d systematic reduction of F2 /F2 , the so-called nuclear shadowing. A small enhancement is seen at 0.1 < x < 0.2. The dip at 0.3 < x < 0.8 is often referred to as the traditional “EMC effect”. For x > 0.8 the observed enhancement of the nuclear structure function is associated with nuclear Fermi motion. Finally, note again that nuclear structure functions can extend beyond x = 1, the kinematic limit for scattering from free nucleons. Shadowing region • Measurements of E665 [76,77,78] at Fermilab and NMC [71,75,79,80,81,82] at CERN provide detailed and systematic information about the x- and A-dependence of the A d structure function ratios F2 /F2 . Nuclear targets ranging from He to Pb have been used. A sample of data for several nuclei is shown in Fig.3.2. While most experiments −4 Xe d cover the region x > 10 , the E665 collaboration provides data for F2 /F2 [76] down to x 2 10−5. Given the kinematic constraints in fixed target experiments, the small " · 30 EMC effect in deuteron 896 G. PILLER, W. MELNITCHOUK, AND A. W. THOMAS 54 2 2 2 ∗ A␮␯␣͑k,q͒ϭi͑q ⑀␮␯k␣ϩ͑k Ϫm ͒⑀␮␯q␣ γ Nuclear “impulse approximation’’ Ϫ2͑k␮⑀kq␣␯Ϫk␯⑀kq␣␮͒͒, ͑8c͒ 2 incoherent scattering A␮␯␣␤͑k,q͒ϭϪim͑q g␮␣g␯␤ϩ2q␣͑k␮g␯␤Ϫk␯g␮␤͒͒. ͑8d͒ N from individual nucleons Here k is the interacting quark four-momentum, and m is its in deuteron ␣ ␤ mass. We use the notation ⑀␮␯kqϵ⑀␮␯␣␤k q . ͑The com- plete forward scattering amplitude would also contain a crossed photon process which we do not consider here, since d in the subsequent model calculations we focus on valence quark distributions.͒ The function (k,p) represents the soft FIG. 1. DIS from a polarized nucleus in the impulse approxima- quark-nucleon interaction. SinceHone is calculating the tion. The nucleus, virtual nucleon, and photon momenta are denoted imaginary part of the forward scattering amplitude, the inte- by P, p, and q, respectively,d and S stands for the nuclear spin N δ(off)F d x gration over the quark momentum k is constrained by ␦ vector. The upper blob Frepresents2 (x) the= truncateddy antisymmetricfN/d(y) F2 (x/y) + 2 ( ) ˆ ! functions which put both the scattered quark and the nonin- nucleon tensor G␮␯ , while the lower one corresponds to the polar- ˆ teracting spectator system on-mass-shell: ized nucleon-nucleus amplitude A. 4 2 2 2 2 ␭ ␳ d k 2␲␦͓͑kϩq͒ Ϫm ͔2␲␦͓͑pϪk͒ ϪmS͔ H␣␤ϭ͑p␣k␤Ϫp␤k␣͒␴␭␳p k f1ϩ͑p␣␴␭␤Ϫp␤␴␭␣͒ dk˜ϵ , nucleon momentum distribution off-shell correction ͑2␲͒4 ͑k2Ϫm2͒2 ␭ ␭ ␭ ␭ ϫ͑p f2ϩk f3͒ϩ͑k␣␴␭␤Ϫk␤␴␭␣͒͑p f4ϩk f5͒ ͑9͒ ␭ ␳ ϩ␴␣␤f6ϩ⑀␭␳␣␤p k ␥5͑p” f7ϩk”f8͒ where m2ϭ(pϪk)2 is the invariant mass squared of the S ϩ⑀ ␥ ␥␳͑p␭f ϩk␭f ͒, ͑13b͒ spectator system. ␭␳␣␤ 5 9 10 Taking the trace over the quark spin indices we find where the functions g and f are scalar functions of 1•••9 1•••10 ␣ ␣␤ Tr͓ r␮␯͔ϭA␮␯␣H ϩA␮␯␣␤H , ͑10͒ p and k.
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