Evolving Dark Energy with w =/ -1 Page: 4 of 5
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To obtain a small value for p4, we require f' < f M.
The pseudo-Goldstone boson 0'+(f'/f)o then acquires a
mass p2/f' Ho, while 0- (f'/f)o' remains an exactly
massless Goldstone boson. Therefore, at this point there
is no candidate for the dynamical quintessence field.
The situation is radically altered if some additional ex-
plicit symmetry breaking interaction is added, V, giving
a mass to 0' that is > p2/f'. In this case the determinant
of the pseudo-Goldstone-boson mass matrix no longer
vanishes, so that the previously massless Goldstone boson
acquires a mass from (6): m9 p2/f Ho. Thus
dynamical quintessence theories naturally emerge from
theories having the explicit symmetry breaking structure
U(1)0 x U(1)x -> U(1)+x A 0, (7)
with the mass of the dark energy field emerging at the
final stage of explicit symmetry breaking.
The form of V is itself highly constrained, since
radiative corrections involving both A and V must not
introduce further operators that give a large mass to 0.
To avoid this, the explicit symmetry breaking parameter
in V should be dimensionful. For example, the case of
G X4 and V yX4 + h.c. clearly does not work.1
4. Hidden Axions and Seesaw Cosmology To
illustrate these ideas, and to see how seesaw cosmology
can solve the "Dark Energy Why Now?" problem,
we consider models with an axion in a hidden sector.
Quintessence axions have been considered previously for
dark energy [11, 12], but not in the context of seesaw
The general idea is as follows. Suppose that the
fundamental scale of supersymmetry breaking in nature
is of order of the TeV scale, v. Any sector of the
theory that feels this supersymmetry breaking only
indirectly via gravity mediation will have an effective
scale of supersymmetry breaking at the seesaw scale
T v2/M. We suppose that such a hidden sector has
a supersymmetric QCD-like gauge interaction acting on
chiral superfields Q and QC. Supersymmetry breaking
1 An important question is whether theories of the form
my vzvfe'Iu/fu lead to acceptable potentials for dark energy
once the three neutrino fields vi are integrated out. If n,, are
treated as parameters, one obtains a potential of the form of (6)
with p identified as m . This would be a very interesting
understanding of the size of dark energy. However, the simplest
such theories do not work: the neutrino mass is not a parameter
but depends on electroweak symmetry breaking m = m~ (h),
and radiative corrections above the weak scale with internal
Higgs fields, h, destroy the radiative stability of the potential.
The schizon models of  avoid this by introducing multiple
Higgs doublets at the weak scale. But, even in this case, the
mass parameters that mix the various Higgs doublets must be
set to the weak scale by hand they cannot arise from vacuum
expectation values of other fields. The successful supersymmetric
prediction for the weak mixing angle is also destroyed.
leads to the corresponding squarks and gluinos acquiring
a mass of order m, changing the beta function for the
gauge coupling and triggering strong dynamics at a scale
A not far below m. A simple example for this behavior
arises if the hidden sector is in a conformal window
above mh. We assume that supersymmetry breaking also
triggers a mass term for the quarks. If this sector has a
Peccei-Quinn symmetry spontaneously broken at f near
the Planck scale, then the interaction between the axion,
0, and the quarks at the scale A has the form
Lax = mq qqC 'f + h.c.
so that, comparing with (5), AG = mqqqc. The U(1)O
symmetry is the Peccei-Quinn symmetry, U(1)pQ, and is
broken near the Planck scale, while the U(1), symmetry
is the axial U(1) symmetry, U(1)A, carried by the
quark bilinear qqC. The interaction (8) explicitly breaks
U(1)pQ x U(1)A to the diagonal subgroup. We assume
that the mass of at least one quark flavor in (8) is $ A,
so that a condensate forms, (qqc) A3e&'/A, generating
the potential (6) with 0' becoming the hidden sector i'
and f' A.
The additional explicit symmetry breaking necessary
for a naturally light quintessence field, V in (7), is auto-
matic: it is the gauge anomaly that breaks U(1)A giving
the n' a mass of order A. Since this explicit symmetry
breaking comes from an anomaly and involves the scale
A, unlike dimensionless symmetry breaking parameters,
it does not lead to further radiative instabilities of the
mass of the dark energy field. The axion field 0 is the
dark energy field, and obtains a mass from the potential
(6) with p4 mqA3. Since A and mg are both close to
T, the scale p is given by the seesaw p Th m v2/M,
solving the "Dark Energy Why Now?" problem. The
then leads to the desired result (2) for a seesaw cosmology
solution of the "Quintessence Why Now?" problem.
It is straightforward to write a complete set of inter-
actions for the above hidden sector. As an example,
consider the supersymmetric interaction Lagrangian
fint 2 (sS
f2) + ZWW)
SJd40 (ZMSQQ + ZZ(QQ+QctQc)) (10)
where all coupling constants, color and flavor indices have
been omitted. The chiral superfield Z is the spurion
for supersymmetry breaking with Fz/M - m v2/M.
The interactions of (10) possess U(1)pQ x U(1)B x U(1)R
symmetry, where U(1)B is the baryon symmetry acting
on Q and QC and U(1)R the R symmetry under which q
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Hall, Lawrence J.; Nomura, Yasunori & Oliver, Steven J. Evolving Dark Energy with w =/ -1, article, March 31, 2005; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc928064/m1/4/: accessed April 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.