SUSY breaking by a metastable ground state: Why the early Universe preferred the non-supersymmetric vacuum

0.23 MB·English

by  Steven A. Abel

hep-th/0610334
PreprinttypesetinJHEPstyle-PAPERVERSION
IPPP/06/76
DCPT/06/152
7
0
0 SUSY breaking by a metastable ground state: Why the early
2

n Universe preferred the non-supersymmetric vacuum
a
J

6
1

3
v
4 Steven A. Abel, Chong-Sun Chu, Joerg Jaeckel and Valentin V. Khoze
3
3 Centre for Particle Theory, University of Durham, Durham, DH1 3LE, UK
0
[email protected], [email protected], [email protected],
1
6 [email protected]
0
/
h
t
-
p
e Abstract:Supersymmetrybreakinginametastablevacuumisre-examinedinacosmologicalcontext.
h
It is shown that thermal effects generically drive the Universe to the metastable minimum even if it
:
v begins in the supersymmetry-preservingone. This is a generic feature of the ISS models of metastable
i
X supersymmetry breaking due to the fact that SUSY preserving vacua contain fewer light degrees of
r freedom than the metastable ground state at the origin. These models of metastable SUSY breaking
a
are thus placed on an equal footing with the more usual dynamical SUSY breaking scenarios.
1. Introduction
Ithas recently been noted that dynamical supersymmetrybreakingin a long-lived metastable vacuum
isaninterestingalternativetotheusualdynamicalsupersymmetrybreakingscenario[1]1. Itwasshown
that a microscopic asymptotically free theory can, for certain choices of parameters, be described by
a Seiberg dual theory in the infra-red which has a metastable minimum at the origin. Moreover, the
global supersymmetric minima are also located where the macroscopic theory is well under control,
anditcould beshownthatthetunnellingratetothetrueglobal minimacould bemadeparametrically
small. This class of theories has recently attracted a lot of interest [3–14]. It opens up new avenues
for SUSY phenomenology, but one open question is that of naturalness; can one explain the fact that
the Universe today resides in a metastable minimum? Are these models more, or less, natural than
the usual scenario of dynamical supersymmetry breaking?
Inthispaperwepointoutthatendingupinsuchametastableminimumisinfactverynaturaland
even generic in the context of a thermal Universe. The theory at the metastable minimum has (quite
generally)moremasslessandnearlymasslessdegreesoffreedomthanatthesupersymmetricminimum,
an unusualconsequence of the fact that the SUSYpreserving vacua appear non-perturbatively2 in the
Intriligator-Seiberg-Shih (ISS) model. We will show that due to these relatively light states, at high
temperatures (greater than thesupersymmetrybreakingscale), the metastable minimumat the origin
becomestheglobalminimum. Moreover, wewillseethateven ifthefieldsbegininthesupersymmetric
minimum,highenoughtemperatures willthermally drivethem totheSUSYbreakingminimumwhere
they remain trapped as the Universe cools. More precisely, we consider a scenario where, at the end
of inflation, the Universe is very cold and we may assume that it is in the energetically preferred
supersymmetric vacuum. Then the Universe reheats. If the temperature is high enough it will then
automatically evolve towards the SUSY breaking state as we will show in the following. Thus these
models offer an alternative and appealing explanation for why supersymmetry is broken: the early
Universe was driven to a supersymmetry breaking, metastable minimum by thermal effects.3
In general we think of the settings where the ISS model forms a sector of the full theory which
includes a supersymmetric Standard Model (MSSM). Supersymmetry of the full theory is broken if
theISSsector endsupinthemetastable non-supersymmetricvacuum. For thefulltheory tobedriven
tothis SUSYbreakingvacuumby thermaleffects we, of course, have to assumethatthe relevant fields
of the supersymmetry breaking sector (ISS), of the MSSM sector, and of the messenger sector were at
some time in thermal equilibrium. This is the case if the SUSY breaking scenario is gauge mediation,
directmediation, or even avisible sector breaking. (Onthe other hand,a SUSYbreaking sector which
couples to the MSSM only gravitationally would remain out of thermal equilibrium.)
1In a simple modified O’Raifeartaigh model the possibility of SUSY breaking in a metastable vacuum has been
considered already a while ago in [2]. It is interesting to note that one of the motivations for this work was that the
Universemay get stuck in such a metastable state.
2In other words, supersymmetry is restored dynamically.
3If the initial conditions are such that the Universe starts in the supersymmetry breaking state (the case considered
in Refs. [15,16] subsequentto this paper), it will stay there (cf. also [1,17]).
– 1 –
The fact that the Universe can be driven to a metastable minimum by thermal effects is well
known in the context of charge and colour breaking minima in the MSSM [18,19]. However these
models differ from our case in two respects. First, it is in the global minimum away from the origin
that a symmetry, namely supersymmetry, is being restored in our scenario. In addition the effective
potential is extremely flat because the SUSY preserving true vacua are generated dynamically. It is
thereforeautomatically very sensitive tofinitetemperatureeffects, whilsthavingametastable vacuum
that is extremely long lived.
In this paper it will be sufficient to consider the ISS metastable SUSY breaking sector on its own.
The thermal transition to the non-supersymmetric vacuum will be driven by the above mismatch
between the numbers of light degrees of freedom in the two vacua of the ISS model.
2. The ISS model at zero temperature
2.1 Set-up of the model
TheIntriligator-Seiberg-Shih model[1]isdescribedbyasupersymmetricSU(N)gaugetheorycoupled
toN flavoursofchiralsuperfieldsϕc andϕ˜i transforminginthefundamentalandtheanti-fundamental
f i c
representations of the gauge group; c = 1,...,N and i = 1,...,N . There is also an N N chiral
f f f
×
superfield Φi which is a gauge singlet. The number of flavours is taken to be large, N > 3N, such
j f
that the β-function for the gauge coupling is positive,
b = 3N N < 0 (2.1)
0 f
−
the theory is free in the IR and strongly coupled in the UV where it develops a Landau pole at the
energy-scale Λ . The Wilsonian gauge coupling is
L
E Nf−3N
e−8π2/g2(E) = (2.2)
Λ
(cid:18) L(cid:19)
The condition (2.1) ensures that the theory is weakly coupled at scales E Λ , thus its low-energy
L
≪
dynamics as well as the vacuum structure is under control. In particular, this guarantees a robust
understanding of the theory in the metastable SUSY breaking vacuum found in [1]. This is one of the
key features of the ISS model(s).
One notes that this formulation of the theory can only provide a low-energy effective field theory
description due to the lack of the asymptotic freedom in (2.1),(2.2). At energy scales of order Λ
L
and above, this effective description breaks down and one should use instead a different (microscopic)
description of the theory, assuming it exists. Fortunately – and this is the second key feature of the
ISS construction [1] – the ultraviolet completion of this effective theory is known and is provided by
its Seiberg dual formulation [20–22]. The microscopic description of the ISS model is = 1 super-
N
symmetric QCD with the gauge group SU(N ) and N flavours of fundamental and anti-fundamental
c f
– 2 –
quarksQ and Q˜i.Thenumberof colours in themicroscopic theory isN = N N andthenumberof
i c f
−
flavours N is thesame in both descriptions. It is required to bein the range N +1 N < 3N . The
f c ≤ f 2 c
lower limit on N ensures that Eq. (2.1) holds, while the bmicro coefficient of the microscopic theory is
f 0
positive (the β-function is negative and the microscopic theory is asymptotically free)
3
bmicro = 3N N , N < bmicro 2N 1 (2.3)
0 c − f 2 c 0 ≤ c −
This microscopic formulation of the ISS model will be referred to as the SU(N ), N microscopic
c f
Seiberg dual. It is weakly coupled in the UV and strongly coupled in the IR. However, we already
mentioned that the vacuum structure of the theory and the supersymmetry breaking by a metastable
vacuum should be considered in the low-energy effective description of the theory, which is IR free.
This description of the ISS model is known as the SU(N), N macroscopic Seiberg dual formulation.
f
From now on we will always assume this macroscopic dual description of the ISS model. For the
purposes of this paper, the microscopic dual description is only necessary to guarantee the existence
of the theory above the Landau pole Λ .
L
The tree-level superpotential of the ISS model is given by
W = hTr ϕΦϕ˜ hµ2Tr Φ (2.4)
cl Nf − Nf
whereh andµ are constants. Theusualholomorphicity arguments imply thatthe superpotential(2.4)
receives no corrections in perturbation theory. However, there is a non-perturbative contribution to
the full superpotential of the theory, W = W +W , which is generated dynamically. W was
cl dyn dyn
determined in [1] and is given by
1
det Φ N
W = N hNf Nf (2.5)
dyn ΛNf−3N!
L
This dynamical superpotential is exact, its form is uniquely determined by the symmetries of the
theory and it is generated by instanton-like configurations.
The authors of [1] have studied the vacuum structure of the theory and established the existence
of the metastable vacuum vac characterised by
+
| i
1l
ϕ = ϕ˜T = µ N , Φ = 0 , V = (N N)h2µ4 (2.6)
+ f
h i h i 0Nf−N ! h i − | |
where V is the classical energy density in this vacuum. Supersymmetry is broken since V > 0. In
+ +
this vacuum the SU(N) gauge group is Higgsed by the vevs of ϕ and ϕ˜ and the gauge degrees of
freedom are massive with m = gµ.
gauge
This supersymmetry breaking vacuum vac originates from the so-called rank condition, which
+
| i
implies that there are no solutions to the F-flatness equation F = 0 for the classical superpotential
Φj
i
W in (2.4),
cl
F = h(ϕ˜jϕc µ2δJ) = 0 (2.7)
Φji c i − i 6
– 3 –
The non-perturbative superpotential of (2.5) gives negligible contributions to the effective potential
around this vacuum and can be ignored there. It was further argued in [1] that the vacuum (2.6) has
no tachyonic directions, is classically stable, and quantum-mechanically is long-lived.
InadditiontothemetastableSUSYbreakingvacuum vac ,theauthorsof[1]havealsoidentified
+
| i
the SUSY preserving stable vacuum4 vac ,
0
| i
ϕ = ϕ˜T = 0 , Φ = Φ = µγ 1l , V = 0 (2.8)
h i h i h i 0 0 Nf 0
where V is the energy density in this vacuum and
0
Nf−3N −1 µ
γ0 = hǫ Nf−N , and ǫ := 1 (2.9)
Λ ≪
(cid:18) (cid:19) L
Thisvacuumwasdeterminedin[1]bysolvingtheF-flatness conditionsforthecompletesuperpotential
W = W +W of the theory. In the vicinity of vac the non-perturbative superpotential of (2.5)
cl dyn 0
| i
is essential as it alleviates the rank condition (2.7) and allows to solve F = 0 equations. Thus,
Φj
i
the appearance of the SUSY preserving vacuum (2.8) can be interpreted in our macroscopic dual
description as a non-perturbative or dynamical restoration of supersymmetry [1].
When in the next Section we putthe ISS theory at high temperature it will beimportant to know
the number of light and heavy degrees of freedom of the theory as we interpolate between vac and
+
| i
vac . The macroscopic description of the ISS model is meaningful as long as we work at energy
0
| i
scales much smaller than the cut-off scale Λ . The requirement that ǫ = µ/Λ 1 is a condition
L L
≪
that the gauge theory is weakly coupled at the scale µ which is the natural scale of the macroscopic
theory. Equation (2.9) implies that γ 1 or in other words that there is a natural separation of
0
≫
scales µ Φ = µγ Λ dictated by ǫ 1. The masses of order hΦ will be treated as heavy and
0 0 L 0
≪ ≪ ≪
all other masses suppressed with respect to hΦ by a positive power of ǫ we will refer to as light. In
0
the SUSYpreserving vacuum vac the heavy degrees of freedom are the N flavours of ϕ and ϕ˜, they
0 f
| i
get the tree-level masses m = hΦ via (2.4). All other degrees of freedom in vac are light.5
ϕ 0 0
| i
In the vacuum vac all of the original degrees of freedom of the ISS theory are light. Hence we
+
| i
note for future reference that the SUSY preserving vacuum vac has fewer light degrees of freedom
0
| i
than the SUSY breaking vacuum vac . The mismatch is given by the N flavours of ϕ and ϕ˜. The
+ f
| i
factthatthemetastableSUSYbreakingvacuumhasfewerdegreesoffreedomthanthesupersymmetric
vacuum is an important feature of the ISS model. As we will explain below, this property will give us
a necessary condition for the thermal Universe to end up in the metastable vacuum in the first place.
4Infact thereare precisely Nf −N =Nc ofsuch vacuadifferingbythephasee2πi/(Nf−N) asrequiredbytheWitten
index of themicroscopic Seiberg dual formulation.
5Gauge degrees of freedom in |vaci are confined, but the appropriate mass gap, given by the gaugino condensate,
0
is parametrically (in ǫ) smaller than m = hΦ . Thus the gauge degrees of freedom can still be counted as light for
ϕ 0
sufficiently high temperatures.
– 4 –
2.2 Effective potential
Having presented the general set-up, let us now turn to the effective potential between the two vacua
vac and vac of theISSmodel. This can bedetermined in its entirety as follows. We parameterise
+ 0
| i | i
the path interpolating between the two vacua (2.6) and (2.8) in field space via
1l
ϕ(σ) = ϕ˜T(σ) = σµ N , Φ(γ)= γµ1l , 0 γ γ , 1 σ 0 (2.10)
0Nf−N ! Nf ≤ ≤ 0 ≥ ≥
Since the Ka¨hler potential in the free magnetic phase in the IR is that of the classical theory, the
effective potential V can be determined from knowing the superpotential of the theory (2.4), (2.5).
First we will calculate the classical potential V along path (2.10) and ignoring the non-perturbative
cl
contribution W . Using (2.4) we find,
dyn
1
V (γ,σ) = Tr Φϕ2+Tr ϕ˜Φ 2+Tr ϕ˜ ϕ µ2δ 2 = µ 4(2Nγ2σ2+N(σ2 1)2+N N)
h2 cl Nf| | Nf| | Nf| i j − ij| | | − f −
| |
(2.11)
This expression for V (γ,σ) is extremized for σ = 1 γ2 or σ = 0. We substitute the first solution
cl
−
for σ into V for the range of 0 γ 1, and the second solution, σ = 0 for 1 γ γ . We thus
cl ≤ ≤ p ≤ ≤ 0
obtain
1 N N + 2Nγ2(1 1γ2) 0 γ 1
V (γ) = f − − 2 ≤ ≤ (2.12)
h2µ4 cl N 1 γ γ
( f 0
| | ≤ ≤
This potential is accurate for γ 1, where the potential is rising. In order to address the flat regime
≤
1 γ γ we need to include the contribution of the non-perturbative W . This gives
0 dyn
≤ ≤
1 2
V = Tr ∂ N hNf detNfΦ N hµ2δ = h2µ4 N γ NfN−N 1 2 (2.13)
Nf (cid:12)(cid:12)∂Φij ΛNLf−3N! − ij(cid:12)(cid:12) | | f(cid:18) γ0 − (cid:19)
(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
Combining these ex(cid:12)pressions we have the final result f(cid:12)or the effective potential interpolating between
(cid:12) (cid:12)
the two vacua:
N N + 2Nγ2(1 1γ2) 0 γ 1
f − − 2 ≤ ≤
1
Vˆ (γ) V(γ) =  (2.14)
T=0 ≡ |h2µ4| T=0  Nf γγ0 NfN−N −1 2 1 ≤ γ
(cid:18) (cid:19)
 (cid:0) (cid:1)

For our forthcoming thermal applications we have included the subscript T = 0 to indicate that this
effective potential is calculated at zero temperature. We plot the zero-temperature effective potential
(2.14) in Figure 1. The key features of this effective potential are (1) the large distance between the
two vacua, γ 1, and (2) the slow rise of the potential to the left of the SUSY preserving vacuum.
0
≫
(For esthetic reasons γ in Figure 1. is actually chosen to be rather small, γ = γ = 7.5.)
0 0
We note that thematching between the two regimes at γ = 1 in (2.14) holds upto an insignificant
Nf−N
1/γ N correction which is small since γ 1. This tiny mismatch (which can be seen in Figure 1.)
0 0 ≫
– 5 –
10
8
6
4
2
0 2 4 6 8 10
Figure 1: Zero temperature effective potential Vˆ (γ) of Eq. (2.14) as a function of γ =Φ/µ. For the SUSY
T=0
preserving vacuum vac we chose γ =γ =7.5. The SUSY breaking metastable minimum vac is always at
0 0 +
| i | i
γ =0, and the top of the barrier is always at γ =1. We have taken the minimal allowed values for N and N ,
f
N =2, N =7.
f
can be easily corrected in the derivation of V by using the full superpotential in both regimes, rather
than neglecting W at γ 1. However, it will not change anything in our considerations. Other
dyn
≤
corrections which we have dropped from the final expression include perturbative corrections to the
Kahler potential. These effects are also expected to be small at scales much below the cut-off Λ .
L
The authors of [1] have already estimated the tunnelling rate from the metastable vac to the
+
| i
supersymmetric vacuum vac by approximating the potential in Figure 1 in terms of a triangle. The
0
| i
action of the bounce solution in the triangular potential is of the form,
2π2 N3 Φ 4 h−6
S4D = 0 1 for ǫ 1 (2.15)
bounce 3h2 Nf2 (cid:18) µ (cid:19) ∼ ǫ4(Nf−3N)/(Nf−N) ≫ ≪
Thus as argued in [1] it is always possible, by choosing sufficiently small ǫ, to ensure that the decay
time of the metastable vacuum to the SUSY ground state is much longer than the age of the Universe.
On closer inspection the constraints imposedby this condition are in any case very weak. In order
toestimate therequiredvalueof ǫ,notethattheEuclideanaction S4D gives thefalsevacuumdecay
bounce
rate per unit volume as
Γ4/V = D4e−Sb4Dounce. (2.16)
D isthedeterminantcoefficientandisirrelevanttothediscussion. TodeterminewhethertheUniverse
4
couldhavedecayed(atleastonce)initslifetime,wemultiplytheabovebythespace-timevolumeofthe
past light-cone of the observable Universe. This results in a boundto have no decay of roughly [19,23]
S4D > 400. (2.17)
bounce ∼
This translates into an extremely weak lower bound on Φ ,
0
1
Φ N 2 4
0 > 3√h f . (2.18)
µ ∼ N3
(cid:18) (cid:19) (cid:18) (cid:19)
– 6 –
Note that the bound is on the relative width to height of the bump. Thus it is indeed simply the
flatness of the potential which protects the metastable vacuum against decay. In terms of ǫ = µ/Λ
L
the bound (2.18) reads
1
1− 2N 1 N3 4
ǫ Nf−N . . (2.19)
3h√h N 2
(cid:18) f (cid:19)
It is interesting to note that this very weak bound (2.18), (2.19) becomes strong when expressed in
terms of ǫ itself in the minimal case of N = 2 and N = 7,
f
1 5 15
ǫ . 1 8 4 4.3 10−4 1 2 1. (2.20)
3h√h 49 ! ≃ · h ≪
(cid:18) (cid:19) (cid:18) (cid:19)
In the following Section we will explain why at high temperatures the Universe has ended up in the
metastable non-supersymmetric vacuum in the first place.
3. Dynamical evolution at finite temperature
3.1 The shape of the effective potential at finite temperature
The effective potential at finite temperature along the Φ direction is governed by the following well-
known expression [24]:
T4 ∞
V (Φ) = V (Φ)+ n dqq2ln 1 exp( q2+m2(Φ)/T2) (3.1)
T T=0 2π2 ± i ∓ − i
Xi Z0 (cid:18) q (cid:19)
The first term on the right hand side is the zero temperature value of the effective potential; for the
ISS model we have determined it in Eq. (2.14) and in Figure 1. The second term is the purely thermal
correction (which vanishes at T = 0) and it is determined at one-loop in perturbation theory. The
n denote the numbers of degrees of freedom present in the theory6 and the summation is over all of
i
these degrees of freedom. The upper sign is for bosons and the lower one for fermions. Finally, m (Φ)
i
denote the masses of these degrees of freedom induced by the vevs of the field Φ.
We have noted in the previous Section that as we interpolate from the metastable vacuum to the
supersymmetricone, theN flavours of ϕ andϕ˜acquire masses m = hΦ = hµγ and become heavy at
f ϕ
large values of Φ. To a good approximation all other degrees of freedom can be counted as essentially
massless.7 As we are not interested in the overall additive (T dependent, but field independent)
constant in the thermal potential, we need to include in (3.1) only those variables whose masses vary
6Weylfermions and complex scalars each count as n=2.
7In the companion paper [17] we will refine this analysis by including contributions of gauge degrees of freedom to
(3.1). Thiswilllead toan evenlower boundforthereheattemperature,T ,necessary fortheUniversetoendupinthe
R
non-supersymmetric |vaci .
+
– 7 –
10
0
-10
-20
-30
0 2 4 6 8
Figure 2: Thermaleffective potential (3.2)for different values ofthe temperature. Going from bottom to top,
the red line corresponds to the temperature Θ & Θ where we have only one vacuum at γ = 0. The orange
crit
line corresponds to Θ Θ where the second vacuum appears and the classical rolling stops. The green line
crit
≈
is in the interval Θ < Θ < Θ where one could hope to tunnel under the barrier. The blue line is at
degen crit
Θ Θ wherethetwovacuabecomedegenerate. Finally,theblacklinegivesthezerotemperaturepotential
degen
∼
where the non-supersymmetric vacuum at the origin becomes metastable.
from being light in one vacuum to being heavy in the other. These are precisely the ϕ’s and ϕ˜’s. This
implies8
h2 ∞
Vˆ (γ) = Vˆ (γ)+ Θ4 4NN dqq2ln 1 exp( q2+γ2/Θ2) . (3.2)
Θ Θ=0 2π2 ± f ∓ −
X± Z0 (cid:16) p (cid:17)
Theabove expression is written in terms of a dimensionless variable γ = Φ/µ and we have also defined
a rescaled temperature
Θ = T/hµ (3.3)
| |
In Figure 2 we plot the thermal potential (3.2) for a few characteristic values of temperature.
In general one can consider a very wide range of temperatures 0 T Λ corresponding to
L
≤ ≪
0 Θ 1/(hǫ). From Fig. 2 where we have plotted the effective potential at various temperatures
≤ ≪
we immediately make out several interesting temperatures. At high enough temperature there is only
one minimum at the origin and one would expect to be able to roll down classically to the non-
supersymmetric minimum vac , as will be discussed in Section 3.2.3. Below Θ = Θ a second
+ crit
| i
minimum forms. It will turn out, that for our ISS potential this critical temperature will actually be
less than γ . In the following analysis, we will consider the range of temperatures hµ < T hΦ
0 0
≤
corresponding to 1 < Θ γ .
0
≤
At temperatures below Θ the SUSY breaking minimum at the origin still has lower free energy
crit
then vac . We will calculate the bubble nucleation rate to the true vacuum vac in Section 3.2.2.
0 +
| i | i
Another distinguished temperature Θ is when the SUSY preserving vacuum vac becomes de-
degen 0
| i
generate with the non-supersymmetric vacuum vac . For a generic potential one would expect that
+
| i
8There are only two terms in the sum in (3.2): one for bosons (+), and one for fermions (-). The prefactor 4NN
f
countsthetotal numberof bosonic degrees of freedom in ϕa and ϕ˜i.
i a
– 8 –
somewhere between these two scales there is a temperature Θ∗ (Θdegen < Θ∗ Θcrit) where the rate
≤
of bubble nucleation turns from large to small. Our main goal is to determine the temperature where
the transitions (classical or tunnelling ones) from vac to vac stop.
0 +
| i | i
3.2 Transition to a SUSY breaking vacuum and the lower bound on T
R
In this subsection we discuss the two possibilities for the field to evolve to the SUSY breaking vacuum
vac , classical evolution and bubble nucleation.
+
| i
3.2.1 Classical evolution to the SUSY breaking vacuum: Estimate of T
crit
It is clear from our expression for the thermal potential that at sufficiently high temperatures there
is only one minimum, namely vac , and the SUSY preserving minimum disappears. This is caused
+
| i
by N flavours of ϕ and ϕ˜ becoming heavy away from the origin. The disappearance of vac is most
f 0
| i
easily seen analytically in the limit of very high temperatures γ Θ where the thermal potential
≪
grows as Θ2γ2 in the γ i.e. Φ direction,
Vˆ Vˆ (γ) NN Θ2γ2 constΘ4 , Θ γ = Φ/µ (3.4)
Θ Θ=0 f
− ∼ − ≫
Thus there is a critical temperature Θ such that at Θ > Θ there is only one minimum, and at
crit crit
Θ < Θ the supersymmetry preserving second minimum starts to materialise.
crit
A better estimate can be obtained by using the approximation,
∞ γ 3 γ
dqq2ln 1 exp( q2+γ2/Θ2) Θ4 2 exp , for γ Θ (3.5)
± ∓ − ∼ 2πΘ −Θ ≫
Z0 (cid:16) p (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
for the integral in the Θ dependent part of Eq. (3.2). Comparing the derivatives (first and second) of
the Θ-dependent part and the Θ-independent part of the effective potential Vˆ in the vicinity of γ
Θ 0
we get an estimate,
γ0 √2π32 Nf 2
Θ , C = 1 (3.6)
crit ≈ log γ4/C Nh2 − N
0 (cid:18) (cid:19)
which we have also confirmed numerically.
(cid:0) (cid:1)
Although the height of the barrier in the zero temperature potential is only of the order of µ
where µ γ µ = hΦ the temperature necessary to erode the second minimum is only slightly
0 0
≪
(logarithmically) smaller than hΦ due to the exponential suppression exp( γ/Θ) apparent in Eqs.
0
∼ −
(3.2) and (3.5).
IntheearlyUniversethesituationisnotstaticandthetemperaturedecreasesduetotheexpansion
of the Universe. So one should check that even in the absence of a second minimum the field has time
enough to evolve to vac before the temperature drops below Θ . We will comment on this issue
+ crit
| i
more in the next subsection. Here, we just point out that the temperature drops on a time scale
M /T2 1/T.
Pl
≫
– 9 –