%% slacpub7010: page file slacpub7010004.tcx.
%% section 4 The Physical (Pseudo)Goldstone Bosons [slacpub7010004 in slacpub7010004: slacpub7010005]
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\section{\usemenu{slacpub7010::context::slacpub7010004}{The Physical (Pseudo)Goldstone Bosons}}\label{section::slacpub7010004}
%Now consider the properties of the physical Goldstone
%bosons.
The physical
pseudoGoldstone bosons are related to the phases
%on the magnitude of contributions to the potential (\ref{vacterms})
by $\phi_{\alpha} = G_{\alpha} / f_{\alpha}$.
%where
%$\phi_{\alpha}$ is one of the two independent phases in (\ref{combinations}).
The decay constants $f_{\alpha}$ are
essentially the expectation values for fields in the
SUSY breaking sector which transform under the symmetries.
All the Goldstone bosons couple to the visible sector
through dimensionful couplings with a suppression of
$1/f$,
\beq
W = \mu e^{i Q_{\mu \alpha} \phi_{\alpha} / f_{\alpha} } H_u H_d
\eq
$$
{\cal L} =  \frac{1}{2} m_{\lambda}
e^{i Q_{\lambda \alpha} \phi_{\alpha} / f_{\alpha} }
\lambda \lambda
 A e^{i Q_{A \alpha} \phi_{\alpha} / f_{\alpha} }
\left( h_u Q H_u \bar{u}
 h_d Q H_d \bar{d}
 h_e L H_d \bar{e} \right)
$$
\beq
 m_{12}^2 e^{i Q_{ud \alpha} \phi_{\alpha} / f_{\alpha}}
H_u H_d
{}~+~ h.c.
\label{goldcouplings}
\eq
where the $Q_{i \alpha}$ depend on the global charge assignments in
the SUSY breaking sector.
For $f$ above the weak scale the Goldstone bosons are
essentially ``invisible'' to laboratory experiments.
The masses for the Goldstone bosons corresponding to SUSY CP violating
phases depend on the magnitude of (\docLink{slacpub7010003.tcx}[vacterms]{4}) and the decay
constant.
The mass for the linear combination lifted by the marginal operators
is then $m_G \sim \left( \sqrt{ \alpha / 4 \pi} \right) m_W^2 / f$,
where $\alpha/ 4 \pi$ counts the loop factor, while
the mass for the combination lifted by the relevant operator is
$m_G \sim m_W \Lambda / f$.
In the universal case with four dynamical phases,
due to the
$U(1)_{PQ}$ and $U(1)_{RPQ}$ symmetries of the $\mu$ and
soft terms, the
two linear combinations of phases orthogonal to the physical
SUSY CP violating phases do not receive a
perturbative potential from coupling
with the visible sector (\docLink{slacpub7010003.tcx}[vacterms]{4}).
One linear combination is anomaly free with respect to QCD, and
so receives a potential only from any explicit breaking from short
distance physics.
The other linear combination is anomalous and receives a
potential from the QCD topological charge density.
If this symmetry is respected by the short distance physics to
high enough order in irrelevant operators, the Goldstone boson
just acts as an invisible axion with mass
$m \sim m_{\pi} f_{\pi} / f$,
thereby solving the strong CP
problem.
The PecceiQuinn mechanism can therefore be wedded with the
proposal to solve the SUSY CP problem by postulating global
symmetries in the SUSY breaking sector.
Depending on the mechanism which transmits SUSY breaking to the
visible sector, $f$ could be anywhere between just above the
weak scale to the Planck scale.
One possibility is a renormalizable hidden sector in which SUSY is
broken in the flat space limit, but transmitted to the visible
sector by gravitational strength interactions.
The scalar expectation values in the hidden sector are
then of order the SUSY breaking
scale, $f \sim M_I \sim \sqrt{m_W M_p}$.
If, as suggested above, both the axion and the pseudoGoldstone bosons
responsible for relaxing the SUSY CP phases all arise from
spontaneously broken global symmetries in the SUSY breaking sector,
then apparently this type of hidden sector naturally
gives a decay constant
in the ``axion window'' allowed by astrophysical and cosmological
bounds on the axion \cite{26}.
{}From (\docLink{slacpub7010004.tcx}[goldcouplings]{8}) it is apparent that this axion couples
both as a hadronic axion \cite{12} through the gluino mass term and
as a DFSZ axion \cite{13} through the scalar Higgs term.
Note that the small coupling introduced in the original DFSZ
models by hand appears here automatically as $m_W/f$.
With this type of hidden sector
the pseudoGoldstone bosons associated with the SUSY CP violating
phases do not lead to excessive cooling of astrophysical systems, and
are heavier than the axion and
therefore not overproduced in the early universe.
With a renormalizable hidden sector it turns out that
$A$ terms arise only from Kahler potential couplings and
are always real \cite{27}.
In this case there are only three possible dynamical phases
in the universal case, thereby eliminating the
(potentially massless) anomaly free linear combination.
One may be tempted to identify one of the required Goldstone bosons
with the $R$ axion which plays a role in all known
models of dynamical SUSY breaking based on a nonperturbative
superpotential.
However, for a renormalizable hidden sector, cancelation of
the cosmological constant by adjustment of the superpotential, explicitly
breaks the $R$ symmetry \cite{28}.
It is also worth noting that in renormalizable hidden sector models
the heavier pseudoGoldstone boson
has a sizeable mass, $m \sim M_I$.
This is because of the sensitivity of the relevant operator
to physics at the scale $\Lambda \sim M_p$.
The mass in other types of SUSY breaking sectors is parametrically
less than the intrinsic SUSY breaking scale.
Another possibility is a nonrenormalizable hidden sector in which
SUSY is restored in the flat space limit.
In this case the fields can have $f \sim M_p$.
With a nonrenormalizable hidden sector
it is possible to cancel the cosmological constant by adjusting the
Kahler potential without explicitly violating $R$ symmetry.
It it is then possible in principle to identify
the $R$ axion with one of the Goldstone bosons.
If both the axion and pseudoGoldstone bosons arise from
spontaneously broken global symmetries in this
type of hidden sector, they will generally be overproduced
in the early universe.
They may be diluted by a period of very late inflation
however \cite{29}.
One interesting consequence of the relaxation mechanism for
the SUSY CP phases %$f \sim M_p$
is the possibility
of a long range force. %from the pseudoGoldstone
%boson which receives a mass from the relevant operators.
%$m \sim m_W^2 / M_p$.
If the minimum of the potential does not align precisely with a
CP conserving point, $\phi_0 \neq 0$ or $\pi$,
the interactions (\docLink{slacpub7010004.tcx}[goldcouplings]{8}) lead to scalar couplings
of the pseudoGoldstone bosons
to matter proportional to mass.
This gives rise to a coherent potential between macroscopic
bodies
\beq
V \simeq  g^2 \phi_0^2 \frac{m_i m_j}{4 \pi f^2} \frac{e^{mr}}{r}
\eq
where
$g \sim {\cal O}(1)$ is a dimensionless coupling which depends on the
charges appearing in (\docLink{slacpub7010004.tcx}[goldcouplings]{8}),
$\phi_0$ is the minimum of the phase potential mod $\pi$, and
$m^{1}$ is the Compton wavelength.
The (lighter) Goldstone boson, which receives a mass from the marginal
operators, gives the longest range force.
For $f \sim M_p$ the Compton wavelength is ${\cal O}(10^{1}  100)$ cm,
and is weaker than gravity by roughly the factor $\phi_0^2$.
As argued above, the shift in the potential from long distance physics
can be quite small in the universal case,
but short distance physics or nonuniversality can
in principle disturb the alignment.
Notice that for $\phi_0 \neq 0$
the magnitude of CP odd observables, such as
electric dipole moments, is correlated with the strength of the long
range force.
This may be the best laboratory signal for the relaxation mechanism.
%For a renormalizable hidden sector with $f \sim M_I \sim \sqrt{m_W M_p}$
%this gives masses of order $m_W^2 / M_I$ and $M_I$ respectively.
%The shift of the minimum of the lighter of these Goldstone bosons
%due to Planck suppressed operators which explicitly break the
%associated global symmetry {\it and} CP symmetry is
%$\delta \phi \sim f^{n}/(M_p^{n4} m_W^4)$ where $n$ is the dimension
%of the operator.
%For a renormalizable hidden sector, requiring that this not
%destroy the alignment
%implies $n > 8$.
%Although the two SUSY CP phases obtain potentials from
%the terms (\ref{vacterms}) the two remaining phases
%in (\ref{muterm}) and (\ref{softterms}) do not.
%If these phases
%DFSZ....
%small coupling to $H_u H_d$ introduced by hand, is
%just a result of the hierarchy between the weak and
%hidden sector SUSY breaking scales. a result
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