HIP-2002-08
Aeck-Dine mechanism and Q-balls
along SUSY at directions
Asko Jokinen
Helsinki Institute of Physics
P.O. Box 64 (Gustaf Hallstromin katu 2)
FIN-00014, University of Helsinki, FINLAND
ACADEMIC DISSERTATION
To be presented, with the permission of the Faculty of Science of the
University of Helsinki, for public criticism in Auditorium E204 of
Physicum on January 14, 2003, at 12 o'clock
Helsinki 2002
ISBN 952-10-0596-3(print)
ISBN952-10-0597-1 (pdf)
Yliopistopaino
Helsinki2002
This thesis is based on the research done at the Theoretical Physics Division of the
Department of Physical Sciences at the University of Helsinki and at the Helsinki
Institute of Physics. The work has been mainly funded by the Academy of Finland
under the contract 101-35224 and the Helsinki Institute of Physics. The trips have
been supportedby the Magnus Ehrnrooth Foundation.
IwishtothankmythesissupervisorprofessorKariEnqvistforthemanydiscussions
andcollaboration. Withouthisadvice, andfundingarrangedbyhim,Iwouldnothave
been able to complete this thesis.
I also thank the referees of this thesis, professor Jukka Maalampi and Doc. Kari
Rummukainen for useful comments and corrections. I would also like to thank Doc.
Hannu Kurki-Suonio for hiscomments onthe thesis.
I thank my fellow students Janne Hogdahl, Vesa Muhonen, Martin Sloth, Antti
Vaihkonen, Jussi Valiviita and Dr. Syksy Rasanen for all the useful discussions on
physics.
I thank my parents Pauli and Pirkko Jokinen for support during the years of
studying in the University. The rst version of the thesis was completed in 6th of
October2002,sothethesishonoursthe76thbirthdayofmygrandfatherRistoJokinen.
Helsinki2002
AskoJokinen
This thesis contains four research papers and an introduction, which provides the
necessary backroundandalsocontainsanewanalyticalsolutiontotheQ-ballequation
of at potentialsinthe Appendix.
At the classical level supersymmetric gauge theories have a large degeneracy of
vacua of the scalar potential. The vacuum scalar eld congurations are called at
directionsormodulields. Thedegeneracyremainsunbrokentoarbitraryperturbative
order. Degeneracy can be lifted by supersymmetry breaking eects or by adding
suitabletermstothesuperpotential. Thentheatdirectionhasanon-trivialpotential
that determinesits dynamics.
Theatdirectionshaveanumberofcosmologicalconsequences. WithintheMSSM
at directionsthatcarrybaryonand/orleptonnumbercanprovideameansforbaryo-
genesis by forming coherently rotating condensates, Aeck-Dine condensates, which
eventually decay into Standard Model fermions. If R -parity is conserved, then the
decay products include the stable Lightest Scalar Particles, which can exist as dark
matter today.
When one includes radiative corrections, the mass term grows slower than 2
for some at directions. As a consequence the AD condensate fragments into non-
topological solitons called Q-balls, which are minimum energy congurations with a
conservednon-zerocharge. Q-ballsthemselvescancarrybaryonnumberandprotectit
fromtheeects oftheelectroweakphasetransition. Dependingonthe SUSYbreaking
mechanismtheQ-ballscanbestableandhencecontributetothedarkmatter. Q-balls
formed out of at directioncondensates, that are not connected toSM elds, can act
as self-interacting dark matter.
In the thesis we have studied the instabilitiesof the at directions within MSSM;
the formation of the ADcondensates and their properties in both gravity and gauge
mediated SUSY breaking scenario; the fragmentation of the Aeck-Dine condensate
toQ-ballsand thethermalizationoftheproducedQ-balldistribution;and constraints
for Q-balls being the self-interacting darkmatter.
Acknowledgements i
Abstract ii
Contents iii
List of Papers v
1 Introduction 1
2 The basics of Supersymmetry 6
2.1 Superelds . . . 6
2.2 Supersymmetric gauge theories . . . 7
2.3 Supergravity . . . 9
2.4 Minimal Supersymmetric Standard Model: MSSM . . . 10
2.5 SUSY breaking . . . 11
3 Flat directions 14 3.1 Flat directions in general . . . 14
3.2 Flat directions in termsof gauge invariant polynomials . . . 15
3.3 Flat directions of the MSSM . . . 20
3.4 Liftingthe at directions . . . 23
4 Cosmological evolution of the SUSY at directions 29 4.1 Formationof the ADcondensate . . . 29
4.2 Radiativecorrections . . . 35
4.3 Instability of the ADcondensate. . . 37
4.4 Fragmentation of the AD condensate . . . 39
5 Q-balls 42 5.1 Solitons . . . 42
5.2 Q-balls inany spacedimension D . . . 43
5.4 Thick-wall approximation . . . 48
5.5 Q-balls inthe MSSM . . . 49
5.6 Decay and scattering of Q-balls . . . 51
5.7 Thermally distributedQ-balls . . . 53
6 Q-balls in cosmology 56 6.1 Baryogenesis. . . 56
6.2 Dark matter . . . 57
6.3 Problems of Cold Dark Matter . . . 58
6.4 Self-Interacting Dark Matter . . . 59
6.5 Q-balls asSelf-Interacting Dark Matter . . . 60
7 Conclusions 62
8 Appendix 64
References 69
I K.Enqvist, A. Jokinen and J. McDonald,
Flat Direction Condensate Instabilitiesin the MSSM,
Phys. Lett. B483 (2000)191.
II K.Enqvist, A. Jokinen, T. Multamaki and I. Vilja,
Numerical simulations of fragmentation of the Aeck-Dine condensate,
Phys. Rev.D63 (2001) 083501.
III K.Enqvist, A. Jokinen, T. Multamaki and I. Vilja,
Constraints on self-interacting Q-balldark matter,
Phys. Lett. B526 (2002)9.
IV A. Jokinen,
Analytical and numericalproperties of Aeck-Dine condensate formation,
ArXiv: hep-ph/0204086 (submitted toPhys.Rev.D).
The long standing probleminthe study of cosmology has been baryogenesis. Cosmo-
logical nucleosynthesis has given us strict bounds on the observed baryon-to-photon
ratio =n
B
=n
,wheren
B
isthe baryonnumberdensity andn
isthe photonnumber
density. This has been narrowed down to 1:210 10
6:3 10 10
[1]. It is a
huge theoretical challenge to achieve a mechanism of baryogenesis that produces the
prescribed . The conditions,that are necessary forbaryogenesis, were long agogiven
bySakharov [2]: 1)Violationofbaryonnumber,2)violationofC andCP and 3)non-
equilibrium conditions. In the Standard Model (SM) of particle physics baryogenesis
ispossibleonlythroughthe electroweakanomaly. However, this hasbeen ruledoutby
the lattice studiesonthe electroweak phasetransition[3{6]and the LEPexperiments
onHiggsmass[7]. Itisstillpossiblethattheelectroweakmechanismwould workinan
extension of the SM such as the Minimal Supersymmetric Standard Model (MSSM),
but this is on the edge of being ruled out, too [8,9]. In [9] it was argued that if the
Higgs mass is larger than 120 GeV , the electroweak baryogenesis within MSSM does
not produce the observed baryon asymmetry. The current bound is m
H
>
115 GeV
[10].
In 1985 Aeck and Dine [11] proposed a mechanism for baryogenesis based on
scalar elds. The ideacan bepresented simplyby consideringaLagrangiandensityof
a complexscalareld,whichhas aglobal U(1)-symmetry(suchasthebaryonnumber
or the lepton number) 1
L=(@
)(@
) V(jj): (1)
From the usual denition of the Noether current related to Eq. (1) one obtains the
charge density
q =i(
_
_
)= _
' 2
; (2)
where we have writtenthe complexscalar eld = 1
p
2 'e
i
, where'; are real scalar
elds. The second form shows clearly that in order tohave anon-zero charge density
has to acquireanon-zero vacuumexpectationvalue j<>j>0 andhas torotate
1
Weusethemetric
=diag(1,-1,-1,-1),with;=0;1;2;3.
around the origin with 6= 0. Naturally one has to include some kind of symmetry
breaking into Eq. (1), for example a U(1)-violating term in the potential. Then one
can calculate how the charge density evolves in an expanding universe with help of
equations of motion related to Eq. (1), with the possible symmetry breaking terms
taken intoaccount, toobtain
_
q+3Hq+
@V
@
=0 ,
@(qR 3
)
@t
= R
3
@V
@
; (3)
where H = _
R =Ris the Hubble parameter, R is the scale factor and qR 3
is the charge
density in the co-moving volume.
Naturallyoneasks: whatcouldthe scalareldbe? IntheMSSM therearemany
scalar elds: squarks, sleptons and Higgses. Natural candidates are the squarks and
sleptons, whichgiverise tobaryogenesisandleptogenesis,respectively. Inleptogenesis
theleptonnumberistransformedintobaryonnumberthroughsphalerontransitionsat
the electroweak phasetransition. Forthis reasonwedonotmakeadierencebetween
the baryonand the leptonnumber, but just consider a U(1)charge in general.
Thereisalsotheissueofwhatkindchargeviolationcanbeincludedinthepotential.
At the renormalizable level B- and L-violating interactions are forbidden in the SM.
They are also forbidden in the MSSM if R -parity is conserved. If R -parity is not
conserved,itispossibletohaveinteractionsviolatingthebaryonnumber. However, R -
parityviolatingtermscauseproblemswithprotondecayunlessthecouplingconstants
are ne-tuned [12]. Therefore the violationhastobeinducedby anon-renormalizable
operator. If a scalar eld acquires a large expectation value, which it has toin order
for the non-renormalizableoperatortobeeective,allthe elds itinteractswith gain
a mass g <>, whereg is the corresponding couplingto the at direction.
Insupersymmetricgaugetheoriesthereisarichvacuumdegeneracyattheclassical
level. The scalar potential, which is a sum of the so-called F-terms and D-terms,
vanishes identically for some eld congurations i.e. along certain "at directions".
The space of allsuchat directions is calledthe modulispace andthe masslesschiral
superelds whose expectation values parameterize the at directions are also known
as moduli [13,14]. The at directions gain a mass comparable to the soft SUSY
radiative corrections they receive to arbitrary perturbative order [15{17]. However,
non-perturbative eects do produce corrections to the at directions, see e.g. [17].
SinceonlySUSYbreakingandnon-renormalizabletermsproduceaneectivepotential
for the at direction, it is possible for the at direction to acquire a large vacuum
expectationvalue (VEV)in the earlyuniverse.
If the at direction contains elds that carry baryon or lepton number, then by
introducingthe B and/or Lviolation through non-renormalizableoperators itispos-
sible tohaveaconsistentmodelforbaryogenesis. Ingeneralabaryon-to-entropyratio
of O(1) can be produced [14]. Even if the at directions were not carrying baryon
number, as in various extensions of MSSM, the extra at directions can have other
cosmological consequences. Forinstance, they can bethe darkmatter. Therefore at
directions are interesting on their own right. One should also note that a non-zero
VEV of ascalar eld spontaneously breaks gauge symmetry.
The zero mode of a rotating scalar eld forms a coherent condensate. The be-
haviour of the condensate is dependent on the mass term. If the mass term is the
usual tree-level m 2
jj 2
, on the average the condensate has zero pressure and eventu-
ally the condensatewould decaythrough thermalscattering. However, ifwe take into
consideration radiative corrections such as a logarithmic correction in case of grav-
ity mediated SUSY breaking [18] or the almost at potential of the gauge mediated
SUSY breaking [19], a non-zero pressure is induced to the condensate. If the poten-
tial grows slower than 2
, the pressure is negative [20]. Within MSSM with gravity
mediated SUSY breaking some at directions are unstable and some are not, when
radiativecorrectionsareincluded, ashasbeenshownbysolvingrenormalizationgroup
equations numerically in Paper I [21]. The pressure also depends on the orbit of the
eld. For pure oscillation the pressure reaches its maximum absolute value, whereas
for circular orbit the pressure vanishes due to the fact that on the circular orbit ki-
netic and potential energy are equal, which was pointed out in Paper IV [22]. In an
expanding universe the orbit cannotbe strictly circular due tothe dissipation caused
by expansion. Thus, the pressure is always negative for some at directions. Then
the condensate unstable with respect to spatial perturbations [23]. Eventually these
perturbationsfragmentthe condensate.
For potentials that grow slower than 2
for some range of and which describe
elds that are charged under a U(1)-symmetry, there exists a new kind minimum
of energy. The minimum energy conguration in the sector of conserved charge is
a non-topological soliton called the Q-ball [24]. Since, as it turns out, at the time
of condensate fragmentation the at direction potential is dominated by the U(1)-
symmetricpart,thechargedcondensatefragmentswillformQ-balls. Q-ballformation
hasbeenseeninseveralsimulations[25{29]amongthemtheonepresentedinthePaper
II [28]. Actuallyeven anoscillatingcondensate fragments intoQ-balls, forming equal
amounts of Q-balls and anti-Q-balls [27]. Because Q-balls are quite robust objects
and donot decay easilythrough thermalscattering, the baryonnumberinsidewillbe
protected fromsphaleron erasure untilQ-balls eventually decay.
AlthoughQ-ballsarestablewith respect todecay intotheirown quanta,therecan
exist decay channels to other scalars or to fermions. The fermion decay channel is
interesting inthat the Q-ball decays intofermions by evaporation fromthe surface of
the Q-ball [31]. It isalso possible that large Q-ballsare completelely stable, inwhich
case they might appear as dark matter (DM) [23]. This can happen in the gauge
mediated SUSY breaking scenariowhere the Q-ballmass increases as Q 3=4
[32]. If Q-
ball DMis baryonic, it seems that itcan onlyform asmallfraction of DM. However,
inextensions ofMSSM newatdirections canact asasourceforQ-ballDM,but they
canalsobeasourceofproblems[30]. RecentlyithasbeensuggestedthatQ-ballscould
act as self-interacting dark matter (SIDM) [33]. In Paper III [34] we have calculated
constraints under which Q-ballscan act as SIDM.
Recently the ADmechanism to produce Q-ballshas been applied to ination[35,
36]. In that approach the inaton mass receives radiative corrections which give rise
toanegativepressureoftheinatoncondensate. HencetheinatonfragmentsintoQ-
balls inthesame mannerasinthe ADmechanism. HenceQ-ballsdecayintofermions
through surface evaporation only, the reheating temperature afterination decreases
Q-balls may also have other consequences relevant for cosmology. For more in-
formation about these see the recent review by Enqvist and Mazumdar [37] and the
references cited therein.
The thesis is organized as follows: In Section 2 we present a short review of su-
persymmetry inorder tohave the relevant denitions used throughout the thesis. In
Section 3aquitethorough introductiontothe at directions isgiven. Theresults are
applied to the MSSM. The lifting of the at directions, i.e. generating a potential
for the at direction, is discussed at the end of the section. In Section 4 cosmologi-
cal evolution of a at direction is addressed, where it is shown how the eld evolves
during andafterination. Wethen show howthenegativepressure arises anddiscuss
the growth of perturbations, which is caused by the negative pressure. In Section 5
general properties of Q-ballsare presented and anew analyticalapproximationof the
Q-ballproleinthegaugemediatedscenarioisderived. Detailsofthiscalculationcan
be found in the Appendix. We also consider the thermalization of the Q-ball distri-
bution. In Section 6 we briey describe cosmological applications of Q-balls such as
baryogenesis and dark matter. In Section 7we giveour conclusions.
In thisSectionwegiveabriefreviewof thevariousaspects ofsupersymmetryrelevant
for the thesis. There exists excellent monographs and reviews on supersymmetry
[12,17]; Haber and Kane [38] deal specically with the MSSM. Here we follow the
notations of Nilles'review [12].
2.1 Superelds
In supersymmetric theories the elds are gathered into multiplets called superelds.
Thescalars,vectorsandspinorsarejustdierentcomponentsofasupereld(insuper-
gravity the graviton and gravitino are in the same supereld). There are practically
two kinds of superelds: chiral and vector superelds (and a metric supereld in su-
pergravity) [12,17].
The covariant derivatives are dened by
D
=
@
@
+i
_
_
@
; (4)
D
_
=
@
@
_
i
_
@
; (5)
where
;
_
are Grassmann parameters.
Withthehelp ofthe covariantderivatives, Eq. (4), wecan denechiral superelds
D
_
L
=0; D
R
=0; (6)
where
L (
R
)isaleft-handed(right-handed)chiralsupereld. Theleft-handedchiral
supereld can bewritten inthe form
L
(y;)=(y)+ (y)+F(y); (7)
where ; F are complex scalar elds, is a Weyl spinor and y
= x
+i
. The
chiralsupereldthuscontainstwospin-0bosonsandaspin-1/2fermion. Theauxiliary
eld F transforms as a total derivative under SUSY transformations. Hence the F
term of a chiral supereld can be used for a supersymmetric action[12].
Withtherealitycondition V =V onegets avectorsupereld,whose components
are
V(x;;
)=(1+ 1
4
@
@
)C +(i+ 1
2
@
)+
i
2
(M +iN)+
( i
+ 1
2
@
)
i
2
(M iN)
V
+i
i
+
1
2
D; (8)
where C; M; N; Dare real scalar elds, ; are Weylspinors and V
is areal vector
eld. This multipletcontains spin-0, spin-1/2and spin-1elds. Now Dtransforms as
a total derivative and can be used in the Lagrangian density. A simplication arises
in the Wess-Zumino gauge,where C ===M =N =0.
2.2 Supersymmetric gauge theories
Gauge invariant supersymmetric actions in the Wess-Zumino gauge are of the form
[12]
S= Z
d 4
x[L
D +L
F
+h:c:]; (9)
where
L
D
= Z
d 2
d 2
y
e gV
=
(D
)
y
(D
)+i
D y
+F y
F +i p
2g(
y
)+g
y
D (10)
with
D
=@
+igV
(11)
and
L
F
= Z
d 2
W()+ 1
32g 2
W A
W A
(12)
where W() is the superpotential, which is at most tri-linear for a renormalizable
theory, and W A
is the eld strength tensor supereld; g is the gauge coupling. The
eld strength tensor isdened as
W
= 8
<
:
D 2
e gV
D
e
gV
; (non abelian)
D 2
D
V; (abelian):
(13)
W A
(y;) =
D 2
D
V
A
+igf ABC
D(D
V
B
)V C
= 4i A
(y) 4
D
A
(y) 2i(
)
V A
(y)+4
_ D
A_
(y);(14)
where
V A
= @
V
A
@
V
A
gf
ABC
V B
V
C
D
A_
= @
A_
gf ABC
V B
C_
y
= x
+i
: (15)
The eld strength term leads toa Lagrangian
L
V
= 1
32g 2
Z
d 2
W A
W A
+h:c:
= 1
4 V
A
V
A
i A
D
A
+ 1
2 D
A
D A
; (16)
where
V A
=@
V
A
@
V
A
+igf ABC
V B
V
C
: (17)
The scalar potentialgained fromthe action Eq. (9)is
V = X
i F
y
i F
i +
1
2 X
A D
A
D A
; (18)
where F and D terms are dened by [12]
F y
i
=
@W()
@
i
; D
A
= X
i g
A
y
i T
A
i
+; (19)
where
i
are the scalar components of the corresponding chiral superelds
i , g
A are
the gaugecouplingsofthecorrespondinggaugegroup,and isaFayet-Iliopoulosterm
whichvanishesfornon-abeliansymmetrybutcanbenon-zeroforanabeliansymmetry
group.
In principle dierent superelds can carry dierent representations of the same
gauge group, in which case one has to use the corresponding form of T A
in Eq. (19)
tothe representation. Forinstance, the generatorsofthe complexconjugaterepresen-
tation are the negatives of the transposed generators (T A
) T
. The at directions are
the eld congurations for which Eq. (18) vanishes.
The eld congurations that are at in globally supersymmetric limit are no longer
at when the SUSY is made local i.e. in supergravity. Since supergravity is a non-
renormalizable theory, one has to include non-renormalizable terms in the superpo-
tential, and kinetic terms, too. Then the most general Lagrangian that is globally
supersymmetric and gaugeinvariantis
L = Z
d 2
d 2
J
y
e gV
;
+ Z
d 2
W()+f
AB ()W
A
W B
+h:c:
; (20)
where f
AB
() is an arbitrary function of the chiral superelds. It transforms like a
chiralsupereldunderSUSYandisasymmetricproductoftwoadjointrepresentations
with respect tothe gaugegroup. If the theory is renormalizable,then f
AB
()=Æ
AB .
J is real supereld and leads to a renormalizable theory only if J = y
e gV
. The
superpotential W() is a chiral supereld which is a polynomial of degree less or
equal tothree, if the theory isrenormalizable.
The coupling of supergravity to matter is very complicated in the most general
form. We givehere only the bosonic part of the Lagrangian; for the rest see [12]:
e 1
L
B
= M
4
p e
G
3+G
k (G
1
)
k
l G
l
1
2 M
4
p (Ref
1
AB )(G
i
T Aj
i
j )(G
k
T Bl
k
l )
1
4 (Ref
AB )V
A
V
B
+ i
4 (Imf
AB )
V A
V
B
M 2
p G
i
j (D
i )(D
j
) 1
2M 2
p
R; (21)
where M
p
= 2:4 10 18
GeV is the reduced Planck mass, R is the Ricci scalar, the
covariant derivatives D
are covariant with respect to gravity and gauge group and
G
k
=@G=@
k
,G k
=@G=@
k
andG l
k
=@ 2
G=@
l
@
k
. Thereal functionofthescalar
elds G(
y
;), usually called the Kahlerpotential,is dened by
G=3log
J
3
log(jWj 2
): (22)
Usuallyone gives the Kahler potential inthe form
G(
y
;)=
K(
y
;)
M 2
p
log
jWj 2
M 6
p
; (23)
dynamics we rewrite the scalar part of the Lagrangian Eq. (21) using Eq. (23) as
e 1
L= 1
M 2
p (D
i )K
i
j (D
j
) V; (24)
where
V = e K
M 2
p
(D
i W)
(K 1
)
i
j (D
j
W) 3
M 2
p jWj
2
+ 1
2 (Ref
1
AB )D
A
D B
; (25)
D j
W = W
j
+ K
j
M 2
p
W; (26)
D A
= K
i
T Aj
i
j
+; (27)
where is the Fayet-Iliopoulos term. The choice of K i
j
= Æ i
j
leads tominimal kinetic
terms. However, with a non-renormalizable theory, such as supergravity, one should
take intoaccount the possibility of non-minimalkinetic terms.
2.4 Minimal Supersymmetric Standard Model: MSSM
MSSM is the minimal supersymmetric extension of the familiar Standard Model of
particle physics. This means that there exist matter superelds, which are chiral,
as follows: quark doublets Q A
i
transforming as (3;2;
1
6
) under SU(3)
C
SU(2)
L
U(1)
Y
, quark singlets u
iA (
3;1;
2
3 ) and
d
iA (
3;1;
1
3
), lepton doublets L
i
(1;2;
1
2 ),
lepton singlets e
i
(1;1;1), Higgs doublets H
u
(1;2;
1
2
) and H
d
(1;2;
1
2
). Gauge
elds areinvectorsuperelds. TherearethreevectorsupereldmultipletsV A
; A
; B
correspondingrespectivelytoSU(3)
C
SU(2)
L
andU(1)
Y
. TheF partof thepotential
is obtained from the superpotential
W = X
ij
y ij
u
Q
A
i H
u u
jA +y
ij
d
Q
A
i H
d
d
jA +y
ij
e
L
i H
d e
j
+
H
u H
d
; (28)
where y ij
u;d;e
are the Yukawa matrices and M
W
. It is also possible to add the
following terms tothe superpotential Eq. (28)
~
W =
Q
A
L
d
A +
L
L
e+
ABC
u
A
d
B
d
C
: (29)
explicitlythe baryonand leptonnumber. Their couplingswould have tobeextremely
small in order to be compatible with experimental limits. The terms in Eq. (29)
are usually left out by imposing a discrete symmetry called R -parity, where R =
( 1)
3B+L+2S
. Weusually ignore Eq. (29).
The eld strengths related tothe gaugeelds are obtained fromEq. (14) as
W A
=
D 2
D
V
A
+ig
3 f
ABC
D 2
(D
V
B
)V C
;
W
=
D 2
D
W
+ig
2
Æ
D 2
(D
W
)W Æ
;
B
=
D 2
D
B: (30)
The pure gaugeLagrangian is formed as inEq. (12) and reads
L
V
= 1
32 (
1
g 2
3 W
A
W A
+
1
g 2
2 W
W
+
1
g 2
1 B
B
+h:c:)
F
: (31)
The interaction between gauge and mattermultipletsis given by Eq. (16) sothat
L
=
"
X
i
Q y
i
exp (ig
3 T
A
V A
+ig
2
W
+ 1
6 ig
1 B)Q
i
+ u y
i
exp ( ig
3 (T
A
) T
V A
2
3 g
1 B)u
i +
d y
i
exp ( ig
3 (T
A
) T
V A
+ 1
3 g
1 B)
d
i
+ L
y
i
exp (ig
2
W
1
2 ig
1 B)L
i +e
y
i
exp(ig
1 B)e
i
+ H
y
u
exp (ig
2
W
+ 1
2 ig
1 B)H
u +H
y
d
exp (ig
2
W
1
2 ig
1 B)H
d
D
: (32)
2.5 SUSY breaking
From experiments we know that supersymmetry is necessarily broken in our world.
Therefore one needstounderstand howSUSYisbroken. One wouldalsoliketoretain
some of the properties of SUSY (such as the cancellation of quadratic divergences)
while breaking it. This can be achieved by the soft SUSYbreaking terms [39]
V
soft
= m
2
H
u jH
u j
2
+m 2
H
d jH
d j
2
+m 2
L jLj
2
+m 2
e jej
2
+m 2
Q jQj
2
+m 2
u juj
2
+m 2
d j
dj 2
+ A
u m
3=2 y
ij
u H
u Q
a
i u
ja +A
d m
3=2 y
ij
d H
d Q
a
i
d
ja +A
e m
3=2 y
ij
e H
d L
i e
j
+Bm
3=2 H
u H
d +h:c:
+ 1
2 M
1
1
1 +
1
2 M
2
2
2 +
1
2 M
3
3
3
; (33)
and
i
are thegauginos. However, theoriginofthe breakingtermsisleftunexplained.
Another approach would be to break SUSY spontaneously. However, a non-su-
persymmetric vacuum has positive energy, which can be cosmologically problematic
(although nowadays a positive cosmological constant is experimentally favourable).
More problems arise from the supertrace relations [12]. For instance, a pure F-term
breaking in renormalizabletheoriesyields attree level [40]
STrM 2
= X
J( 1) 2J
(2J +1)TrM 2
J
=0; (34)
where M 2
J
isthe mass matrixofallparticleswithspin J. Becauseof Eq. (34)making
the superpartners of the usual fermions heavy enough is very diÆcult. Therefore the
only option seems tobe the soft SUSY breakingterms.
The origin of the soft terms is usually assumed to be in a sector that is \hid-
den" to us. The hidden sector then couples to our \visible" sector via loops or non-
renormalizablecouplings thus avoidingthe supertrace constraint. Thereare ofcourse
other constraints on the breaking terms in the Lagrangian; for example the avour
changingneutral currents have to besuppressed [7].
Onehiddensectormechanismisbasedonthe gravity mediationofSUSYbreaking
[12] fromthe hiddento the visiblesector. A minimal Kahler potentialfor the hidden
and visiblesector eldsgives rise totherequired softSUSYbreaking parameters[12].
The typicalmass scale of the superpartners is given by the gravitinomass m
3=2 .
Another type of mechanism is based on gauge mediation [41]. In this case the
supersymmetric partners of theSM receive the dominantpartof theirmass viagauge
interactions with the hidden sector. For instance, a superpotential which includes a
term
W =X
+:::; (35)
where is a supereld with SM couplings but does not belong to the spectrum of
MSSM, and X a SMsinglet supereldin the hiddensector thatgets anon-zero VEV
in the D- and F-directions. The scalar VEV of X gives masses to the fermionic
component of . The masses of the scalar components come from the VEVs of the
the 1-looplevelwith m<F
X
>=<X >. The scalars obtainmasses m at
the 2-looplevel. The tri-linearsoft terms alsoarise atthe 2-looplevel.
Flat directions,alsocalledthemodulispace,aresupersymmetric minimaof thescalar
potential. Since SUSY is spontaneously broken, supersymmetric at directions cor-
respond to eld congurations whose D and F terms vanish, see Eqs. (18,19). In
reality the at directions are lifted by supersymmetry breaking eects. This means
that the degeneracy along a at directionis broken and a potentialfor the at direc-
tion is generated. Another mechanism is to add new non-renormalizable terms into
the superpotential. Thecouplingtosupergravity inducesSUSY breakinginthe Early
Universe, which provides its own contribution to the eective potential of the at
direction.
3.1 Flat directions in general
We start with general considerations. Let us take N chiral superelds X
i
, which
transform under some gauge group G as a (in general reducible) representation in
whichthegenerators oftheLiegaugealgebraarematricesT A
. Inprincipleallthe at
directions of the modelcan befound by solving directlythe constraints(ignoringnow
the Fayet-Iliopoulosterm)
D A
= X
ij x
i T
A
ij x
j
=0; (36)
F
x
i
=
@W(x)
@x
i
=0 (37)
wherex
i
arethe scalarcomponentsofthesupereldsX
i
. Werstdescribeanexample
of a direct solutionto Eq. (36).
Example 3.1 SU(N) gauge theory with squarks, q A
i
with i = 1;:::;m and A =
1;:::;N, in N and anti-squarks, q
j
A
with j =1;:::;n and
A=1;:::;N, in
N [13].
One can solve the D-term condition Eq. (36) by introducing a basis for the Lie
algebra of SU(N). It consists of traceless Hermiteanmatrices, and we write
(T A
B )
C
D
=Æ A
D Æ
C
B 1
N Æ
A
B Æ
C
D
; A;
B;
C; D=1;:::;N: (38)
The complex conjugate representation is generated by (T
B
) = (T
B
) . Using the
denition of generators Eq. (38) inEq. (36) one obtainsaftersome manipulation(see
[13] for details)
m
X
i=1 q
i
B q
A
i n
X
j=1 q
A
j q
j
B
= 1
N Æ
A
B N
X
C=1
"
m
X
i=1 jq
C
i j
2 n
X
j=1 jq
jC j
2
#
kÆ A
B
; (39)
where k is a constant. Eq. (39) is an orthogonality constraint with respect to the
U(m;n)-metric of N vectorsof the form(q A
1
;:::;q A
m
;q A
1
;:::;q A
n
) with A=1;:::;N.
Here the N vectors allhavethe same norm.
3.2 Flat directions in terms of gauge invariant polynomials
SolvingEq. (36)directlyisquitesimpleinthecaseofSU(N)symmetryinthedening
representation. However, one always hasto givethe formof thegenerators beforeone
can start to solve Eq. (36). For representations other than the fundamental one the
form of the generators is more complicated, not to mention other gauge groups. For
exampleinSU(5)GUTtheparticlesareinrepresentations10and
5. Whendiscussing
the AD mechanism, we would have to introduce a non-renormalizable operator that
liftstheatdirection. Onehastoaddsomepolynomialtothesuperpotential,whoseF-
term liftsthe degeneracy ofthe solutionof Eqs. (36,37). Adirect solutionofEq. (36)
doesnot yield sucha polynomial which has to be found separately.
Another way of parameterizing the at directions relies on the correspondence
of the D at directions and gauge invariant holomorphic polynomials of the chiral
superelds X
i
[13,42{47]. The smaller moduli space of D and F at directions is
parameterized by the same basis of monomialsand redundancy constraints as the D
atdirectionsaloneandsubjectedtotheadditionalconstraintsfollowingfromF
x
i
=0.
One should note that this discussion applies only for gauge groups without Fayet-
Iliopoulos terms. The case with non-zero Fayet-Iliopoulos term should be considered
separately.
We give simplearguments why gauge invariant polynomials produce D at direc-
tions, withoutgoingintothe technicaldetailswhichcanbefound in[43{47]. LetI(X)
the scalar componenttransforms inthe following way:
ÆI(x)= X
i
@I(x)
@x
i Æx
i
= X
ij
@I(x)
@x
i
A
T A
ij x
j
=0; (40)
where a
are innitesimal parameters. Eq. (40) vanishes by virtue of the gauge in-
variance of the polynomial. Note that Eq. (40) resembles the D atness constraint
Eq. (36). As rst notedin [13], Eq. (40) isequivalent toEq. (36),if
@I(x)
@x
i
=C(x
i )
(41)
for someC 6=0. Thereforeif thereisagaugeinvariantpolynomialsatisfyingEq. (40),
then the D-term, Eq.(36), automatically vanishes. The reverse of this statement,i.e.
thatalltheD-atdirectionscanbeparameterizedbygaugeinvariantpolynomials,was
rst conjectured in[13]. The proof of this conjecture requires a considerable amount
of mathematics. We just outlinethe methodof the proof.
\Proof of the conjecture"
As notedin Eq. (39), the D-atness constraint isactually anorthogonality condi-
tion withrespect tothecomplexscalarproduct<x;y>=
P
i x
i y
i
. Inthisrespect the
vector x is orthogonal to T A
x at x if the D-term vanishes. T A
are generators of the
Lie algebra, which from the point of view of dierential geometryare tangents of the
curves in the Lie group, of which all the elements inthe Lie group can be generated.
The orbit of x, fgxjg 2 G
CI
g, under the action of the complexied Lie group formsa
surface in the representation space.
2
So x is now orthogonal to the orbit of x at the
points where the D-term vanishes. Surfaces generated by an action of the complexi-
ed Liegroup can be parameterized as I(x)=C, where I is apolynomial and C isa
constant, meaning that the surfaces are algebraic[48]. The complex conjugate of the
2
ThecomplexiedLiealgebraisgeneratedbyT A
andiT A
,whereT A
aregeneratorsofLiegroup,
andthevanishingD-termvanisheswithgeneratorsofthecomplexiedLiealgebraifitvanisheswith
Lie algebra. ThecomplexiedLiegroupisacquiredbyexponentiatingthecomplexiedLiealgebra.
The need forthe complexication arisesbecausefor examplewith SU(N)oneobtains jxj=jgxj
for all g 2 SU(N) and x 2 C N
. From this follows that < x;T A
x >= 0 for all x 2 C N
and
T A
2Lie(SU(N)). Complexicationxesthisambiguity.
i i
I(x) = C. So the points where (rI(x))
is parallel to x are the points where the
D-termvanishes. In summary,the points where D-termvanishesare the same as the
pointsx, which are orthogonal totheir orbit underthe complexied Liegroup, which
in turnare the same pointswhere the gradientisparallel tothecomplex conjugateof
the eld, rI(x)=Cx
with C 6=0. Hence the conjecture is\proven".
3
We next give a few examples of how to apply Eq. (41) in the case of an SU(N)
group before givingthe complete catalogue of the basis of gauge invariantmonomials
of MSSM in Section 3.3. The only thing one needs in order to apply the method of
[13] is thelistof the gaugeinvarianttensorsof the Liegroup. These can begenerated
assums,productsand contractionsoftheso-calledprimitivetensorsofthegroup,and
they are listed in [49].
Example 3.2 Abelian U(1) gaugetheory with two charged superelds
+
and .
This is a rather trivial example, but it is useful as a rst example. The D at
directions inthis case are obtained simplyby directlysolving Eq. (36)
D=j
+ j
2
j j 2
=0)j
+
j=j j; (42)
where
is the scalar component of
. Now we show how to use gauge invariant
polynomials. All the U(1) gauge invariant polynomialsof this example are generated
by the product
+
. Allthe powers ofthis product actually parameterizethe same
at direction, so for simplicity we take I(
+
; ) I(
+
) =
+
. Now the
conditions of Eq. (41)become
@I
@
+
= =C
+
@I
@
=
+
=C
: (43)
One shouldnow note that it isessentialin Eq. (41) that C isthe same for allderiva-
tives. Eq. (43) is easy to solve by multiplying the rst equation with
+
and the
3
Severalmathematicaldetailshavebeenomittedinthe\proof". Themostimportantoneiswhy
allthesurfacescan beparameterizedasI(x)=C,whichcanbefoundat[48].
some powerof thepolynomialI,then theF-termconstraintEq. (37)producesaterm
proportional to the left hand side of Eq. (43) and thus causes the right hand side to
vanish. This isa genericfeature with gauge invariant polynomials.
Example 3.3 Non-abelian SU(N) gauge theory with matter superelds A
i and
j
A
transforming according to the dening N representation and complex conjugate
N
representation,respectively.
Thiswas alreadysolved inExample3.1, but herewedemonstrate theuse ofgauge
invariant tensors. The primitive tensors of SU(N) are the Kronecker delta Æ A
B and
the Levi-Civita tensor
A
1 A
N
;
A
1
A
N
in N dimensions [49]. An extra simplication
is provided by the factthat the product ofLevi-Civita tensors produces a generalized
Kronecker delta
A1A
N
A1
A
N
=Æ
A
1
A
N
A
1 A
N
=Æ
A
1
A
1 Æ
A
N
A
N
(permutations): (44)
Letusassumethat therearem mattersuperelds A
i
; i=1;:::;m andn anti-matter
superelds
j
A
; j =1;:::;n,whichwecontractinallpossiblewayswiththeprimitive
tensors. Thus the generating monomialsare
(
i
j
)
A
i Æ
A
A
j
A
;
(
i1
i
N
)
A1A
N
A
1
i1
A
N
i
N
;
(
i
1
i
N
)
A1
A
N
i1
A1
i
N
A
N
: (45)
Let us rst study the gauge invariant monomial (
i
j
). The D atness constraint
Eq. (36) nowreads
A
i
= C
A
j
jA
= C
iA
; A =1;:::;N; (46)
where A
i (
j
A
)isthescalarcomponentof A
i (
j
A
). BysolvingCfrombothequations
one can see that Eq. (46) is the same as Eq. (39) with a zero norm with respect to
U(1;1). Bysubstituting
jA
fromthe secondequationtotherstone obtainsjCj=1.
Nowthe at directioncan be parameterized by N complex scalar elds A
i
and areal
phaseÆ
c
. Wecanfurtherreducethenumberofindependenteldsbychoosingagauge.
WithanSU(N)transformationonecanchooseN 1ofthecomponentsof
i
tovanish
i.e.
A
i
=0for A=2;:::;N. What isleft isessentially acomplexscalar eld 1
i
,
a real phase Æ
C and
j1
= e iÆ
C
. One can still transform the phase Æ
C
away in such
a way that
j1
= 1
i
=. So nally,the at direction(
i
j
) isparameterized by one
complex scalar eld up togauge transformations.
Let usnext takeI(
i
1
;:::;
i
N
)=(
i
1
i
N
). All the avour indices i
j
have to
be dierent, i.e. i
j 6= i
k
, for j 6= k because of the anti-symmetrization in Eq. (45).
Becauseofthisthenumberofavours,m,hastobelargerthanorequaltothenumber
of colours, N. Now the D atnessconstraint Eq. (41) reads
A
1 A
N
A1
i
1
A
k 1
i
k 1
A
k +1
i
k +1
A
N
i
N
=C
A
k
i
k
; k =1;:::;N: (47)
By multiplying Eq. (47) with A
k
i
k
and summingover A
k
one obtains
(
i1
i
N
)=Cj
ij j
2
; for all i
j
=1;:::;m: (48)
Hence one sees that all the vectors
ij
have the same length, j
ij
j jj. Let us now
multiplyEq. (47) with A
k
i
j
and sum over A
k
, wherek 6=j. One then obtains
C <
ij
;
i
k
>=
A1A
N
A
1
i
1
A
k 1
i
k 1
A
k
i
j
A
k +1
i
k +1
A
j
i
j
A
N
i
N
=0; (49)
because A
j
indices are antisymmetrized. Thus all the elds have the same length
and are orthogonal. There is one more constraint, which follows when one takes the
absolute value squared of Eq. (47) and sums over A
k
. One obtains
jCj 2
j
i
k j
2
= X
A;B
A
1 A
k 1 A
k A
k +1 A
N
A
1
i
1
A
k 1
i
k 1
A
k +1
i
k +1
A
N
i
N
B
1 B
k 1 A
k B
k +1 B
N
B1
i
1
B
k 1
i
k 1
B
k +1
i
k +1
B
N
i
N
= Æ B
1 B
k 1 B
k +1 B
N
A
1 A
k 1 A
k +1 A
N
A1
i
1
A
k 1
i
k 1
A
k +1
i
k +1
A
N
i
N
B1
i
1
B
k 1
i
k 1
B
k +1
i
k +1
B
N
i
N
= j
i
1 j
2
j
i
k 1 j
2
j
i
k +1 j
2
j
i
N j
2
=jj 2(N 1)
; (50)
from which follows that jCj = jj . In the third line only the combination with
A
1
= B
1
;:::;A
N
= B
N
survives and all the other combinations vanish because of
Eq. (49). Altogether the at direction is parameterized by N complex scalar elds
whichareorthogonalandhavethesamenormalization<
ij
;
i
k
>= jj 2
Æ
jk
. Choosing
a gauge one can take
T
i
j
=(00jje iÆj
00) (51)
and x N 1 of the phases Æ
j
; j = 2;:::;N to be equal to Æ
1
arg. The phase
of C is determined fromEq. (47) by insertingEq. (51). One then obtains Æ
C
= P
Æ
j .
Hence againtheat directionisparameterizedby onecomplexscalareldsuchthat
k
i
j
= Æ k
j
. The at direction(
i
1
i
N
) is similar. These at directions can again
belifted by adding asuitable monomialtothe superpotential.
3.3 Flat directions of the MSSM
IntheMSSMtherearemanyatdirectionsattherenormalizablelevel. However,these
are only approximately at, since supersymmetry breaking lifts the at directions.
There are also other eects that lift the at directions, as will be discussed in the
Section 3.4. Inthe present Sectionwe justcatalogue the basis of atdirections of the
MSSM in the globally supersymmetric limit.
TheeldcontentofMSSM wasdiscussedinSection2.4. Here weapplythemethod
of Section3.2to the SU(3)
C
SU(2)
L
U(1)
Y
symmetry group of MSSM. We form
the basisof gaugeinvariantmonomialsbyrstformingSU(3) invariantcombinations,
then out of these SU(2)invariant combinationsand nally from the SU(3)SU(2)
invariantmonomialstheU(1)invariantcombinations. ThenbyapplyingtheF atness
constraints one obtains the basis of gauge invariant monomials of the MSSM. This
analysis was carriedout by Gherghetta,Kolda and Martin [50]. Here we only review
the main points. The nal basis of at directions can be found in Table 1 at the end
of this Section.
UnderSU(3)
C
the chiral supereldstransform assinglets(e
i
;L
i
; H
u
; H
d
), triplets
(Q
i
) or antitriplets (u
i
;
d
i
). As mentioned in Section3.1, the gauge invarianttensors
of SU(3) are
ABC
; and Æ
A
. Contracting these with the quark superelds one
obtains
(Q
i q
j
) Q
A
i q
jA (Q
i q
j
); (52)
(Q
i Q
j Q
k
)
ABC Q
A
i Q
B
j Q
C
k
; (53)
(q
i q
j q
k
)
ABC
q
iA q
jB q
kC
; (54)
where q
i
= u
i
;
d
i
. These SU(3) -invariant products of chiral superelds in Eq. (52)
generate a reducible representation of SU(2) in general. The doublet representation
can bereducedintoirreducibleparts. Theproductof (Q
i q
j
)naturallygivesadoublet,
since q
j
is a singlet under SU(2). The product (q
i q
j q
k
) is also a singlet under SU(2)
and thusinvariantwith respect toit. The product (Q
i Q
j Q
k
)reduces as
222=2+2+4: (55)
Here 4isasymmetricrepresentationunderSU(2)andthus(Q
i Q
j Q
k )
4
hastobeanti-
symmetric inits family indices. Therefore thereis a unique SU(3)
C
-singletmade out
of three Q's, which is a4 under SU(2)
L
,namely
(Q
i Q
j Q
k )
()
4
Q A
i Q
B
j Q
C
k
ABC
ijk
: (56)
The remaining combinations are SU(2)
L
doublets, where one pair of Q's has been
antisymmetrized, i.e. contracted with
,and can bewritten inthe form
(Q
i Q
j Q
k )
Q A
i Q
B
j Q
C
k
ABC
; (57)
which is subject tothe constraint that not all three of the family indices are allowed
to be the same due to the anti-symmetrization. In Eq. (57) there are two other pos-
sibilities to contract with
toproduce a 2 under SU(2). Withthe antiquarkelds
q'sthesituationissimpler,sincetheyare SU(2)-singlets,i.e. invariantunderSU(2)
L .
The rest of the superelds are either doublets orsinglets under SU(2).
NowtheSU(3)
C
SU(2)
L
atdirectionscanbeformedbycombining(Q
i Q
j Q
k )
4 ,
(Q
i Q
j Q
k )
, (Q
i q
j )
, L
, H
u
and H
d
into SU(2) -singlets. This is achieved by con-
tracting all the terms with
, for example
(Q
i Q
j Q
k )
L
m
. Then it is already
Y
combinations, for example
H
d L
i e
j
. For detailssee [50].
Letus now consider a couple of examplesof the eld congurations of the MSSM
D at directions such as
Q
A
i L
j
d
kA and
H
u L
i
, which lead tothe eld congu-
rations (up to agauge transformation)
Q 1
i
= 0
@ 1
p
3
0 1
A
; L
j
= 0
@ 0
1
p
3
1
A
;
d 1
k
= 1
p
3
;
H
u
= 0
@ 1
p
2
0 1
A
; L
i
= 0
@ 0
1
p
2
1
A
: (58)
We now have to apply the constraints coming from F-terms, Eq. (37), using the
superpotentialof the MSSM given inEq. (28), to the D at monomials. We require
F
Hu
= H
d +y
ij
u Q
i u
j
=0; (59)
F
H
d
= H
u +y
ij
d Q
i
d
j +y
ij
e L
i e
j
=0; (60)
F a
Q
i
= y ij
u H
u u a
j +y
ij
d H
d
d a
j
=0; (61)
F
L
i
= y ij
e H
d e
j
=0; (62)
F a
u
i
= y ji
u H
u Q
a
j
=0; (63)
F a
d
i
= y ji
d H
d Q
a
j
=0; (64)
F
e
i
= y ji
e H
d L
j
=0: (65)
Assuming that the Yukawa matrices, y ij
u;d;e
,are non-singular we cancancel them from
the constraints Eqs. (62)-(65). When contracting the constraints (59)-(65) to form
gauge-invariantcombinations one can see that some of the D-at directions are con-
strained to vanish. In that case wesay that the at directionis lifted. Eqs. (62)-(65)
show that all the at directions, where H
d
is contracted with any other eld except
H
u
, vanish. From the example Eq. (58) one can see that Q
i L
j
d
k
is F at, too, but
L
i H
d e
j
vanishes by Eqs. (62) and (65) as it contains H
d .
One should note that in general M
W
, i.e. it is of the same order as the soft
SUSY breaking masses. Therefore we regard a direction at, if it is at up to the
when classifying the at directions of the MSSM and includingit inthe terms lifting
the at directions.
One could also include the R -parity violating terms given in Eq. (29) to the su-
perpotential. In that case Eqs. (59)-(65) would become more complicated and the
structure of at directions would change. However, R -parity violating terms break
explicitly the baryonand lepton number conservation atthe renormalizableleveland
would lead toa rapid protondecay unless the couplings are ne-tuned.
Table 1: Basis of at directions of the MSSM
Flatdir. B L Flatdir. B L Flat dir. B L
LH
u
-1 QQQL 0 uuuee 1
H
u H
d
0 QuQ
d 0 QuQue 1
u
d
d -1 QuLe 0 QQQQu 1
LLe -1 uu
de 0 (QQQ)
4
LLLe -1
Q
dL -1
d
d
dLL -3 uu
dQ
dQ
d -1
3.4 Lifting the at directions
So far we have described the method of how to nd all the at directions given a
gauge group and the matterelds. Since the at directions correspond tocontinuous
degeneracy of vacua, one gets an innite number of possibilities for gauge symme-
try breaking. Supersymmetry breaking eects and non-renormalizableterms lift this
degeneracy.
As pointed out before, soft SUSY breakingterms lift the at directions. However,
the F-term constraints Eqs. (59)-(65) make the A-terms in the soft SUSY breaking
potentialtovanishintheatdirection. ThereforeonlythemasstermsandtheB-term
can lift the at directions with
V
soft
= m
2
Hu jH
u j
2
+m 2
H
d jH
d j
2
+m 2
L jLj
2
+m 2
e jej
2
+m 2
Q jQj
2
+m 2
u juj
2
+m 2
d j
dj 2
+(Bm
3=2
H
u H
d
+h:c:)+ 2
jH
u j
2
+ 2
jH
d j
2
: (66)
The B-term is important only for the H
u H
d
at direction. The mass terms coming
fromthe-termofthesuperpotentialhavebeenincludedhere,sincethey wereignored
in section3.3. However, this is importantonlyfor the at directions H
u
L and H
u H
d .
The softterms in Eq. (66) generallyproduce a mass term for the at direction
V()=m 2
jj
2
; where m
2
= N
X
i=1 a
2
i m
2
i
; (67)
when there are N elds gaining a VEV along a at direction. Normalization is such
that
i
= p
a
i and
P
a 2
i
=1.
Theeasiestwaytoliftaat directionistoaddanoperator,whichisformed ofthe
gauge invariantpolynomial describing the at direction. In that case the polynomial
will become of the form I() n
, if I is composed of a monomial of degree n. If
the at directionisdescribed by arenormalizableoperator,then onehas toraise itto
some powertoobtainanon-renormalizableoperator,sincethe formofrenormalizable
operatorsinthe superpotentialisrestricted. Ofcourseitispossiblethat anotherkind
of operator of lower degree can accomplish the lifting, too. In general one obtains a
superpotentialterm of the form
W =
dM d 3
d
(68)
where d =nk,and M issome large mass scale such asthe GUTor Planck scale. We
generically identify M withthe Planck scale M
p
=2:410 18
GeV.
There is also another type of operator, which lifts the at direction. It consists
of a eld not in the at direction and some number of elds which make up the at
direction:
W =
M d 3
d 1
: (69)
For a superpotential of this form F is non-zero along the at direction. In the at
space limit,with minimal kinetic terms, the lowest order contributions of either type
of superpotentialterm, i.e. Eqs. (68) and (69), giverise toa potential
V()=
@W()
@
2
= jj
2
M 2(d 3)
jj 2(d 1)
; (70)
dominate the potential.
We already considered SUSY breaking. However, if we consider the general form
of the hiddensector SUSY breaking,one has A-terms of the form[12,14,51]
V()=Am
3=2
W()= Am
3=2
M d 3
p
d
; (71)
where the superpotentialis of the formEq. (68).
Inthe earlyuniversethereare eects, whichproduceterms tothe scalarpotential,
that are dominating over the soft SUSY breaking terms. In a radiation dominated
era the relevant scale of excitations is the temperature of the universe. On the other
hand, during the inationary era the scale of quantum deSitter uctuations is given
by the Hubble constant. When a at direction acquires a large VEV, the non-at
directions gain a mass,m
? y
u;d;e
j<>j,from Yukawaand gauge couplings inthe
superpotentialEq. (28). Alsothe gauge particlesgain masses, m
g
gj<>jwhere
g isthegauge coupling,through super-Higgsmechanism,sincethe gaugesymmetry is
broken along the at direction. ForlargeVEVs the masses of the non-atmodes and
gauge particles become heavier than the excitation scales. This causes the non-at
directions toquickly settle ontotheir minimum values, andthey eectively decouple.
Therefore allthe dynamicstakes place inthe modulispace, when the elds are large.
In general one at direction may not give a mass to all other directions. In that
case there can be other directions gaining a VEV, too. In terms of gauge invariant
polynomialsthis means that new terms are addedto the polynomial.
The most important source of SUSY breaking in the Early Universe is the nite
energy density of the Universe [14,51]. If the energy density has a non-zero expecta-
tion value, it implies that the supercharge does not annihilatethe vacuum state and
supersymmetry is thus broken. During the inationary epoch the vacuum energy is
positive by denition and SUSY is broken. During the post-inationary epoch the
energy density is dominated by inaton oscillationsso that the vacuum energy aver-
aged over time is non-vanishing. In a radiationdominated era SUSY is also broken,
since the occupation numbers of bosons and fermions in the thermal backround are
deSitteructuations,whichgivedierentoccupationnumbersforbosonsandfermions.
However, the thermal and quantum eects are in generalless important atlarge eld
values than the SUSYbreaking due tonite energy, althoughthis may not always be
the case, especiallyfor the thermaleects.
A simple way of generating a mass term for the at directionin the global super-
symmetric limitis toadd tothe Kahlerpotential,J ofEq. (20),acontributionof the
form (we ignorethe vector superelds here)
ÆJ = 1
M 2
p Z
d 2
d 2
I y
I y
; (72)
where I is a eld that dominates the energy density of the universe and is the
canonically normalized at direction. If I dominates the energy density, then
R
d 2
d 2
I y
I. Using the Friedmann equation = 3M 2
p H
2
one can see that Eq. (72)
generates a mass term for the scalar proportionalto H 2
. So, for H > m
the
nite energy SUSY breaking is moreimportantthan soft SUSY breaking.
Since Planck scale operators are discussed, supergravity interactions should be
included. The general scalar potential for supergravity is given in Eq. (25). The
generalstudy ofinducingapotentialfortheatdirectionswasdonein[14]forF-term
inationand in[52] forD-termination(F and Dterminationingeneralhavebeen
discussed in [55]. Here we only givea simple example leadingto the simplest formof
potentialinthe atdirections. WeaddtotheminimalKahlerpotentialaninteraction
that producesa mass of order Hubble scale:
K(;I)=I y
I+ y
+
M 2
p I
y
I y
; (73)
where jj 1 and I; are scalar parts of the superelds. We also assume that the
superpotential can be split intotwo parts W = W(I)+W() related tothe inaton
and the at direction, respectively.
AssumingI; M
p
weexpandtheF-termpartofthepotentialEq. (25)toobtain
V
F
= jW
I j
2 3
M 2
p
jW(I)j 2
+ I
y
I
M 2
p jW
I j
2
+jW
j
2
+
1
M 2
p jW
I j
2
y
+ 1
M 2
p
[W(I)
(W
3W())+h:c:]: (74)
dominatesthe inatonpotentialoneobtainsfromtheFriedmannequationW
I
M
p H
and W(I) M 2
p
H. The third term in that case gives a Hubble-squared correction
to the inatonmass, whichcauses problems for slow-rollconditions. The fourth term
gives the F-term part for the potential of the at direction. The renormalizablepart
of the superpotential Eq. (28) does not contribute, because it vanishes along the at
direction. Therst termonthe secondrowgivestheHubblesquaredcorrectiontothe
mass ofthe atdirection. ThesecondtermproducesA-typeterms,whichareoforder
Hubble. Altogether one obtains the following potential for the at direction, where
the soft SUSYbreaking terms Eqs. (66,71) have alsobeen taken into account:
V
F
()=m 2
jj
2
+c
H H
2
jj 2
+
Am
3=2 +aH
dM d 3
p
d
+h:c:
+ jj
2
M 2(d 3)
p
jj 2(d 1)
; (75)
whereA; a; 1areingeneralcomplexconstants,c
H
1isrealanditssigndepends
on . In particular,the induced Hubble squared mass is negative if >1.
The low-energy expansion of the D-term part of the potential Eq.(25) with the
KahlerpotentialEq. (73)isdoneinthesameway. Firstwend thepotentialtoorder
M 2
p ,
V
D
= 1
2 Ref
1
AB (I
y
T A
I+)(I y
T B
I+)+
M 2
p
y
(I y
T A
I+I y
T B
I)+2(I y
T A
I)(I y
T B
I)
+O(M 4
p
): (76)
The Fayet-Iliopoulos contribution, , is included here and it is understood that it is
non-zero only if the symmetry group is U(1), which is the typical scenario in the D-
term ination [52]. For a U(1) symmetry, the generators, T A
, are just the charges
of the elds. The D-terms of donot contribute, because they vanish by denition,
see Eq. (36). If the inaton potential is dominated by the D-term and in particular
by its Fayet-Iliopoulos term, the at direction receives a negligible correction to its
mass. If, on the other hand, the potential is dominated by the terms that do not
depend on,whichis the caseafter ination,thenthe at directionreceivesanorder
Hubble correction to its mass. Another dierencewith respect to F-term ination is
that there are no A-terms in the D-term ination case. However, the superpotential
SUSY breaking terms. Sonallywith D term inationthe potential reads
V
D
()=m 2
jj
2
+c
H H
2
jj 2
+
Am
3=2
dM d 3
p
d
+h:c:
+ jj
2
M 2(d 3)
p
jj 2(d 1)
; (77)
wherec
H
vanishesduringinationandc
H
1afterination[52] withc
H
<0if <0.
