An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 18.2

Operator Products and Effective Vertices — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · operator-products-effective-vertices

Problem

Parity-violating deep inelastic form factor. In this problem, we first motivate the presence of additional deep inelastic form factors that are proportional to *differences* of quark and antiquark distribution functions. Then we define these functions formally and work out their properties. (a) Analyze neutrino-proton scattering following the method used at the beginning of Section 18.5. Define

J_+^\mu = \bar{u}\gamma^\mu\left(\frac{1-\gamma^5}{2}\right)d, \qquad J_-^\mu = \bar{d}\gamma^\mu\left(\frac{1-\gamma^5}{2}\right)u.

Let

W^{\mu\nu(\nu)} = 2i \int d^4x\, e^{iq\cdot x} \langle P|\, T\{J_-^\mu(x) J_+^\nu(0)\}\,|P\rangle,

averaged over the proton spin. Show that the cross section for deep inelastic neutrino scattering can be computed from

W^{\mu\nu(\nu)}

according to

\frac{d^2\sigma}{dxdy}(\nu p \to \mu^- X) = \frac{G_F^2 y}{2\pi^2} \cdot \mathbf{Im}\big[(k_\mu k'_\nu + k_\nu k'_\mu - g_{\mu\nu} k\cdot k' - i\epsilon_{\mu\nu}{}^{\alpha\beta} k'_\alpha k_\beta) W^{\mu\nu(\nu)}(P,q)\big].

(b) Show that any term in

W^{\mu\nu(\nu)}

proportional to

q^\mu

q^\nu

gives zero when contracted with the lepton momentum tensor in the formula above. Thus we can expand

W^{\mu\nu(\nu)}

with three scalar form factors,

W^{\mu\nu(\nu)} = -g^{\mu\nu} W_1^{(\nu)} + P^\mu P^\nu W_2^{(\nu)} + i\epsilon^{\mu\nu\lambda\sigma} P_\lambda q_\sigma W_3^{(\nu)} + \cdots,

where the additional terms do not contribute to the deep inelastic cross section. Find the formula for the deep inelastic cross section in terms of the imaginary parts of

W_1^{(\nu)}

W_2^{(\nu)}

, and

W_3^{(\nu)}

. (c) Evaluate the form factors

W_i^{(\nu)}

in the parton model, and show that

\mathrm{Im}\, W_1^{(\nu)} = \pi(f_d(x) + f_{\bar{u}}(x)),

\mathrm{Im}\, W_2^{(\nu)} = \frac{4\pi}{ys} x(f_d(x) + f_{\bar{u}}(x)),

\mathrm{Im}\, W_3^{(\nu)} = \frac{2\pi}{ys}(f_d(x) - f_{\bar{u}}(x)).

Insert these expressions into the formula derived in part (b) and show that the result reproduces the first line of Eq. (17.35). (d) This analysis motivates the following definition: For a single quark flavor

f

, let

J_{fL}^\mu = \bar{f}\gamma^\mu\left(\frac{1-\gamma^5}{2}\right)f.

Define

W_{fL}^{\mu\nu} = 2i \int d^4x\, e^{iq\cdot x} \langle P|\,T\{J_{fL}^\mu(x) J_{fL}^\nu(0)\}\,|P\rangle.

Decompose this tensor according to

W_{fL}^{\mu\nu} = -g^{\mu\nu} W_{1fL} + P^\mu P^\nu W_{2fL} + i\epsilon^{\mu\nu\lambda\sigma} P_\lambda q_\sigma W_{3fL} + \cdots,

where the remaining terms are proportional to

q^\mu

q^\nu

. Evaluate the

W_{iL}

in the parton model. Show that the quantities

W_{1fL}

and

W_{2fL}

reproduce the expressions for

W_{1f}

and

W_{2f}

given by Eqs. (18.120) and (18.144), and that

W_{3fL}

is given by

\mathrm{Im}\, W_{3fL} = \frac{2\pi}{ys}(f_f(x) - f_{\bar{f}}(x)).

(e) Compute the operator product of the currents in the expression for

W_{fL}^{\mu\nu}

, and write the terms in this product that involve twist-2 operators. Show that the expressions for

W_{1fL}

and

W_{2fL}

that follow from this analysis reproduce the expressions for

W_{1f}

and

W_{2f}

given by Eqs. (18.144) and (18.145). Find the corresponding expression for

W_{3fL}

. (f) Define the parton distribution

f_f^-

by the relation

f_f^-(x, Q^2) = \frac{ys}{2\pi}\,\mathrm{Im}\, W_{3fL}(x, Q^2).

Show that, by virtue of this definition, the distribution function

f_f^-

satisfies the sum rule (18.155) for odd

n

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