The quark magnetic moment is modelled as

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Symmetry, groups and the quark model · Symmetry, groups and the quark model

Symmetry and Flavour, Ch The constituent quark model: wave functions, magnetic moments and hyperfine masses

Problem

The quark magnetic moment is modelled as μq=Qqe/(2mq)\mu_q = Q_qe/(2m_q), and the constituent quark model gives μp=13(4μu−μd)\mu_p = \tfrac13(4\mu_u-\mu_d) for the proton, with the neutron result obtained by exchanging u↔du\leftrightarrow d. Take equal constituent masses mu=md=mm_u=m_d=m. ((a)) Using fractional quark charges Qu=+23Q_u=+\tfrac23, Qd=−13Q_d=-\tfrac13, evaluate μp\mu_p and μn\mu_n in terms of e/me/m, and hence confirm the ratio μp/μn\mu_p/\mu_n. Compare with the measured ratio 2.793/(−1.913)2.793/(-1.913). ((b)) Using the same fractional quark charges, evaluate R=Nc∑qQq2R=N_c\sum_qQ_q^2 for Nc=3N_c=3 colours below the charm threshold, where only u,d,su,d,s quarks contribute. ((c)) The measured RR below the charm threshold is about 2.22.2. Explain in one or two sentences why the magnetic-moment ratio of part (a) cannot by itself fix the number of colours NcN_c, while the value of RR in part (b) can.

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