Srednicki — Quantum Field Theory · Problem 83.6

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · chiral-symmetry-breaking

Problem

Consider the case of three light quark flavors, with masses

m_u

m_d

, and

m_s

. a) Find the masses-squared of the eight pseudogoldstone bosons. Take the limit

m_{u,d} \ll m_s

, and drop terms that are of order

m_{u,d}^2/m_s

. b) Assume that

m^2_{\pi^\pm}

and

m^2_{K^\pm}

each receive an electromagnetic contribution; to zeroth order in the quark masses, this contribution is the same for both, but the comparatively large strange quark mass results in an electromagnetic contribution to

m^2_{K^\pm}

that is roughly twice as large as the electromagnetic contribution

\Delta m^2_{\mathrm{EM}}

m^2_{\pi^\pm}

. Use the observed masses of the

\pi^\pm

\pi^0

K^\pm

, and

K^0

to compute

m_u v^3/f_\pi^2

m_d v^3/f_\pi^2

m_s v^3/f_\pi^2

, and

\Delta m^2_{\mathrm{EM}}

. c) Compute the quark mass ratios

m_u/m_d

and

m_s/m_d

. d) Use your results from part (b) to predict the

\eta

mass. How good is your prediction?

Related theory

Order Parameters And Symmetry Breaking

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