Stars as Laboratories for Fundamental Physics - MPP Theory Group

526 Chapter 14
experimental limit (Tab. 14.1), however, indicates |Θ| < ∼ 10 −9 , surprisingly
small in view of the phase δ = 3.3×10 −3 which appears in the
Cabbibo-Kobayashi-Maskawa matrix Eq. (7.6) and which explains the
observed CP-violating effects in the K ◦ -K ◦ system.
Even worse, QCD alone produces a term like Eq. (14.1) because of
the nontrivial topological structure of its ground state (Callan, Dashen,
and Gross 1976; Jackiw and Rebbi 1976; t’Hooft 1976a,b). The coefficient
Θ QCD is a parameter characterizing the “Θ-vacuum.” It is mapped
onto itself by a transformation Θ QCD → Θ QCD + 2π so that different
ground states are characterized by values in the range 0 ≤ Θ QCD < 2π.
The phase of the quark mass matrix and the QCD-vacuum together
yield Θ ≡ Θ QCD + arg det M q as a compound coefficient for Eq. (14.1).
The experimental bounds then translate into
∣
∣
∣∣
∣Θ QCD + arg det M q
< ∼ 10 −9 . (14.2)
The CP-Problem of strong interactions consists of the smallness of Θ
which implies that the numbers Θ QCD and arg det M q are either separately
very small, or cancel each other with very high accuracy. However,
both are expected to be of order unity, or perhaps of order δ in
the case of arg det M q , and completely unrelated to each other.
14.2 The Peccei-Quinn Mechanism
14.2.1 Generic Features
An attempt to explain the smallness of Θ may be overambitious as
long as we do not have an understanding of the origin of the Yukawa
couplings that went into M q and of the other seemingly arbitrary parameters
of the standard model. Still, whatever determines these “constants
of nature,” the strong CP-problem can be elegantly explained
by the existence of a new physical field, the axion field, which allows
Θ to vanish dynamically (Peccei and Quinn 1977a,b; Weinberg 1978;
Wilczek 1978). In this scheme, the CP-violating Lagrangian Eq. (14.1)
is literally switched off by its own force.
To this end the new field a(x) must be a pseudoscalar which couples
to gluons according to Eq. (14.1) with Θ replaced by −a/f a . The
constant 90 f a with the dimension of an energy is the Peccei-Quinn scale
90 In the literature one often finds f a /N, F a /N, v PQ /N etc. for what I call f a . It
was stressed, e.g. by Georgi, Kaplan, and Randall (1986) that a discussion of the
generic properties of all axion models does not require a specification of the modeldependent
integer N which can be conveniently absorbed in the definition of f a .