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1 + Wz(v, qZ) mZ ( Pp -
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The cross section is expressed as
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d(J (XZ 0 dO'dE' = 4E z sin 4 (0/2) cos "2 Wz
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Experimental conditions, sin z (0/2) 1, make it difficult to extract Wl'
The amplitude W~, is related to the scattering of polarized virtual photons off a nucleon. Call and (is the transverse and longitudinal cross sections (with (is = 0 at q2 = 0). Show that
ASYMPTOTIC BEHAVIOR
W2(V,q)=-S-2-1 nam
+ q2
(13-104)
-V 2/2
2(TS+(TT)
where in (TT and (Ts the flux factor is conventionally computed as if the photons were real with the same energy. Derive from these expressions positivity conditions for WI and W 2. For elastic scattering (w = 1) derive the following relations with the electric and magnetic form factors, G E(q2) and GM (q2):
q2 W{I(V,q2) = - 2m Glt(q2)i5(2v
+ q2)
(13-105)
We recall from Chap. 11 that the scattering of neutrinos involves a third structure function W 3 The cross section for neutrinos (v) or anti neutrinos (0 reads [compare with Eq. (11-101)]
dO.' dE'
- - = - - - - - = G2
n dv dq2
d(T(v,VJ
EE'm d(T(v,VJ
2n 2
(e Wi',VJ + 2 sin () wlv,VJ + - - s i n eW (v. VJ ) E + E' cos
2 2 2 -
(13-106a)
2 d(T(V,V) m -E [ V W G - (1-y)-2+xy2mWI+x ( 1-- vW ] y) _ 3 --= dxdy 2n m 2 m (13-106b)
in terms of the variable
y=--=E mE
E- E'
(13-107)
The extra contribution represents the interference between the vector and axial parts of the current. In the high-energy limit where the mass of the final lepton (electron or muon) has been neglected, the nonconserved part of the current has disappeared.
The experimental results show that beyond the domain of resonance excitation the cross section remains sizable in the very inelastic region, for - qZ and v very large. For fixed qZ the integral over v is comparable with the Mott cross section on a point nucleon, Accurate measurements at different angles allow us to separate Wi and Wz to obtain the ratio R = (JS/(JT with (Js and (JT defined according to (13-104). Its value is small, R ~ 0.15. The most striking phenomenon is, however, the scaling property anticipated by Bjorken and Feynman. In the very inelastic domain the dimensionless quantities 2m Wi and vWz/m tend to nontrivial functions of the scaling variable w = 2v/ - qZ. This is shown on Fig. 13-8 where vWz/m is plotted versus the modified variable w' = (2v + mZ)j_qZ ~ w. Analogous results hold in the case of neutrino scattering, These phenomena suggest an interpretation of the scatterer in terms of pointlike nearly noninteracting constituents, most of which have spin i (which corresponds to a ratio R equal to zero). Such a phenomenological parton model, as it has been called by Bjorken and Feynman, suggests a number of sum rules in agreement with the data. An asymptotically free-field theoretic model gives a coherent framework where the constituents are the fundamental quanta: the quark and gauge gluons.
QUANTUM FIELD THEORY
0.35 0.30 0.25
0.20 0.15 0.10 0.05 0.0
3 w'
Figure 13-8 The function vW2 /m plotted versus the modified scaling variable OJ' = (2v + m 2)/ - q2. The value of R is of the order of 0.18. The data are from SLAC and are commented in the report of R. E. Taylor at the Palerme Conference of 1975. The values of - q2/m 2 being large, resonances at small OJ' have been washed out.
13-4-2 Light-Cone Dynamics
Let us investigate more closely the kinematical domain which is probed in these deep inelastic experiments. At high energy, we test the singularities of the commutator [JI'(X)' J.(O)]. Singularities occur at short distance (x ~ 0) or for lightlike separation (Xl ~ 0). For the commutator which vanishes outside the light cone, short distances in Minkowski space are time-like. To study such a region would require us to look in momentum space for large q and small w. The physical boundaries are, however, w ~ 1. In the experimental conditions we can write q = An - qo where n is light-like and A large. As ..1,-4 00, - ql ~ 2A(n qo)-4 00, v ~ A(n p) -400, and w ~ p. n/qo n tends to a finite limit. Qualitatively, we expect to probe a region where (n x) ~ 1/..1, in such a way that Xl ~ 0(1/..1,) ~ 0(1/ - ql). Thus we are looking at the singular structure of current commutators in the neighborhood of the light cone. It is therefore attractive to formulate the parton model hypothesis in the so-called infinite momentum frame (Fig. 13-9).
The dynamical evolution is considered starting from an hyperplane containing a light-like direction suited to the case where energies and longitudinal momenta are large and comparable. Without entering into details, let us sketch the intuitive reasoning of the parton model. Let P be the magnitude of the large longitudinal proton momentum in the center of mass frame, for instance; its four-momentum is then approximately p "" (P, 0, 0, Pl. The target is thought of as an assembly of N elementary constituents, to be treated during their electromagnetic interactions as free particles with longitudinal momentum XiP and negligible transverse momentum with respect to ~. The virtual photon four-momentum will be approximated in this frame by q "" (v/2P, ~, 0, -v/2P). The Feynman rules in the infinite momentum frame become analogous to those of nonrelativistic perturbation theory. The contribution of the ith constituent to the scattering cross section will thus be proportional to
Qr ;
b(E' - E - qo)
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