Srednicki — Quantum Field Theory · Problem 97.5

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · su5-yukawa-renormalization-quark-lepton-masses

Problem

Consider the Yukawa couplings for the the down quark and charged lepton of one generation,

\mathcal{L}_{\text{Yuk}} = -Z_y y \varepsilon^{ij}\varphi_i \ell_j \bar{e} - Z_{y'} y' \varepsilon^{ij}\varphi_i q_{\alpha j} \bar{d}^\alpha , \tag{97.47}

where we have included renormalizing factors. a) Consider one-loop contributions from gauge-boson exchange to

Z_{y'}/Z_y

. Show that the only contributions of this type that do not cancel in the ratio are those where the gauge boson connects the two fermion lines. b) Show that, in Lorenz gauge, these contributions to

Z_{y'}

and

Z_y

are given by

Z_m

in spinor electrodynamics in Lorenz gauge, with a replacement analogous to eq. (97.44) that you should specify. c) Let

r \equiv y'/y

, and compute the anomalous dimension of

r

. d) Take

r(M_\mathrm{X}) = 1

, and evaluate

r(M_Z)

. For simplicity, take the top quark mass equal to

M_Z

. e) Below

M_Z

, treat the top quark as heavy, and neglect the small electromagnetic contribution to the anomalous dimension of

r

. Compute

r(m_b)

, where

m_b = m_b(m_b) = 4.3\,\text{GeV}

is the bottom quark mass parameter. Use your results to predict the tau lepton mass. How does your prediction compare with its observed value,

m_\tau = 1.8\,\text{GeV}

Browse the problem library