Axions
153
where qR and qL are the chiral components of q. is anomalous [6] (see also
chapter 4 [33.34]) (see [5] for an account of this). In fact. for massless quarks,
a/IfS) = N I g~ G Q GQ/L"
(5.46)
/L
16:n'2 /LIJ
where NI is the number of flavours (= 2NG). Thus. as in (4.126), such a O-term
can be removed by performing the V(I)A transformation (5.44) with
()
a=---.
(5.47)
2NI
With all quarks massless, the f)-term is not physical. since it can be removed
by an (unobservable) VO)A transformation. However. this is not the end of the
problem. since quarks are not massless. The contribution of the mass terms to the
QCD Lagrangian is
Cm = -qLiMijqRj - qRiMjjqLj
(5.48)
where i. j = I, ... , NI label the quark flavours, and Mij is the mass matrix. The
effect of the V(I)A transformation (5.44) on M is
V(l)A : M ..... e 2ia M
Mt ..... e- 2ia Mt.
(5.49)
Thus. if M was initially Hermitian. so that there are no yss in the mass terms,
it is no longer so after the transformation and the transformed Lagrangian has
reacquired the P and T non-conserving interactions which the transformation
(5.44) with a satisfying (5.47) sought to remove. The quantity 8 defined by
8 = 0 + 2N I arg(det M)
(5.50)
is invariant under V(I)A transformations and parametrizes the T-violation in the
(strong) QCD Lagrangian: 8 is the effective QCD vacuum angle in the basis
where all quark masses are real, positive and Y5 free. If non-zero, it contributes
to the neutron electric dipole moment dn and the measured upper bound on this
requires [7]
8 '" < IO-IO
(5.51 )
.
The outstanding question, then, is why 8 is so small. This is the 'strong CP
problem'.
For each value of the parameter 8, we have a different QCD theory and there
is no a priori reason why one (very small or zero) value is preferred over another.
A possible escape from this is that 8 is the expectation value of a field 8 (x), whose
VEV is determined dynamically by an effective potential, as happens when the
electroweak symmetry is spontaneously broken by the Higgs-doublet field. Then
it is conceivable that, at the minimum of the effective potential, (j = o.
153
where qR and qL are the chiral components of q. is anomalous [6] (see also
chapter 4 [33.34]) (see [5] for an account of this). In fact. for massless quarks,
a/IfS) = N I g~ G Q GQ/L"
(5.46)
/L
16:n'2 /LIJ
where NI is the number of flavours (= 2NG). Thus. as in (4.126), such a O-term
can be removed by performing the V(I)A transformation (5.44) with
()
a=---.
(5.47)
2NI
With all quarks massless, the f)-term is not physical. since it can be removed
by an (unobservable) VO)A transformation. However. this is not the end of the
problem. since quarks are not massless. The contribution of the mass terms to the
QCD Lagrangian is
Cm = -qLiMijqRj - qRiMjjqLj
(5.48)
where i. j = I, ... , NI label the quark flavours, and Mij is the mass matrix. The
effect of the V(I)A transformation (5.44) on M is
V(l)A : M ..... e 2ia M
Mt ..... e- 2ia Mt.
(5.49)
Thus. if M was initially Hermitian. so that there are no yss in the mass terms,
it is no longer so after the transformation and the transformed Lagrangian has
reacquired the P and T non-conserving interactions which the transformation
(5.44) with a satisfying (5.47) sought to remove. The quantity 8 defined by
8 = 0 + 2N I arg(det M)
(5.50)
is invariant under V(I)A transformations and parametrizes the T-violation in the
(strong) QCD Lagrangian: 8 is the effective QCD vacuum angle in the basis
where all quark masses are real, positive and Y5 free. If non-zero, it contributes
to the neutron electric dipole moment dn and the measured upper bound on this
requires [7]
8 '" < IO-IO
(5.51 )
.
The outstanding question, then, is why 8 is so small. This is the 'strong CP
problem'.
For each value of the parameter 8, we have a different QCD theory and there
is no a priori reason why one (very small or zero) value is preferred over another.
A possible escape from this is that 8 is the expectation value of a field 8 (x), whose
VEV is determined dynamically by an effective potential, as happens when the
electroweak symmetry is spontaneously broken by the Higgs-doublet field. Then
it is conceivable that, at the minimum of the effective potential, (j = o.