Srednicki — Quantum Field Theory · Problem 94.2

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · axion-peccei-quinn-mechanism

Problem

Consider adding to the Standard Model a massless quark, represented by a pair of Weyl fermions

\chi

and

\xi

in the

\mathbf{3}

and

\bar{\mathbf{3}}

representations of SU(3). Also add a complex scalar

\Phi

in the singlet representation. Assume that these fields have a Yukawa interaction of the form

\mathcal{L}_{\mathrm{Yuk}} = y\Phi\chi\xi + \mathrm{h.c.}

, where

y

is the Yukawa coupling constant. Assume that the scalar potential

V(\Phi)

depends only on

\Phi^\dagger\Phi

. a) Show that the lagrangian is invariant under a Peccei--Quinn transformation

\chi \to e^{i\alpha}\chi

\xi \to e^{i\alpha}\xi

\Phi \to e^{-2i\alpha}\Phi

, all other fields unchanged. b) Show that this global

\mathrm{U}(1)_{\mathrm{PQ}}

symmetry is anomalous, and that

\theta \to \theta + 2\alpha

under a

\mathrm{U}(1)_{\mathrm{PQ}}

transformation. c) Suppose that

V(\Phi)

has its minimum at

|\Phi| = f/\sqrt{2}

, with

f \neq 0

. Show that this gives a mass to the quark we introduced. d) Write

\Phi = 2^{-1/2}(f + \rho)e^{ia/f}

, where

\rho

and

a

are fields. Argue that, in eq.\,(94.10), we should replace

\theta

with

\theta + a/f

, and add to eq.\,(94.9) a kinetic term

-\frac{1}{2}\partial^\mu a\partial_\mu a

for the

a

field. d) Show that the minimum of

V(U)

, defined in eq.\,(94.11), is at

U = I

and

a = -f\theta

. Show that

P

and

CP

are conserved at this minimum. e) The particle corresponding to the

a

field is the axion; compute its mass, assuming

f \gg f_\pi

. f) Note that if

f

is large, the extra quark becomes very heavy, and the axion becomes very light. Show that couplings of the axion to the hadrons are all suppressed by a factor of

1/f

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