206 16. Structure functions with i = NC or CC as before. In the M 2 /Q2 → 0 limit, Eq. (16.8) and Eq. (16.14)maybewrittenintheform d2σi � 2πα2 = dxdy xyQ2 ηi Y+F i 2 ∓ Y−xF i 3 − y2F i � L , d2Δσi � 4πα2 = ηi −Y+g dxdy xyQ2 i 4 ∓ Y−2xg i 1 + y2g i � L , (16.16) with i = NC or CC, where Y± =1± (1 − y) 2 and F i L = F i 2 − 2xF i 1 , gi L = gi 4 − 2xgi 5 . (16.17) In the naive quark-parton model, the analogy with the Callan-Gross relations [6] F i L = 0, are the Dicus relations [7] gi L = 0. Therefore, there are only two independent polarized structure functions: g1 (parity conserving) and g5 (parity violating), in analogy with the unpolarized structure functions F1 and F3. 16.2.1. Structure functions in the quark-parton model : In the quark-parton model [8,9], contributions to the structure functions F i and gi can be expressed in terms of the quark distribution functions q(x, Q2 ) of the proton, where q = u, u, d, d etc.The quantity q(x, Q2 )dx is the number of quarks (or antiquarks) of designated flavor that carry a momentum fraction between x and x + dx of the proton’s momentum in a frame in which the proton momentum is large. For the neutral-current processes ep → eX, � F γ 2 ,FγZ 2 ,F Z � 2 = x � � e q 2 � q 2 2 q , 2eqgV ,gq V + gq A (q + q) , � F γ 3 ,FγZ 3 ,F Z � 3 = � � 0, 2eqg q q A , 2gq V gq � A (q − q) , � g γ 1 ,gγZ 1 ,gZ � 1 = 1 � � e 2 q 2 � q 2 2 q , 2eqgV ,gq V + gq A (Δq +Δq) , � g γ 5 ,gγZ 5 ,gZ � 5 = � � 0,eqg q q A ,gq V gq � A (Δq − Δq) , (16.18) where g q V = ± 1 2 − 2eq sin2 θW and g q A = ± 1 ,with± according to whether 2 q is a u− or d−type quark respectively. The quantity Δq is the difference q ↑−q↓ of the distributions with the quark spin parallel and antiparallel to the proton spin. For the charged-current processes e−p → νX and νp → e + X, the structure functions are: W − W − F2 =2x(u + d + s + c...) , F3 =2(u−d − s + c...) , (16.19) W − − g1 =(Δu +Δd +Δs +Δc...) , gW 5 =(−Δu +Δd +Δs− Δc...) , where only the active flavors are to be kept and where CKM mixing has been neglected. For e + p → νX and νp → e−X, the structure functions F W + ,gW + are obtained by the flavor interchanges d ↔ u, s ↔ c in the expressions for F W − ,gW− . The structure functions for scattering on a neutron are obtained from those of the proton by the interchange u ↔ d. For both the neutral- and charged-current processes, the quark-parton model predicts 2xF i 1 = F i 2 and gi 4 =2xgi 5 . Further discussion may be found in the full Review of <strong>Particle</strong> <strong>Physics</strong>.
F 2 (x,Q 2 ) * 2 i x 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 -1 10 -2 10 -3 16. Structure functions 207 H1 ZEUS BCDMS E665 NMC SLAC Q 2 (GeV 2 10 ) -1 1 10 10 2 10 3 10 4 10 5 10 6 Figure 16.7: The proton structure function F p 2 measured in electromagnetic scattering of positrons on protons (collider experiments ZEUS and H1), in the kinematic domain of the HERA data, for x>0.00006 (cf. Fig. 16.10 for data at smaller x and Q2 ), and for electrons (SLAC) and muons (BCDMS, E665, NMC) on a fixed target. Statistical and systematic errors added in quadrature are shown. The data are plotted as a function of Q2 in bins of fixed x. Some points have been slightly offset in Q2 for clarity. The ZEUS binning in x is used in this plot; all other data are rebinned to the x values of the ZEUS data. For the purpose of plotting, F p 2 has been multiplied by 2ix ,whereixis the number of the x bin, ranging from ix =1(x =0.85) to ix =28(x =0.000063).