Minimal predicts at

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Loops, renormalisation and the renormalisation group · Beyond the Standard Model

Symmetry and Flavour, Ch Grand unification: running couplings, SU(5), proton decay and the hierarchy problem

Problem

Minimal SU(5)SU(5) predicts sin⁡2θW=38\sin^2\theta_W=\tfrac38 at MXM_X, and the chapter's worked example ran that prediction down to MZM_Z with the Standard Model coefficients, finding 0.2080.208 against the measured 0.23120.2312. Suppose instead that the supersymmetric coefficients (b1,b2,b3)=(335,1,−3)(b_1,b_2,b_3)=(\tfrac{33}{5},1,-3) held all the way from MZM_Z to MXM_X. Use αem−1(MZ)=127.95\alpha_{\rm em}^{-1}(M_Z)=127.95, α3−1(MZ)=8.47\alpha_3^{-1}(M_Z)=8.47 and MZ=91.19 GeVM_Z=91.19\,\mathrm{GeV}, and nothing else measured. ((a)) Find the slope with which αem−1−83α3−1\alpha_{\rm em}^{-1}-\tfrac83\alpha_3^{-1} runs, and hence L/2πL/2\pi, where L=ln⁡(MX/MZ)L=\ln(M_X/M_Z), and MXM_X in GeV. ((b)) Find αG−1\alpha_G^{-1}, the predicted α2−1(MZ)\alpha_2^{-1}(M_Z) and the predicted sin⁡2θW\sin^2\theta_W at MZM_Z, and compare with measurement. ((c)) Using the dimensional estimate τp≈MX4/(αG2mp5)\tau_p\approx M_X^4/(\alpha_G^2m_p^5) with mp=0.938 GeVm_p=0.938\,\mathrm{GeV}, which ignores any new decay channels the extension might bring, decide whether this MXM_X survives the Super-Kamiokande bound τ(p→e+π0)>2.4×1034 yr\tau(p\to e^+\pi^0) > 2.4\times10^{34}\,\mathrm{yr}.

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