Srednicki — Quantum Field Theory · Problem 97.3

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · su5-operator-renormalization-proton-decay

Problem

Let us write eq. (97.25) as

\mathcal{L}_{\text{eff}} = -Z_{C_1} C_1 \mathcal{O}_1 - Z_{C_2} C_2 \mathcal{O}_2 + \text{h.c.} , \tag{97.43}

where

\mathcal{O}_1 \equiv \varepsilon^{ij}\varepsilon^{\alpha\beta\gamma}(\ell_i q_{j\gamma})(\bar{d}^\dagger_\alpha \bar{u}^\dagger_\beta)

and

\mathcal{O}_2 \equiv \varepsilon^{ij}\varepsilon^{\alpha\beta\gamma}(\bar{e}^\dagger \bar{u}^\dagger_\gamma)(q_{i\alpha} q_{j\beta})

Z_{C_1}

and

Z_{C_2}

are renormalizing factors, and

C_1

and

C_2

are coefficients that depend on the

\overline{\text{MS}}

renormalization scale

\mu

. At

\mu = M_\mathrm{X}

, we have

C_1(M_\mathrm{X}) = C_2(M_\mathrm{X}) = 4\pi\alpha_5(M_\mathrm{X})/M_\mathrm{X}^2

. a) Working in Lorenz gauge, and using the results of problems 88.7 and 89.5, show that the one-loop contribution to

Z_{C_1}

from gauge-boson exchange is given by

Z_m

in spinor electrodynamics in Lorenz gauge, with

(-1)(+1)e^2 \to \left[0 + \varepsilon^{\alpha'\beta'\gamma}(T^a_3)_{\alpha'}{}^\alpha (T^a_3)_{\beta'}{}^\beta/\varepsilon^{\alpha\beta\gamma}\right] g_3^2

+ \left[\varepsilon^{i'j'}(T^a_2)_{i'}{}^i(T^a_2)_{j'}{}^j/\varepsilon^{ij} + 0\right]g_2^2

+ \left[(-\tfrac{1}{2})(+\tfrac{1}{6}) + (+\tfrac{1}{3})(-\tfrac{2}{3})\right] g_1^2 . \tag{97.44}

Evaluate these coefficients. b) Similarly, compute the one-loop contribution to

Z_{C_2}

from gauge-boson exchange. c) Compute the corresponding anomalous dimensions

\gamma_1

and

\gamma_2

C_1

and

C_2

. d) Compute the numerical values of

C_1(\mu)

and

C_2(\mu)

\mu = 2\,\text{GeV}

. For simplicity, take the top quark mass equal to

M_Z

, and all other quark masses less than

2\,\text{GeV}

. Ignore electromagnetic renormalization below

M_Z