An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 20.4

Electroweak Theory: Basics — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · neutral-current-deep-inelastic-scattering

Problem

20.4 Neutral-current deep inelastic scattering. (a) In Eq. (17.35), we wrote formulae for neutrino and antineutrino deep inelastic scattering with W±W^\pm exchange. Neutrinos and antineutrinos can also scatter by exchanging a Z0Z^0. This process, which leads to a hadronic jet but no observable outgoing lepton, is called the *neutral current* reaction. Compute dσ/dxdyd\sigma/dxdy for neutral current deep inelastic scattering of neutrinos and antineutrinos from protons, accounting for scattering from uu and dd quarks and antiquarks. (b) Next, consider deep inelastic scattering from a nucleus AA with equal numbers of protons and neutrons. For such a target, fu(x)=fd(x)f_u(x) = f_d(x), and similarly for antiquarks. Show that the formulae in part (a) simplify in such a situation. In particular, let RνR^\nu, RνˉR^{\bar{\nu}} be defined as Rν=dσ/dxdy(νA→νX)dσ/dxdy(νA→μ−X),Rνˉ=dσ/dxdy(νˉA→νˉX)dσ/dxdy(νˉA→μ+X).R^\nu = \frac{d\sigma/dxdy(\nu A \to \nu X)}{d\sigma/dxdy(\nu A \to \mu^- X)}, \qquad R^{\bar{\nu}} = \frac{d\sigma/dxdy(\bar{\nu} A \to \bar{\nu} X)}{d\sigma/dxdy(\bar{\nu} A \to \mu^+ X)}. Show that RνR^\nu and RνˉR^{\bar{\nu}} are given by the following simple formulae: Rν=12−sin⁡2θw+59sin⁡4θw(1+r),R^\nu = \frac{1}{2} - \sin^2\theta_w + \frac{5}{9}\sin^4\theta_w(1+r), Rνˉ=12−sin⁡2θw+59sin⁡4θw(1+1r),R^{\bar{\nu}} = \frac{1}{2} - \sin^2\theta_w + \frac{5}{9}\sin^4\theta_w(1+\frac{1}{r}), where r=dσ/dxdy(νˉA→μ+X)dσ/dxdy(νA→μ−X).r = \frac{d\sigma/dxdy(\bar{\nu} A \to \mu^+ X)}{d\sigma/dxdy(\nu A \to \mu^- X)}. These formulae remain true when RνR^\nu and RνˉR^{\bar{\nu}} are redefined to be the ratios of neutral- to charged-current cross sections integrated over the region of xx and yy that is observed in a given experiment. (c) By setting rr equal to the observed value—say, r=0.4r = 0.4—and varying sin⁡2θw\sin^2\theta_w, the relations of part (b) generate a curve in the plane of RνR^\nu versus RνˉR^{\bar{\nu}} that is known as *Weinberg's nose*. Sketch this curve. The observed values of RνR^\nu, RνˉR^{\bar{\nu}} lie close to this curve, near the point corresponding to sin⁡2θw=0.23\sin^2\theta_w = 0.23.

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