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Trieste Meeting of the TMR Network on Physics beyond the SM
PROCEEDINGS
Supersymmetric Quintessence
Francesca Rosati∗
SISSA, via Beirut 2-4, I-34013 Trieste, ITALY
INFN, sezione di Trieste, Padriciano 99, I-34014 Trieste, ITALY
E-mail: rosati@sissa.it
Abstract: Recent data point in the direction of a cosmological constant dominated universe. We
investigate the rˆole of supersymmetric QCD with Nf<N
cas a possible candidate for dynamical
cosmological constant (“quintessence”). We take in full consideration the multi-scalar nature of the
model, allowing for different initial conditions for the Nfindependent scalar VEVs and studying the
coupled system of Nfequations of motion. The issues related to the coupling of the scalars with other
cosmological fields are also addressed.
1. Introduction
Indications for an accelerating universe coming
from redshift-distance measurements of High-Z
Supernovae Ia (SNe Ia) [2, 3], combined with
CMB data [4] and cluster mass distribution [5],
have recently drawn a great deal of attention on
cosmological models with Ωm∼1/3andΩ
Λ∼
2/3, Ωmand ΩΛbeing the fraction densities in
matter and cosmological constant, respectively.
More generally, the rˆole of the cosmological con-
stant in accelerating the universe expansion could
be played by any smooth component with neg-
ative equation of state pQ/ρQ=wQ<
∼−0.6
[6, 7], as in the so-called “quintessence” models
(QCDM) [7], otherwise known as xCDM models
[9].
A natural candidate for quintessence is given
by a rolling scalar field Qwith potential V(Q)
and equation of state
wQ=˙
Q2/2−V(Q)
˙
Q2/2+V(Q),
which – depending on the amount of kinetic en-
ergy – could in principle take any value from −1
to +1. The study of scalar field cosmologies has
shown [10, 11] that for certain potentials there
exist attractor solutions that can be of the “scal-
ing” [12, 13, 14] or “tracker” [15, 16] type; that
∗Report on work done in collaboration with Antonio
Masiero and Massimo Pietroni [1].
means that for a wide range of initial conditions
the scalar field will rapidly join a well defined late
time behavior.
If ρQρB,whereρ
Bis the energy density
of the dominant background (radiation or mat-
ter), the attractor can be studied analytically.
In the case of an exponential potential, V∼
exp (−Q) the solution Q∼ln tis, under very
general conditions, a “scaling” attractor in phase
space characterized by ρQ/ρB∼const [12, 13,
14]. This could potentially solve the so called
“cosmic coincidence” problem, providing a dy-
namical explanation for the order of magnitude
equality between matter and scalar field energy
today. Unfortunately, the equation of state for
this attractor is wQ=wB, which cannot explain
the acceleration of the universe neither during
RD (wrad =1/3) nor during MD (wm=0).
Moreover, Big Bang nucleosynthesis constrain the
field energy density to values much smaller than
the required ∼2/3 [11, 13, 14].
If instead an inverse power-law potential is
considered, V=M4+αQ−α,withα>0, the
attractor solution is Q∼t1−n/m,where
n=3(w
Q+1),m=3(w
B+1);
and the equation of state turns out to be
wQ=wBα−2
α+2 ,
Trieste Meeting of the TMR Network on Physics beyond the SM Francesca Rosati
which is always negative during MD. The ratio
of the energies is no longer constant but scales
as ρQ/ρB∼am−nthus growing during the cos-
mological evolution, since n<m.ρ
Qcould then
have been safely small during nucleosynthesis and
have grown lately up to the phenomenologically
interesting values. These solutions are then good
candidates for quintessence and have been de-
nominated “tracker” in the literature [11, 15, 16].
The inverse power-law potential does not im-
prove the cosmic coincidence problem with re-
spect to the cosmological constant case. Indeed,
the scale Mhas to be fixed from the requirement
that the scalar energy density today is exactly
what is needed. This corresponds to choosing
the desired tracker path. An important differ-
ence exists in this case though. The initial con-
ditions for the physical variable ρQcan vary be-
tween the present critical energy density ρ0
cr and
the background energy density ρBat the time of
beginning [16] (this range can span many tens
of orders of magnitude, depending on the ini-
tial time), and will anyway end on the tracker
path before the present epoch, due to the pres-
ence of an attractor in phase space [15, 16]. On
the contrary, in the cosmological constant case,
the physical variable ρΛis fixed once for all at
the beginning. This allows us to say that in the
quintessence case the fine-tuning issue, even if
still far from solved, is at least weakened.
A great effort has recently been devoted to
find ways to constrain such models with present
and future cosmological data in order to distin-
guish quintessence from Λ models [17, 18]. An
even more ambitious goal is the partial recon-
struction of the scalar field potential from mea-
suring the variation of the equation of state with
increasing redshift [19].
On the other hand, the investigation of quin-
tessence models from the particle physics point
of view is just in a preliminary stage and a real-
istic model is still missing (see for example refs.
[20, 21, 22, 23]). There are two classes of prob-
lems: the construction of a field theory model
with the required scalar potential and the inter-
action of the quintessence field with the stan-
dard model (SM) fields [24]. The former prob-
lem was already considered by Bin´etruy [20], who
pointed out that scalar inverse power law po-
tentials appear in supersymmetric QCD theories
(SQCD) [25] with Nccolors and Nf<N
cfla-
vors. The latter seems the toughest. Indeed
the quintessence field today has typically a mass
of order H0∼10−33eV. Then, in general, it
would mediate long range interactions of grav-
itational strength, which are phenomenologically
unacceptable.
In this talk, both theese issue will be ad-
dressed, following the results obtained in ref. [1].
2. SUSY QCD
As already noted by Bin`etruy [20], supersymmet-
ric QCD theories with Nccolors and Nf<N
c
flavors [25] may give an explicit realization of
a model for quintessence with an inverse power
law scalar potential. The remarkable feature of
these theories is that the superpotential is ex-
actly known non-perturbatively. Moreover, in
the range of field values that will be relevant for
our purposes (see below) quantum corrections to
the K¨ahler potential are under control. As a con-
sequence, we can study the scalar potential and
the field equations of motion of the full quantum
theory, without limiting ourselves to the classical
approximation.
The matter content of the theory is given by
the chiral superfields Qiand Qi(i=1...N
f)
transforming according to the Ncand Ncrep-
resentations of SU(Nc), respectively. In the fol-
lowing, the same symbols will be used for the
superfields Qi,Qi, and their scalar components.
Supersymmetry and anomaly-free global sym-
metries constrain the superpotential to the unique
exact form
W=(N
c−N
f)Λ
(3Nc−Nf)
detT1
Nc−Nf
(2.1)
where the gauge-invariant matrix superfield Tij =
Qi·Qjappears. Λ is the only mass scale of
the theory. It is the supersymmetric analogue of
ΛQCD , the renormalization group invariant scale
at which the gauge coupling of SU(Nc) becomes
non-perturbative. As long as scalar field values
Qi, QiΛ are considered, the theory is in the
weak coupling regime and the canonical form for
the K¨ahler potential may be assumed. The scalar
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Trieste Meeting of the TMR Network on Physics beyond the SM Francesca Rosati
and fermion matter fields have then canonical ki-
netic terms, and the scalar potential is given by
V=
Nf
X
i=1 |FQi|2+|FQi|2+1
2DaDa(2.2)
where FQi=∂W/∂Qi,FQi=∂W/∂Qi,and
D
a=Q
†
i
t
a
Q
i−Q
i
t
a
Q
†
i.(2.3)
The relevant dynamics of the field expectation
values takes place along directions in field space
in which the above D-term vanish, i.e. the per-
turbatively flat directions hQiαi=hQ†
iαi,where
α=1···N
cis the gauge index. At the non-
perturbative level these directions get a non van-
ishing potential from the F-terms in (2.2), which
are zero at any order in perturbation theory.
Gauge and flavor rotations can be used to
diagonalize the hQiαiand put them in the form
hQiαi=hQ†
iαi=qiδiα 1≤α≤Nf
0Nf≤α≤Nc
.
Along these directions, the scalar potential is given
by
v(qi)≡hV(Q
i
,Q
i
)i=2Λ
2a
QN
f
i=1 |qi|4d
Nf
X
j=1
1
|qj|2
,
with
a=3Nc−Nf
Nc−Nf
,d=1
N
c
−N
f
.
In the following, we will be interested in the cos-
mological evolution of the Nfexpectation values
qi,givenby
h¨
Q
i+3H˙
Q
i+∂V
∂Q†
i
i=0 ,i=1, ..., Nf.
In Ref. [20] the same initial conditions for all the
NfVEV’s and their time derivatives were cho-
sen. With this very peculiar choice the evolution
of the system may be described by a single VEV
q(which we take real) with equation of motion
¨q+3H˙q−gΛ
2a
q
2g+1 =0,g=
N
c
+N
f
N
c
−N
f
,(2.4)
thus reproducing exactly the case of a single scalar
field Φ in the potential V=Λ
4+2gΦ−2g/2 consid-
ered in refs. [10, 11, 16]. We will instead consider
the more general case in which different initial
conditions are assigned to different VEV’s, and
the system is described by Nfcoupled differen-
tial equations. Taking for illustration the case
Nf= 2, we will have to solve the equations
¨q1+3H˙q
1−d·q
1Λ
2a
(q
1
q
2
)
2dNc2+N
c
q
2
2
q
2
1=0,
¨q
2+3H˙q
2−d·q
2Λ
2a
(q
1
q
2
)
2dNc2+N
c
q
2
1
q
2
2=0(2.5)
with H2=8π/3M2
P(ρm+ρr+ρQ), where MPis
the Planck mass, ρm(r)is the matter (radiation)
energy density, and ρQ=2(˙q
2
1+˙q
2
2
)+v(q
1,q
2)is
the total field energy.
3. The tracker solution
In analogy with the one-scalar case, we look for
power-law solutions of the form
qtr,i =Ci·tpi,i=1,···,N
f.(3.1)
It is straightforward to verify that – when ρQ
ρB– the only solution of this type is given, for
i=1,···,N
f
,by
p
i≡p=1−r
2,C
i
≡C=
h
X
1−r
Λ
2(3−r)i1/4,
with
X≡4m(1 + r)
(1 −r)2[12 −m(1 + r)] ,
where we have defined
r≡Nf
Nc=1
Nc
,...,1−1
N
c.
This solution is characterized by an equation of
state
wQ=1+r
2w
B−1−r
2.(3.2)
Following the same methods employed in ref.
[11] one can show that the above solution is the
unique stable attractor in the space of solutions
of eqs. (2.5). Then, even if the qi’s start with dif-
ferent initial conditions, there is a region in field
configuration space such that the system evolves
towards the equal fields solutions (3.1), and the
late-time behavior is indistinguishable from the
case considered in ref. [20].
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Trieste Meeting of the TMR Network on Physics beyond the SM Francesca Rosati
The field energy density grows with respect
to the matter energy density as
ρQ
ρm
∼a3(1+r)
2,(3.3)
where ais the scale factor of the universe. The
scalar field energy will then eventually dominate
and the approximations leading to the scaling so-
lution (3.1) will drop, so that a numerical treat-
ment of the field equations is mandatory in order
to describe the phenomenologically relevant late-
time behavior.
The scale Λ can be fixed requiring that the
scalar fields are starting to dominate the energy
density of the universe today and that both have
already reached the tracking behavior. The two
conditions are realized if
v(q0)'ρ0
crit ,v
00(q0)'H2
0,(3.4)
where ρ0
crit =3M
2
PH
2
0/8πand q0are the present
critical density and scalar fields VEV respectively.
Eqs. (3.4) imply
Λ
MP
'3(1 + r)(3 + r)
4π(1 −r)2rNc1+r
2(3−r)1
2rNc
ρ0
crit
M4
P1−r
2(3−r)
q2
0
M2
P
'3
4π
(1 + r)(3 + r)
(1 −r)2
1
rNc
.(3.5)
Depending on the values for Nfand Nc,Λ
and q0/Λ assume widely different values. Λ takes
its lowest possible values in the Nc→∞(N
f
fixed) limit, where it equals 4 ·10−2(h2/N 2
f)1/6
GeV (we have used ρ0
crit/M 4
P=(2.5·10−31h1/2)4).
For fixed Nc, instead, Λ increases as Nfgoes
from 1 to its maximum allowed value, Nf=
1−Nc.ForN
c>
∼
20 and Nfclose to Nc,the
scale Λ exceeds MP.
The accuracy of the determination of Λ given
in (3.5) depends on the present error on the mea-
surements of H0,i.e., typically,
δΛ
Λ=1−r
3−r
δH0
H0
<
∼0.1.
In deriving the scalar potential (2.2) and the
field equations (2.5) we have assumed that the
system is in the weakly coupled regime, so that
the canonical form for the K¨ahler potential may
be considered as a good approximation. This
condition is satisfied as long as the fields’ VEVs
are much larger than the non-perturbative scale
Λ. From eq. (3.5) one can compute the ratio
between the VEVs today and Λ, and see that
it is greater than unity for any Nfas long as
Nc<
∼20.
4. Interaction with the visible sector
The superfields Qiand Qihave been taken as sin-
glets under the SM gauge group. Therefore, they
may interact with the visible sector only gravi-
tationally, i.e. via non-renormalizable operators
suppressed by inverse powers of the Planck mass,
of the form
Zd4θK
j
(φ
†
j
,φ
j)·βji "Q†
iQi
M2
P#,(4.1)
where φjrepresents a generic standard model
superfield. From (3.5) we know that today the
VEV’s qiare typically O(MP), so there is no rea-
son to limit ourselves to the contributions of low-
est order in |Q|2/M 2
P. Rather, we have to con-
sider the full (unknown) functions β’s and the
analogous β’s for the Qi’s. Moreover, the re-
quirement that the scalar fields are on the track-
ing solution today, eqs. (3.4), implies that their
mass is of order ∼H2
0∼10−33 eV.
The exchange of very light fields gives rise to
long-range forces which are constrained by tests
on the equivalence principle, whereas the time
dependence of the VEV’s induces a time varia-
tion of the SM coupling constants [24, 27]. These
kind of considerations set stringent bounds on
the first derivatives of the βji’s and βji’s today,
αji ≡dlog βji x2
i
dxixi=x0
i
,
αji ≡dlog βji x2
i
dx
i
x
i=x
0
i
,
where xi≡qi/MP. To give an example, the best
bound on the time variation of the fine structure
constant comes from the Oklo natural reactor.
It implies that |˙α/α|<10−15 yr−1[28], leading
to the following constraint on the coupling with
the kinetic terms of the electromagnetic vector
superfield V,
αVi
, α
Vi <
∼10−6H0
h˙qiiMP,(4.2)
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Trieste Meeting of the TMR Network on Physics beyond the SM Francesca Rosati
Figure 1: The evolution of the energy densities ρof different cosmological components is given as a funcion
of red-shift. All the energy densities are normalized to the present critical energy density ρ0
cr. Radiation and
matter energy densities are represented by the short-dashed lines, whereas the solid line is the energy density
of the tracker solution discussed in Section 3. The long-dashed line is the evolution of the scalar field energy
density for a solution that reaches the tracker before the present epoch; while the dash-dotted line represents
the evolution for a solution that overshoots the tracker to such an extent that it has not yet had enough time
to re-join the attractor.
where h˙qiiis the average rate of change of qiin
the past 2 ×109yr.
Similar –although generally less stringent–
bounds can be analogously obtained for the cou-
pling with the other standard model superfields
[27]. Therefore, in order to be phenomenologi-
cally viable, any quintessence model should pos-
tulate that all the unknown couplings βji’s and
βji’s have a common minimum close to the ac-
tual value of the qi’s1.
The simplest way to realize this condition
would be via the least coupling principle intro-
duced by Damour and Polyakov for the massless
superstring dilaton in ref. [26], where a universal
coupling between the dilaton and the SM fields
was postulated. In the present context, we will
invoke a similar principle, by postulating that
βji =βand βji =βfor any SM field φjand any
flavor i. For simplicity, we will further assume
β=β.
The decoupling from the visible sector im-
plied by bounds like (4.2) does not necessarily
1An alternative way to suppress long-range interac-
tions, based on an approximate global symmetry, was
proposed in ref. [24].
mean that the interactions between the quintes-
sence sector and the visible one have always been
phenomenologically irrelevant. Indeed, during
radiation domination the VEVs qiwere typically
MPand then very far from the postulated
minimum of the β’s. For such values of the qi’s
the β’s can be approximated as
βQ†Q
M2
P=β0+β1
Q†Q
M2
P
+... (4.3)
where the constants β0and β1are not directly
constrained by (4.2). The coupling between the
(4.3) and the SM kinetic terms, as in (4.1), in-
duces a SUSY breaking mass term for the scalars
of the form [29]
∆L∼H2β1X
i
(|Qi|2+Qi
2),(4.4)
where we have used the fact that during radiation
domination DPjRd4θKj(φ†
j,φ
j)
E∼ρ
rad.
If present, this term would have a very inter-
esting impact on the cosmological evolution of
the fields. First of all one should notice that, un-
like the usual mass terms with time-independent
masses considered in [22], a mass m2∼H2does
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Trieste Meeting of the TMR Network on Physics beyond the SM Francesca Rosati
Figure 2: The effect of taking different initial conditions for the fields, at the same initial total field energy.
Starting with qin
1/qin
2=10
15 the tracker behaviour is not reached today. For comparison we plot the solution
for qin
1/qin
2=1.
not modify the time-dependence of the tracking
solution, which is still the power-law given in eq.
(3.1). Thus, the fine-tuning problems induced
by the requirement that a soft-supersymmetry
breaking mass does not spoil the tracking solu-
tions [22] are not present here.
Secondly, since the Qand Qfields do not
form an isolated system any more, the equation
of state of the quintessence fields is not linked to
the power-law dependence of the tracking solu-
tion. Taking into account the interaction with
the SM fields, represented by H2, we find the
new equation of state during radiation domina-
tion (wB=1/3)
w0
Q=wQ−4β1
1+r
9(1 −r)+6β
1
where wQwas given in eq. (3.2).
From a phenomenological point of view, the
most relevant effect of the presence of mass terms
like (4.4) during radiation domination resides in
the fact that they rise the scalar potential at large
fields values, inducing a (time-dependent) min-
imum. In absence of such terms, if the fields
are initially very far from the origin, they are
not able to catch up with the tracking behav-
ior before the present epoch, and ρQalways re-
mains much smaller than ρB. These are the well-
known ‘undershoot’ solutions considered in ref.
[16]. Instead, when large enough masses (4.4)
are present, the fields are quickly driven towards
the time-dependent minimum and the would-be
undershoot solutions reach the tracking behavior
in time.
The same happens for the would-be ‘over-
shoot’ solutions, [16], in which the fields are ini-
tially very close to the origin, with an energy den-
sity much larger than the tracker one, and are
subsequently pushed to very large values, from
where they will not be able to reach the tracking
solution before the present epoch. Introducing
mass terms like (4.4) prevents the fields to go to
very large values, and keeps them closer to the
traking solution.
In other words, the already large region in
initial condition phase space leading to late-time
tracking behavior, will be enlarged to the full
phase space. In the next section we will discuss
numerical results with and without the super-
symmetry breaking mass (4.4).
5. Numerical results
In this section we illustrate the general results
of the previous sections for the particular case
Nf=2,N
c=6.
In Fig.1 the energy densities vs. redshift are
given. We have chosen the same initial condi-
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Trieste Meeting of the TMR Network on Physics beyond the SM Francesca Rosati
Figure 3: The effect of the interaction with other fields, to be compared with Fig. 1. Adding a term like eq.
(4.4) with β1=0.3 the would-be overshooting solution (dash-dotted line) reaches the tracker in time.
tions for the two VEVs, in order to effectively
reproduce the one-scalar case of eq. (2.4), al-
ready studied in refs. [10, 11, 16]. No interaction
with other fields of the type discussed in the pre-
vious section has been considered.
We see that, depending on the initial energy
density of the scalar fields, the tracker solution
may (long dashed line) or may not (dash-dotted
line) be reached before the present epoch. The
latter case corresponds to the overshoot solutions
discussed in ref. [16], in which the initial scalar
field energy is larger than ρBand the fields are
rapidly pushed to very large values. The under-
shoot region, in which the energy density is al-
ways lower than the tracker one, corresponds to
ρ0
cr ≤ρin
Q≤ρin
tr . Thus, all together, there are
around 35 orders of magnitude in ρin
Qat redshift
z+1 = 10
10 for which the tracker solution is
reached before today. Clearly, the more we go
backwards in time, the larger is the allowed ini-
tial conditions range.
Next, we explore to which extent the two-
field system that we are considering behaves as a
one scalar model with inverse power-law poten-
tial. We have found that, given any initial energy
density such that – for qin
1/qin
2= 1 – the tracker
is joined before today, there exists always a lim-
iting value for the fields’ difference above which
the attractor is not reached in time. In fig. 2 we
plot solutions with the same initial energy den-
sity but different ratios between the initial values
of the two scalar fields.
The effect of the interaction with other fields
discussed in Section 4 is shown in Fig.3. Here,
we have included the mass term (4.4) during ra-
diation domination with β1=0.3 and we have
followed the same procedure as for Fig.1, search-
ing for undershoot and overshoot solutions. The
range of initial energy densities for the solutions
reaching the tracker is now enormously enhanced
since, as we discussed previously, the fields are
now prevented from taking too large values. The
same conclusion holds even if different initial con-
ditions for the two fields are allowed, for the same
reason.
Acknowledgments
I thank Antonio Masiero and Massimo Pietroni
with whom the results reported in this talk were
obtained.
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Trieste Meeting of the TMR Network on Physics beyond the SM Francesca Rosati
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8