# 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

cosmology.

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 [9]. 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 [10] 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 formLax = mq qqC 'f + h.c.

(8)

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

double seesawp2

(9)

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 Lagrangianfint 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 q3

v2

M'

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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.