Nucl.Phys.B v.602

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Nuclear Physics B 602 (2001) 3–22

www.elsevier.nl/locate/npe

A warped supersymmetric standard model

Tony Gherghetta a, Alex Pomarol ba IPT, University of Lausanne, CH-1015 Lausanne, Switzerland

b IFAE, Universitat Autònoma de Barcelona, E-08193 Barcelona, Spain

Received 10 January 2001; accepted 15 May 2001

Abstract

We study the breaking of supersymmetry in five-dimensional (5d) warped spaces, using theRandall–Sundrum model as a prototype. In particular, we present a supersymmetry-breakingmechanism which has a geometrical origin, and consists of imposing different boundary conditionsbetween the fermions and bosons living in the 5d bulk. The scale of supersymmetry breakingis exponentially small due to the warp factor of the AdS metric. We apply this mechanism to asupersymmetric standard model where supersymmetry breaking is transmitted through the AdS bulkto matter fields confined on the Planck brane. This leads to a predictable superparticle mass spectrumwhere the gravitino mass is 10−3 eV and scalar particles receive masses at the one-loop level viabulk gauge interactions. We calculate the mass spectrum in full detail using the 5d AdS propagators.The AdS/CFT correspondence suggests that our 5d warped model is dual to the ordinary 4d MSSMwith a strongly coupled CFT sector responsible for the breaking of supersymmetry. © 2001 ElsevierScience B.V. All rights reserved.

1. Introduction

The standard model is believed to be an effective theory valid up to some energyscale near the electroweak scale. What lies beyond the standard model has been thesubject of active research. Among the possible candidates, there are technicolor theories,supersymmetry, and, recently, extra dimensions [1].

Extra dimensions and supersymmetry present an additional motivation. They couldbe an important ingredient in the underlying theory that includes a quantum descriptionof gravity, and in particular for string theory they play a crucial role. A particularly

interesting extra dimension scenario is the Randall–Sundrum model [2]. In this modelthe extra dimension is compactified in a slice of anti-de-Sitter (AdS) space, and, as a

E-mail address: [email protected] (T. Gherghetta).

0550-3213/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0 5 5 0 -3 2 1 3 (0 1 )0 0 1 2 7 -4

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4 T. Gherghetta, A. Pomarol / Nuclear Physics B 602 (2001) 3–22

consequence, the electroweak scale is generated by an exponential warp factor in the

metric. This model can be supersymmetrized [3–5] providing an interesting alternative to

the minimal supersymmetric standard model (MSSM), and a possible connection to stringtheories [6].

In this article we want to continue the study of supersymmetric extensions of the

standard model living in five dimensions where the extra dimension is compactified as

in the Randall–Sundrum model [3,7]. In particular, we want to study supersymmetry

breaking. A warped extra dimension allows for new ways of breaking supersymmetry. The

particular mechanism that we will consider here consists of imposing different boundary

conditions between the fermions and bosons. This supersymmetrybreaking mechanism has

been previously studied in flat space but not in warped spaces. In warped space this way of

breaking supersymmetry leads to novel phenomenologicalconsequences. For example, the

radius of compactification does not need to be large (TeV−1) as in the case of flat space [8].Therefore it can be consistent with a large cut-off scale that is related with the Planck

scale, M P , or grand unified theories [7,9]. As we will show, the scale of supersymmetry

breaking can be very low (∼ TeV) and this implies a superlight gravitino m3/2 ∼ 10−3 eV.Scalars are massless at tree-level and get masses at the one-loop level. We will study in

detail a “warped” version of the MSSM, where gravity and gauge bosons live in the five-

dimensional (5d) AdS bulk, while matter fields are located on one of the boundaries, the

Planck brane. In this warped MSSM the squark and slepton masses arise at one-loop fromthe gauge interactions and are therefore naturally flavor independent. One of the most

interesting properties of the model is its predictivity of the low-energy mass spectrum.

We will calculate it here in full detail. Although we present the calculation for a particular

extension of the standard model, the calculation of quantum effects in warped spaces that

we present here is much more general and can also be useful for other scenarios.

Another important motivation for the study of the MSSM in a slice of AdS arises from

the AdS/CFT correspondence [10]. This conjecture suggests that these 5d models have a

strongly-coupled 4d dual [11–14]. Therefore, the study of the weakly coupled 5d gravity

theory here will be helpful in understanding supersymmetric 4d theories with a strongly

coupled sector. We will comment later on this duality.

In Section 2 we introduce the Randall–Sundrum compactification and its supersymmet-

ric version. In particular we also analyze the gravitino Kaluza–Klein decomposition since

it is the only field not considered in Ref. [3] (see also [15]). In Section 3, we present the

supersymmetry-breaking mechanism, which is based on imposing “twisted” boundary con-

ditions for fermions in the bulk, and comment on the differences compared with the case

of a flat extra dimension. In Section 4 we introduce a version of the MSSM living in a slice

of AdS and calculate the sparticle mass spectrum, at tree-level and at one-loop level. Wewill also comment on the holographic interpretation of the model. Our concluding remarks

appear in Section 5. Finally in the appendix, we present a detailed calculation of the 5d

propagators in a slice of AdS.

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2. The warped supersymmetric brane-world

We will consider the scenario of Ref. [2], which is based on a nonfactorizable 5dgeometry. The fifth dimension y is compactified on an orbifold, S 1/Z2 of radius R,with 0 y π R. The boundary of the 5d spacetime consists of two 3-branes locatedat the orbifold fixed points y∗ = 0 and y∗ = π R. This configuration with the 5d metricsolution [2]

(1)ds2 = e−2σ ηµν d xµ dxν + dy 2 ≡ gMN d xM dxN ,is a slice of AdS space, where σ = k|y| and 1/k is the AdS curvature radius. The 5dcoordinates are labelled by capital Latin letters, M = (µ, 5) where µ = 0, . . . , 3. Thecomplete supergravity action for this configuration is obtained by including the gravitinoand graviphoton together with the graviton [3–5]. However, for the discussion of supersymmetry breaking it will suffice to only consider the additional gravitino kineticand mass terms, which are given by [3]

S = S 5 + S (0) + S (πR),S 5 =

d 4x

dy

√ −g

× − 12 M 35 R+ i Ψ iM γ MNP DN Ψ iP − i 32 σ Ψ iM γ MN (σ 3)ij Ψ j N − Λ,(2)S (y∗) = d 4x √ −g4 L(y∗) − Λ(y∗),

where g4 is the induced metric on the 3-brane located at y∗, and γ M 1M 2...M n =1n! γ

[M 1 γ M 2 · · · γ M n] is the antisymmetrized product of gamma matrices. We have definedσ = dσ/dy. Supersymmetry automatically ensures the bulk/boundary conditions Λ(0) =−Λ(πR) = −Λ/ k. The action contains the 5d Planck scale M 5, the 5d Ricci scalar R, twosymplectic-Majorana gravitinos, Ψ iM (i = 1, 2), and a bulk cosmological constant Λ. Aty∗ = 0 the effective 4d mass scale is of order of the Planck scale, M 2P M 35 /k, and wewill refer to the brane there as the Planck brane. Similarly, at y∗ = π R the effective mass

scale is of order M P e−π kR

, which is the TeV scale for kR 11. Consequently the 3-branelocated there will be referred to as the TeV brane. The index i labels the fundamentalrepresentation of the SU(2)R automorphism group of the N = 1 supersymmetry algebra infive dimensions. The gravitino supersymmetry transformation is given by [3]

(3)δΨ iM = DM ηi + σ

2 γ M (σ 3)

ij ηj ,

where the symplectic-Majorana spinor ηi is the 5d supersymmetry parameter.Similarly, gauge bosons and matter can be added to the bulk [3,9,16–18]. In a 5d

supersymmetric theory they form part of vector supermultiplets and hypermultiplets. The

behavior of these supermultiplets in the background of Eq. (1) was considered in Ref. [3],where the Kaluza–Klein mass spectrum was also derived. Only the analysis of the gravitinofield was not presented in Ref. [3]. For this reason, we will present below the Kaluza–Kleindecomposition of the gravitino. This will also help to show how the superHiggs mechanism

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operates level-by-level in the Kaluza–Klein modes, and will help to better understand thesupersymmetry breaking mechanism presented in the next section.

2.1. Kaluza–Klein decomposition of the gravitino and the superHiggs mechanism level

by level

Let us start by decomposing the 5d gravitino, Ψ M , and the 5d supersymmetryparameter, η, into 4d Kaluza–Klein fields

Ψ µL,R(xµ, y) =

∞n=0

ψ(n)µL,R(x

µ) f (n)

L,R (y),

Ψ 5 L,R (xµ, y) = ∞

n=0ψ(n)5 L,R(x

µ)f (n)5 L,R (y),

(4)ηL,R (xµ, y) =

∞n=0

η(n)L,R (x

µ)f (n)

L,R (y).

We have dropped the SU(2)R index i , since we need only consider the i = 1 component(the i = 2 component is simply obtained from the symplectic-Majorana condition). Wehave also defined γ 5Ψ L,R = ±Ψ L,R . It is important to note that we have chosen they-dependent wavefunction of the supersymmetry parameter η to be the same as that for

the Kaluza–Klein gravitinos.

2.1.1. Kaluza–Klein modes n = 0The supersymmetry transformation Eq. (3) for i = 1 gives

δΨ µ = ∂µη + σ γ µ

1 − γ 52

η,

(5)δΨ 5 = ∂5η + σ γ 52

η.

Substituting Eq. (4) into Eq. (5) and projecting out the nth mode,1

we find that thesupersymmetry transformation for the nth Kaluza–Klein gravitino mode is given by

(6)δψ (n)µ L = ∂µη(n)L + γ̃ µ∞

k=0ankη

(k)R ,

(7)δψ (n)µ R = ∂µη(n)R ,where γ̃ µ is the 4d Minkowski gamma matrix and the coefficients ank are given by

(8)ank ≡ dy e

−2σ σ f (n)L

(y)f (k)

R

(y).

1 This corresponds to multiplying each side of Eq. (5) by f (n)L,R

, integrating over y, and using the gravitino

orthogonality condition

d y e−σ f (n)L,R

f (m)

L,R = δnm.

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The coefficients ank imply that the supersymmetry transformation of ψ(n)µ L at level n,

depends nontrivially on the complete tower of Kaluza–Klein parameters η(k)R . This effect

is entirely due to the fact that the bulk is AdS. Let us now impose the following relationfor the wavefunctions of Ψ 5:

(9)f (n)5 L,R = 1

mn

±∂5 + 12 σ f (n)L,R ,where mn is the 4d mass of the gravitino Kaluza–Klein mode n, which will be derivedbelow. The condition (9) allows us to write a simple expression for the supersymmetrytransformation of Kaluza–Klein modes ψ (n)5

(10)δψ (n)5 L =

mnη(n)L ,

(11)δψ (n)5 R = −mnη(n)R .This shows that the nth Kaluza–Klein mode of the 5th component of the gravitinotransforms as a Goldstino of the η(n) supersymmetry transformation and that these N = 2supersymmetries are nonlinearly realized. We can now see that the redefined gravitinos

ψ̃(n)µ L ≡ mnψ(n)µ L − ∂µψ(n)5 L + mn ̃γ µ

∞k=0

ankψ

(k)5 R

mk,

(12)ψ̃(n)µ R ≡

mnψ(n)µ R

+∂µψ

(n)5 R ,

are invariant under supersymmetry transformations, and therefore correspond to thephysical fields. On the contrary, the fields ψ (n)5 are gauge dependent and can be eliminated.

This is the superHiggs mechanism. The ψ(n)5 are eaten by the gravitino ψ(n)µ to become

massive.Let us now turn to the Rarita–Schwinger equation for the bulk gravitino, which in the

AdS background reads

(13)γ MNP DN Ψ P − 3

2σ γ MP Ψ P = 0.

Using the redefined gravitino fields (12), the equation of motion (13) simplifies to

(14)γ µνρ ∂ν ψ̃(n)ρ L ,R − mnγ µρψ̃(n)ρR,L = 0,

which represents the 4d massive Rarita–Schwinger equation for the spin 3/2 field ψ̃(n)µ ,and where the y -dependent Kaluza–Klein wavefunctions satisfy

(15)

∂5 + 12 σ

f (n)

L = mneσ f (n)R ,(16)

∂5 − 52 σ

f

(n)R = −mneσ f (n)L .

One can see that the dependence on ψ(n)

5 has dropped out and the equation of motiondepends, as expected, only on ψ̃(n)µ . The solutions of Eqs. (15) and (16) are a special caseof the general solution appearing in Ref. [3]. In fact, defining f̂ (n)L,R = e−σ f (n)L,R one cansee that f̂ (n)L,R corresponds to the wavefunction of a “hatted” fermion of mass m = 3σ /2

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defined in Ref. [3]. Thus, using the results in Ref. [3] and the fact that Ψ µL (Ψ µR ) are

defined even (odd) under the Z2-parity, we obtain the y-dependentgravitino wavefunctions

(17)f (n)L = 1

N ne

32 σ

J 2mn

keσ

+ b(mn)Y 2mn

keσ

,

(18)f (n)R = σ

kN ne

32 σ

J 1

mn

keσ

+ b(mn)Y 1

mn

keσ

,

where J α and Y α are Bessel functions, N n are normalization constants and the coeffi-

cients b(mn) satisfy

(19)b(mn)= −

J 1(mn

k )

Y 1( mnk ),

(20)b(mn) = b

mneπ kR

.

The Kaluza–Klein masses of the gravitinos ψ̃(n)µ can be obtained by solving (20), and forn > 0 they are approximately given by

(21)mn

n + 14

π ke−π kR .

Finally, using (15) and (16), and the fact that Ψ 5 R (Ψ 5 L) are even (odd) under the Z2-parity,

we have from the condition (9)

(22)f (n)5 L = eσ f (n)R ,

(23)f (n)5 R = eσ f (n)L − 2σ

mnf

(n)R .

2.1.2. Massless sector

The y -dependence of the gravitino zero-mode wavefunction is obtained from Eq. (15),

since under the orbifold symmetry, f (0)R is projected out. Thus for the remaining mode, f (0)

L

with m0 = 0, we obtain(24)f (0)L (y) =

1√ N 0

e−12 σ ,

where the normalization factor N 0 = (1 − e−2π kR )/k. This is consistent with the y-dependence of the graviton zero-mode wavefunction, as expected from supersymmetry.

Similarly, η(0)R is projected out and η(0)L whose wavefunction is also given by Eq. (24)

parametrizes the remaining N = 1 supersymmetry of the theory. In fact one can check thatEq. (24) satisfies the Killing spinor condition.

Similarly, for the fifth-component of the gravitino, we have that ψ(0)5 L is projected out

and only ψ (0)5 R remains in the theory. This corresponds to the supersymmetric partner of the

radion, the “radino”. The 4d effective Lagrangian of this field has been recently presentedin Ref. [19].

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3. Supersymmetry breaking in a slice of AdS

Different mechanisms of supersymmetry breaking in brane-world scenarios have beenconsidered in the past. The most popular, based on the Horava–Witten model, correspondsto breaking supersymmetry in a hidden-sector living on a brane located at a finite distancefrom the observable-sector brane [20]. The moduli (e.g., the dilaton and radion) play therole of messengers communicating the supersymmetry-breaking from the hidden to theobservable sector. These scenarios rely on gaugino condensation to occur on the hidden-sector brane in order to explain the hierarchy.

Warped (AdS) spaces allow for new possibilities. First of all, since the hierarchy is nowexplained by the warp factor, one does not need a gaugino condensation in a hidden sectorto be responsible for a small supersymmetry breaking. Supersymmetry can be broken attree-level if it occurs on the TeV brane, and therefore have a stringy origin.

The supersymmetry-breaking mechanism that we will consider here is based onimposing different boundary conditions between fermions and bosons on the TeV brane.This breaks supersymmetry for the bulk fields, and as we shall see, the Kaluza–Klein fermions and bosons receive TeV mass-splittings. The mechanism consists of thefollowing. The 5d bulk fields in the supersymmetric Z2 orbifold can be classified as eitherodd or even fields under the Z2 parity. For the 5d fermions, we have two possibilities todefine the Z2 parity, namely

(25)ψ(−y) = ±γ 5ψ(y).Once a choice is made, this also defines the chirality on the 4d boundary at y ∗ = 0, sinceψ(0) = ±γ 5ψ(0). For the supersymmetric Z2 orbifold the same chirality is chosen on thetwo boundaries at y∗ = 0 and y∗ = π R. In this way only half of the bulk supersymmetry isbroken by the boundaries, leaving an N = 1 supersymmetric theory at the massless level.However, there also exists the possibility to separately define the chirality of fermions onthe two boundaries. For example, the choice

(26)ψ(0) = γ 5ψ(0), ψ(π R) = −γ 5ψ(πR),

corresponds to the following y -dependence(27)ψ(−y) = γ 5ψ(y), ψ(−y + π R) = −γ 5ψ(y + πR).

Thus, under a 2π rotation around the circle S 1, Eq. (27) leads to fermions that areantiperiodic

(28)ψ(y + 2π R) = −ψ(y).As will be shown below, the boundary conditions (26) project out the massless fermionmodes arising from bulk fields. Also supersymmetry is now completely broken since noKilling spinor can be defined. 2

The fermionic boundary conditions (26) have been considered previously in theliterature. If the space of the extra dimension is flat, imposing these boundary conditions

2 This has some similarities with finite temperature which also breaks supersymmetry [21].

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exactly corresponds to breaking supersymmetry by the Scherk–Schwarz mechanism [22].This mechanism has been applied to the MSSM in Refs. [8,23–26]. In the Horava–Witten

theory this was studied in Ref. [27]. However, for warped spaces the fermionic boundaryconditions Eq. (26) do not correspond to the Scherk–Schwarz mechanism, because thisrequires a smooth limit where supersymmetry is restored [22]. In the case of warped spaces,we have not found such a smooth limit. 3

Let us now study the fermionic spectrum with the boundary conditions (26), which wewill refer to as “twisted” boundary conditions. The resulting Kaluza–Klein mass spectrumfor ψL is now determined by (see appendix)

(29)J α−1( mnk )Y α

−1(

mn

k

)= J α (

mnk

eπ kR )

Y α(mn

k

eπ kR ),

where α = |c + 1/2| for a fermion of Dirac-type mass m = cσ and α = 2 for thegravitino. One can easily check that the equation resulting from imposing twisted boundaryconditions on ψR leads to an identical Kaluza–Klein spectrum. The first thing to notice inEq. (29) is that mn = 0 is no longer a solution of the above equation and therefore nomassless fermions are present. Consequently, supersymmetry is now completely broken.One can also see that the Killing spinor, η(0)L , whose wavefunction is identical to Eq. (24),is not consistent with the new boundary conditions. Thus, the only change with respectto Section 2.1 is that Eq. (20) is now replaced by Eq. (29), and the massless sector of

Subsection 2.1.2, is no longer present in the theory. Notice that from Eq. (10) the Goldstinoof the broken N = 1 supersymmetry is ψ (0)5 L .

In the limit mn k and kR 1, the solution of Eq. (29) is given by

(30)mn

n + α2 − 1

4

π ke−π kR .

Comparing with the result for “untwisted” boundary conditions [3], one finds thatthe Kaluza–Klein mass spectrum is shifted by a value that asymptotically approaches1/2(πke−π kR ). This is to be contrasted with the flat case where the shift in the Kaluza–

Klein mass spectrum is 1/(2R).There is an important difference when this type of supersymmetry breaking is realized inwarped spaces compared to the flat case. In flat spaces this type of supersymmetry breakingis global. To see this, let us consider an observer living on one of the branes with the otherbrane sent to infinity (R → ∞). In this limit and in flat space, supersymmetry is restoredbecause the Kaluza–Klein spectrum becomes continuous (the scalar-fermion mass splittingdisappears). This is related to the fact that one can locally (i.e., on either brane) definea supersymmetric theory. Breaking supersymmetry globally (when the extra dimensionis compact, no Killing spinor can be defined in the whole space) leads to the important

3 Similarly, one can also show that the models of Refs. [24,25] with only one Higgs hypermultiplet (instead of two) do not have this smooth limit to a supersymmetric theory. This alternative has recently been considered inRef. [26], where it was shown that the boundary conditions Eq. (26) can also be understood as compactifying ona S 1/(Z2 × Z2) orbifold.

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property that the vacuum energy and the one-loop scalar masses are finite and independentof the cut-off scale [25,28].

In warped spaces the situation is different and the finiteness of the one-loop scalarmasses depends on which particular brane the observable sector lives. Consider first theobservable sector on the Planck brane where the TeV brane is sent off to infinity. In thislimit supersymmetry is restored on the Planck brane because the Kaluza–Klein spectrumbecomes continuous. Therefore the one-loop scalar masses on the Planck brane will befinite. Alternatively, suppose that the observable sector is on the TeV brane. Now, even if we consider the limit where we send the Planck brane away (R → ∞), the Kaluza–Kleinspectrum remains discrete and supersymmetry stays broken. Therefore on the TeV branesupersymmetry is broken and corrections to scalar masses will be sensitive to the ultraviolet

cut-off. Another way to see that supersymmetry is broken by the TeV brane (contrary tothe flat case) is that no Killing spinor can be defined if fermions have twisted boundaryconditions. Even in a noncompact space, the TeV-boundary breaks all the supersymmetries.These expectations will be confirmed in the following sections by the explicit calculationof the one-loop scalar masses in a warped AdS space.

Finally, as an alternative to the supersymmetry-breaking mechanism considered above,there also exists the possibility of breaking supersymmetry by the F -term, F T of theradion field T . This can easily be achieved by turning on a constant term, W , in thesuperpotential localized on the TeV brane. In flat space this is known to generate a vacuumexpectation value (VEV)

F

T ∼ W /M 3

5. In fact in flat space, this corresponds exactly

to the Scherk–Schwarz mechanism [29] or to imposing the twisted boundary conditionfor the fermion as in Eq. (26). However, in a warped space this is not the case, and anonzero F T leads to a new way of breaking supersymmetry. Furthermore, in a warpedspace the VEV of F T , induced by a constant term in the superpotential at the TeV brane, isexponentially suppressed, F T ∼ e−π kR W/M 35 . The tree-level spectrum is easily derived.For the gaugino we have mλ ∼ F T /T ∼ TeV, while for scalars localized on either branetheir masses are zero. The scalar masses are, however, induced at the one-loop level. Thisscenario leads, qualitatively, to the same mass spectrum as the one considered above, andwill not be pursued here.

4. The warped MSSM

Let us now present a candidate MSSM based on the 5d model described above. We willassume that both gravity and gauge fields are in the bulk. Supersymmetry is spontaneouslybroken by imposing twisted boundary conditions (26), on the gravitino and gaugino. TheMSSM matter fields are assumed to be completely localized on the Planck brane. At treelevel, the matter fields are massless and the dominant supersymmetry-breaking effects will

be transmitted to the matter fields on the Planck brane by the 5d gauge interactions. Thus,the soft masses on the Planck brane will arise via radiative corrections and can be computedusing the 5d AdS propagators. On the other hand, the Higgs field can also be assumed tobe a bulk field. However, in this case we will see that radiative corrections to the Higgs

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soft masses are sensitive to TeV-scale physics. Other alternative scenarios will also bediscussed. Since the bulk fields live in a warped space, these models will be referred to as

the “warped MSSM”.Before proceeding to calculate the sparticle spectrum an important comment is in order.

Since the gauge bosons live in a warped extra dimension, the effective 4d coupling is givenby g2 = (g25 k)/(πkR), where g5 is the 5d gauge coupling [9,17]. In order to explain thePlanck–TeV scale hierarchy we need k R 10, which implies that for g25 k 1, we obtaing2 1/30. This is in contradiction with the experimental values of the gauge couplingswhich require g2 ∼ O(1). Therefore, in order to agree with the experimental values onerequires that g25 k 30. This inevitably means that the theory is close to the strong couplingregime at energies E ∼ k. On the TeV brane this corresponds to energies E ∼ ke−π kR . At

these energies the expansion parameter becomes g2

5k/(16π2

) ∼ 0.2. We will assume thatthe effects from the strong coupling regime do not spoil the AdS geometry. Similarly, wewill be able to trust our low-energy predictions, provided that the energy of the processessatisfies E ke−π kR .

4.1. Tree-level masses

If twisted boundary conditions are imposed on the fermions in the bulk, then all the4d fermion modes will receive masses. In particular, the zero mode of the gravitino willreceive a mass whose magnitude can easily be obtained by solving (29) for α

=2:

(31)m3/2 √

8 ke−2π kR .

Thus, for k = M P and ke−π kR = TeV we obtain m3/2 2.8×10−3 eV. This is a superlightgravitino, as compared to the usual gravity-mediated and gauge-mediated scenarios in fourdimensions, and satisfies the usual experimental constraints from cosmology and colliderexperiments [30]. In the warped case the small gravitino mass arises because the couplingof the gravitino to the TeV brane is exponentially suppressed, and therefore it is veryinsensitive to the twisting of boundary conditions on the TeV brane. The higher Kaluza–Klein gravitino modes are approximately given by

(32)mn n + 34π ke−π kR .Notice that compared to the untwisted gravitino Kaluza–Klein spectrum (21), the twistedmass spectrum has indeed shifted approximately by an amount 1/2(πke−π kR ). This shiftis much larger than for the zero mode, because the nonzero Kaluza–Klein gravitino modesare localized near the TeV brane and therefore couple more strongly to the TeV brane ascompared to the gravitino zero mode which is localized near the Planck brane.

Similarly, the tree-level gaugino mass is obtained by solving (29) with α = 1:

(33)mλ 2π kR

ke−π kR .

Thus, for k = M P and ke−π kR = TeV we obtain mλ 0.24 TeV. Notice that unlikethe gravitino zero mode, the gaugino zero mode receives a TeV-scale supersymmetry

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breaking mass. This is because the vector superfield is not localized in the AdS space,and therefore directly couples to the TeV brane, which is the source of the supersymmetry

breaking. Using (30), the higher Kaluza–Klein modes are approximately given by mn (n+1/4)πke−π kR . These masses are obtained at tree-level and we will see that interactionsof boundary fields with the bulk gauge bosons will generate boundary masses at one-loop.Since the mediation of the supersymmetry breaking is due to gauge interactions, the flavorproblem is naturally solved. It is important to note that the theory has a U(1) R symmetry,since the induced masses are of the Dirac-type instead of the Majorana-type. This is aunique property of these theories, and is due to the N = 2 bulk supersymmetry.

It is also possible to add hypermultiplets in the bulk, where the fermions have twistedboundary conditions. In particular, if c = 1/2 then the hypermultiplet is conformal, and the

resulting Kaluza–Klein spectrum is identical to the vector supermultiplet case.

4.2. Radiative corrections on the Planck brane

In order to compute the radiative corrections of the matter fields completely confined onthe Planck brane, let us consider the 5d AdS propagator for the gauge boson and gaugino.The general expression for the 5d propagator in a slice of AdS is derived in the appendix.Using the expression for the vector field Green’s function restricted to the Planck brane(z = z = 1/k), we have

GV

x,1k; x, 1

k

(34)=

d 4p

(2π )4 eip·(x−x

) 1

ip

J 0(ipeπ kR /k)Y 1(ip/k) − Y 0(ipeπ kR /k)J 1(ip/k)

J 0(ipeπ kR /k)Y 0(ip/k) − Y 0(ipeπ kR /k)J 0(ip/k).

In the limit that p ke−π kR we obtain

(35)GV

x,1

k; x, 1

k

− 1

π R

d 4p

(2π )4eip·(x−x

) 1

p2,

which reduces to the usual massless vector field Green’s function in flat space. In particularnotice that by Eq. (35), the charge screening [7,31] is absent in the slice of AdS since thereare no continuum Kaluza–Klein modes. Similarly on the Planck brane, the twisted gauginoGreen’s function defined in the appendix reduces to the form

GF

x,

1

k; x, 1

k

(36)=

d 4p

(2π )4 eip·(x−x

) 1

ip

J 1(ipeπ kR /k)Y 1(ip/k) − Y 1(ipeπ kR /k)J 1(ip/k)

J 1(ipeπ kR /k)Y 0(ip/k) − Y 1(ipeπ kR /k)J 0(ip/k).

In the limit that p ke−π kR and kR 1 the twisted gaugino Green’s function becomes

(37)GF

x,

1

k; x, 1

k

− 1

π R

d 4p

(2π )4eip·(x−x

) 1

p2 − 2π kR

(ke−π kR )2,

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which reduces to a massive gaugino Green’s function where the gaugino mass agreeswith (33). This difference between the gauge boson and gaugino Green’s function

represents the source of supersymmetry breaking on the Planck brane.Note that the vector supermultiplet in the bulk, is also equivalent to a conformal

hypermultiplet (c = 1/2) in the bulk, where the fermion has twisted boundary conditions.The 5-dimensional mass-squared of the scalar is −3k2 + 2k(δ(y) − δ(y − πR)), while themass of the fermion is σ /2 [3]. On the Planck brane (z = z = 1/k) the twisted fermionGreen’s function is the same as Eq. (36), while the scalar field Green’s function is identicalto Eq. (34). In particular, we will also consider the bulk Higgs fields to be conformalhypermultiplets.

The scalar and twisted fermion Green’s function on the Planck brane can be used

to calculate the one-loop contribution to the mass-squared of boundary matter fields.The boundary matter fields couple to the vector supermultiplet in the bulk via gaugeinteractions. The Feynman diagrams for the one-loop mass contributions to the boundaryscalar fields are the same as those in flat space and can be found in Ref. [25]. They give

(38)m2i = 4g2C(Ri )Π(0),where the 4d gauge coupling is given by g2 = g25/(πR) and we have defined

(39)Π(0) = −π R

d 4p

(2π )4

G(V )p − G(F )p

.

The coefficient C(Ri ) is the quadratic Casimir of the representation Ri in the correspond-ing gauge group, and Gp is the 4d Fourier transform of the Green’s function (see appen-dix). Similarly, the boundary matter fields can also couple to a chiral supermultiplet inthe bulk [25]. For a conformal supermultiplet (c = 1/2), where the fermions have twistedboundary conditions we obtain

(40)m2i = 2Y 2Π(0),where Y is the boundary-bulk Yukawa coupling. Assuming that k = M P and ke−π kR =TeV then we obtain π kR = 34.54, and the integral in Eq. (39) can be numerically evaluatedto give

(41)Π(0) = 0.25252π 4

(TeV)2 (0.0360 TeV)2.This result is finite and insensitive to the ultraviolet cut-off for the same reason that wealready explained in the previous section, namely that the supersymmetry breaking islocalized on the TeV brane. In fact, the integration region p ke−π kR = TeV alreadycontributes approximately 90% of the integral in Eq. (39).

Comparing the above result with the flat space case where [25] Π (0) = (0.0367/Rflat)2,we see that the two cases are almost numerically identical for a flat-space radius of Rflat =ke−π kR .

4.2.1. Superparticle spectrum

It is now straightforward to extend the above result to the case of the warped MSSM.Assuming that the squarks and sleptons live on the Planck brane they will receive a one-

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loop contribution from the bulk gauge and Higgs sector (if they are in the bulk). Assumingthe bulk Higgs to be a conformal supermultiplet and following Ref. [25] we obtain for

Π(0) = 16π Π (0)(42)m2Q =

43 α3 + 34 α2 + 160 α1

Π (0) + 12 (αt + αb)Π (0),(43)m2U =

43 α3 + 415 α1

Π(0) + αt Π(0),(44)m2D =

43 α3 + 115 α1

Π (0) + αb Π (0),(45)m2L =

34 α2 + 320 α1

Π(0) + 12 ατ Π (0),(46)m2E = 35 α1 Π(0) + ατ Π(0),

where the bulk gauge contribution is proportional to the gauge couplings α1,2,3 and the

conformal bulk Higgs contribution is proportional to the Yukawa couplings αt,b,τ . Thus toobtain an experimentally allowed soft mass spectrum the scale on the TeV brane shouldbe at least a few TeV. Notice that the dominant corrections are proportional to the gaugecouplings. Thus, the lightest scalar field is the right-handed slepton.

4.3. Radiative corrections in the slice of AdS

Consider a conformal hypermultiplet in the bulk with twisted boundary conditions forthe fermion. The massless scalar mode φ in the hypermultiplet will receive a one-loop

mass contribution due to the breaking of supersymmetry from fields in the bulk withthe twisted boundary conditions. In particular the scalar can couple to the bulk vectorsupermultiplet. This radiative correction can simply be calculated using the 5-dimensionalAdS scalar propagator. For an alternative method to calculate quantum effects in the AdSslice see Ref. [32]. Since the scalar propagates in the bulk we need to integrate over theextra dimension, and the corresponding mass correction is proportional to

Π(0) = −π R

d 4p

(2π )4

π R 0

dy

G(V )p (z,z) − G(F )p (z,z)

(47) Λke−2π kR16π 2

,

where Λ is a Planck-scale cutoff. Unlike the radiative corrections of the boundary fields, itturns out that the radiative correction (47) is not finite, as expected from the arguments of the previous section. In fact, the bulk radiative corrections (47) are linearly divergent. Thisreflects the fact that the bulk fields are propagating in five dimensions and are sensitive tophysics on the TeV brane represented by the cutoff scale Λe−π kR . This behavior is relatedto the fact that the supersymmetry breaking mechanism is localized on the TeV brane andis sensitive to the UV physics. This is different from flat space where the breaking of

supersymmetry is inherently a global effect, and consequently the nonlocality produces afinite result. Although the Higgs soft mass depends on the UV cutoff Λ ∼ k ∼ M 5, its valuewill be of order TeV/(4π ) (due to the warp factor in Eq. (47)), and therefore the modelis fully viable (the hierarchy is not spoiled). However, the dependence on the UV cutoff

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means that a precise calculation of the Higgs mass can only be done by having a completeknowledge of the ultraviolet theory. We must also assume that the Higgs mass will be

negative in order to trigger electroweak symmetry breaking. The fact that the Higgs massis a factor of 4π smaller than a TeV gives a natural explanation for the weak scale beingan order of magnitude smaller than the Kaluza–Klein masses (of order TeV), as requiredby experimental bounds.

Let us finally comment on other possible alternatives. If the Higgs is also confined onthe Planck brane, then its mass will be generated at the one-loop level, with a magnitudesimilar to that of the sleptons (without the Yukawa coupling contribution). Although thiscontribution will be positive, there are sizeable two-loop effects arising from the squarksthat can make the mass-squared negative [25]. Unfortunately, the Higgsino mass cannot

be generated by radiative corrections and we will need to extend the model to include aHiggs singlet whose VEV must induce the Higgsino mass. In the above cases we haverestricted the observable sector to the Planck brane. Nevertheless, many more possibilitiesexist by placing part of the matter in the bulk or on the TeV brane. For example, considerdelocalizing the first two families off the Planck brane by changing their bulk massparameters [3]. In this case the corresponding squarks and sleptons of the first two familieswill have masses larger that those of the third family, a scenario whose phenomenologycan have interesting consequences.

4.4. Relation to 4d strongly-coupled CFT

The AdS/CFT correspondence relates the 5d theory of gravity in AdS to a 4d stronglycoupled conformal field theory (CFT) [10]. In the case of a slice of AdS, a similarcorrespondence can also be formulated [11–14]. The Planck brane in AdS5 correspondsto an ultraviolet cutoff of the 4d CFT and to the gauging of certain global symmetries.For example, in the case we are considering where gravity and the standard model gaugebosons live in the bulk, the corresponding CFT will have the super-Poincaré group gauged(giving rise to gravity) and also the standard model group SU(3) × SU(2)L × U(1)Y (giving rise to the standard model gauge bosons and gauginos). Matter on the Planck brane

corresponds to adding new fields to the CFT which only couple to CFT states via gravityand gauge interactions. On the other hand, the TeV brane corresponds in the dual theoryto a infrared cutoff of the CFT [13,14]. In other words, it corresponds to breaking theconformal symmetry at the TeV scale. The Kaluza–Klein states of the 5d theory correspondto the bound states of the strongly coupled CFT.

This alternate dual description suggests that the supersymmetry-breaking mechanismthat we have discussed represents a class of strongly coupled CFT’s where supersymmetryis broken at the TeV scale. The bound states therefore do not respect supersymmetry andgive rise to a fermion–boson mass splitting. The 5d warped MSSM is then simply the

ordinary 4d MSSM with a strongly coupled CFT sector responsible for the breaking of supersymmetry. The standard model fields coupled to the CFT sector will get tree-levelmasses while those coupled only via gravity or gauge interactions will receive massesat the one-loop level. In our model the CFT sector is charged under the standard model

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gauge group and consequently it implies that the gauginos get masses at tree-level. Noticethat as we mentioned earlier the gaugino mass is of the Dirac-type. This means that

the gaugino has married a fermion bound-state to become massive.4

Since the gauginomass comes from the mixing between the gaugino and the CFT bound-state, the masswill be proportional to

g2bCFT/(8π 2) = 1/

√ π kR , where we have used the AdS/CFT

relation [13] g25 k = 8π 2/bCFT and g2 = g25/(πR). This agrees with Eq. (33). Similarly, thesmallness of the gravitino mass (of order 10−3 eV) is also easy to understand in the CFTpicture. The gravitino coupling to the CFT sector is suppressed by 1/M P , so its mass willbe of order TeV2/M P ∼ 10−3 eV.

Although the CFT picture is useful for understanding some qualitative aspects of thetheory, it is practically useless for obtaining quantitative predictions since the theory is

strongly coupled. In this sense, the 5d gravitational theory in a slice of AdS representsa very useful tool since it allows one to calculate the particle spectrum, which wouldotherwise be unknown from the CFT side.

5. Conclusion

In this paper we have presented a supersymmetric 5d theory in warped space wheresupersymmetry is spontaneously broken by imposing different boundary conditionsbetween the fermion and bosons. While this is reminiscent of the Scherk–Schwarz

mechanism in flat space, we have argued that in a warped space this is a novel way of breaking supersymmetry. Unlike the flat-space case where the supersymmetry-breakingmechanism is a global effect, the twisted boundary conditions in the warped space lead toa local supersymmetry breaking effect on the TeV brane.

A particularly interesting model is the warped MSSM, where matter is confined on thePlanck brane, and gravity and gauge fields propagate in the 5d bulk. The gravitino andgaugino receive tree-level masses from the twisted boundary conditions. In particular, thetree-level mass of the gravitino is ∼ 10−3 eV and the gaugino mass ∼ TeV. The one-loopradiative corrections to the squarks and sleptons confined to the Planck brane are finite

and insensitive to the UV cutoff. This simply reflects the fact that the supersymmetry-breaking is localized on the TeV brane, at a finite distance away from the Planck brane.The one-loop radiative corrections from the bulk gauge fields are proportional to the gaugecouplings and thus naturally solve the flavor problem. If the Higgs sector is also includedin the bulk, then the one-loop radiative corrections also give a contribution proportionalto the Yukawa couplings. However, in this case the radiative corrections to the Higgs softmass are not finite. This is in contrast to the flat-space case, and is due to the fact that thebulk Higgs directly couples to the supersymmetry breaking effects on the TeV brane.

By the AdS/CFT correspondence, the warped supersymmetric standard model can beinterpreted in terms of a strongly coupled CFT, where supersymmetry (and conformalsymmetry) are broken at the TeV scale. Thus, the warped MSSM is simply the ordinary

4 A Majorana-type mass would correspond, for example, to a breaking of supersymmetry (in the 5d dual) by anonzero F T .

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4d MSSM with a strongly coupled CFT which is responsible for breaking supersymmetry.The fact that there exists a weakly coupled 5d gravity dual, allows us to calculate the mass

spectrum. This provides a powerful tool in obtaining information about the dynamics of this class of strongly coupled CFT’s, and is worthy of further investigation.

Acknowledgements

We wish to thank Emilian Dudas and Dan Waldram for helpful discussions. One of us(T.G.) acknowledges the Aspen Center for Physics where part of this work was done. Thework of T.G. is supported by the FNRS, contract no. 21-55560.98, while that of A.P. ispartially supported by the CICYT Research Project AEN99-0766.

Appendix A. 5D propagators in a slice of AdS

Let us consider the propagation of bulk fields in a slice of AdS. We will follow thederivation of the Green’s function presented in Refs. [12,33], except that we will extendthe previous results to the case of arbitrary bulk fields in the two-brane scenario. The resultfor the bulk scalar has also recently been given in Ref. [34].

As shown in Ref. [3] the equation of motion for bulk fields Φ = {V µ, φ , e−2σ ψL,R}, canbe conveniently written as a second-order differential equation. Thus, introducing a sourcefunction J , one obtains

(A.1)

e2σ ηµν ∂µ∂ν + esσ ∂5

e−sσ ∂5− M 2ΦΦ(x,y) =J (x,y),

where the parameter s = {2, 4, 1}, and the 5d masses are [3] M 2Φ = {0, ak2 + bσ ,c(c ± 1)k2 ∓ cσ }. The corresponding Green’s function for (A.1) can then be defined as

(A.2)Φ(x,y) =

d 4x dy √ −gG(x,y; x , y)e(4−s)kyJ (x, y ),

provided thatJ = {

J µ, J φ , γ µ∂µJ R,L

±∂5J L,R

−(c

±1)σ J L,R

}, where J µ, J φ and J L,R

are the source terms for the bulk vector, scalar, and fermion, respectively. It is convenientto introduce the variable z = eky /k. In these coordinates the Planck brane is located atz∗ = 1/k and the TeV brane at z∗ = eπ kR /k. If we now take the 4d Fourier transform of the Green’s function

(A.3)G(x,z; x, z) =

d 4p

(2π)4eip·(x−x

)Gp(z,z),

then the Fourier component Gp(z,z) must satisfy the equation

(A.4)∂2z + 1 − sz ∂z − p2 − M 2

Φ(kz)2Gp(z,z) = (kz)s−1δ(z − z),

where M 2Φ = {0, ak2,c(c ± 1)k2}. If we define Gp(z,z) = (k2zz)s/2Gp(z,z), thenEq. (A.4) simply becomes the Bessel equation

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(A.5)

∂2z +

1

z∂z − p2 − α

2

z2

Gp(z,z

) = (kz)−1δ(z − z),

where α = (s/2)2 + M 2Φ /k2. The standard procedure for solving Eq. (A.5) is to use thesolution to the homogeneous equation in the regions z < z and z > z, and then imposematching conditions at z = z. Thus writing

(A.6)Gp(z,z) = θ (z − z)G> + θ (z − z)G(z)J αipeπ kR /kH (1)α (ipz) − H (1)α ipeπ kR /kJ α(ipz),where H (1)α = J α + iY α is the Hankel function of the 1st kind of order α , and J α , Y α areBessel functions. If the boundary condition for the Green’s function, G) is even onthe Planck brane (TeV brane) then [3]

(A.9)J α (z) = (−r + s/2)J α(z) + zJ α(z),where the parameter r = {0, b, ∓c}, while if the boundary condition is odd then [3]

(A.10)J α (z) = J α (z),and similarly for H (1)α . Note that in the presence of the boundary mass terms parametrizedby r , the even boundary condition is equivalent to imposing the modified Neumanncondition, (∂z − rσ )Gp(z,z)|z=z∗ = 0, while the odd boundary condition is equivalentto imposing the Dirichlet condition, Gp (z,z)|z=z∗ = 0.

The unknown functions A(z) are determined by imposing matchingconditions at z = z. Continuity of Gp at z = z leads to the condition

(A.11)

G>

z=z =

G<

z=z ,

while the discontinuity in ∂zGp gives the condition(A.12)

∂zG> − ∂zG and G

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× J α (ipeπ kR /k)H (1)α (ipz>) − H (1)α (ipeπ kR /k)J α (ipz>)J α (ipeπ kR /k)

H (1)

α (ip/k)

− H

(1)α (ipeπ kR /k)

J α (ip/k)

(A.15)× J α(ip/k)H (1)α (ipz

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Nuclear Physics B 602 (2001) 23–38

www.elsevier.nl/locate/npe

Supersymmetric triplet Higgs model of neutrinomasses and leptogenesis

Thomas Hambye a,b, Ernest Ma c, Utpal Sarkar d

a Centre de Physique Théorique, CNRS Luminy, 13288 Marseille, Franceb INFN — Laboratori Nazionali di Frascati, 00044 Frascati, Italyc Department of Physics, University of California, Riverside, CA 92521, USA

d Physical Research Laboratory, Ahmedabad 380 009, India

Received 16 November 2000; accepted 8 March 2001

Abstract

We construct a supersymmetric version of the triplet Higgs model for neutrino masses, which cangenerate a baryon asymmetry of the Universe through lepton-number violation and is consistent with

the gravitino constraints. © 2001 Published by Elsevier Science B.V.

PACS: 12.60.Fr; 12.60.Jv; 14.60.Pq; 98.80.Cq

1. Introduction

The first definite evidence for physics beyond the standard model came from therecent evidence for the mass of the neutrinos. The atmospheric neutrino anomaly [1], asobserved by the SuperKamiokandeexperiment, has established that there is a mass-squareddifference between the muon neutrino and the tau neutrino. On the other hand, the solarneutrino problem [2] implies a mass-squared difference between the electron neutrino andthe other two active neutrinos. Hence it has now been established that at least two neutrinosare massive.

The mass-squared differences between the different generations of neutrinos have tobe very small, but the mixing angles large, to explain the atmospheric and solar neutrinoanomalies. The required masses for the neutrinos are several orders of magnitude smallerthan those of other fermions, which are all Dirac particles. The smallness of the neutrinomass is naturally explained if the neutrinos are Majorana particles [3], hence lepton number

is not conserved and that should be due to some physics beyond the standard model.There are several motivations for lepton-number violation in Nature [4]. In addition, the

E-mail address: [email protected] (T. Hambye).

0550-3213/01/$ – see front matter © 2001 Published by Elsevier Science B.V.PII: S0 5 5 0 -3 2 1 3 (0 1 )0 0 1 0 9 -2

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24 T. Hambye et al. / Nuclear Physics B 602 (2001) 23–38

associated lepton-number violation may have the added virtue of accounting for the presentobserved baryon asymmetry of the Universe. One model of neutrino mass having this virtue

is the triplet Higgs model [5]. In the nonsupersymmetric case, this model has been studiedin detail and found to share all the interesting features of other models of neutrino mass.Moreover, in theories with large extra dimensions [6], this mechanism happens to be theonly one which gives Majorana (rather than Dirac) masses to the neutrinos [7]. In thisarticle we will study the supersymmetric version of this model.

In the supersymmetric version of the triplet Higgs model, there are several new aspects.Similar to the requirement of two Higgs doublets in the supersymmetric extension of thestandard model, we now have two Higgs triplets. Only one of them couples to the leptons,but it can acquire a vacuum expectation value (vev) only if the other Higgs triplet is present.

This is related to the fact that a mass term in the superpotential requires two triplet Higgssuperfields, which are of course also necessary for anomaly cancellation. This mass termconnecting the two triplet superfields in the superpotential also allows a trilinear couplingto exist between two scalar doublets and the scalar triplet which couples to leptons, whichis necessary for neutrino mass as well as leptogenesis [5]. In the present supersymmetricversion of the triplet Higgs model, we must consider the decays of both heavy triplets. Notethat supersymmetry is not yet broken at this energy scale. There are now also several newdiagrams which contribute to the CP violation.

Another important feature of the supersymmetric model comes from the constraints of nucleosynthesis. In supersymmetric models there is a strong bound on the scale of inflationfrom nucleosynthesis due to the gravitino problem [8]. This means that baryogenesis has tooccur at temperatures below about 1011 GeV. On the other hand, in the triplet Higgs model,the gauge interactions of the triplet Higgs scalars and fermions bring their number densitiesto equilibrium at temperatures below ∼ 1012 GeV. This naive order-of-magnitude estimatethus implies that the supersymmetric triplet Higgs model of leptogenesis is probably notconsistent with the gravitino constraints [9]. However, detailed calculations give severalpossible ways out of this problem. In the following we will consider the cases where thispotential problem is first ignored and then taken into account. We point out here that thesupersymmetric triplet Higgs model can evade this problem of gravitinos when the massesof the triplet Higgs superfields are moderately degenerate.

In Section 2 we introduce the model and describe its consequences for neutrino masses.Then in Section 3 we calculate the amount of CP violation in the decays of the triplet Higgsscalars and fermions which can generate a lepton asymmetry of the Universe. In Section 4we solve the Boltzmann equations to calculate the evolution of the lepton asymmetry andpresent our results. In Section 5 the gravitino problem is discussed. Finally in Section 6 wesummarize and conclude.

2. The model

The Majorana masses of the neutrinos can be generated by extending the standard modelto include a triplet Higgs scalar [10], which acquires a small vev and couples to two leptons.

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T. Hambye et al. / Nuclear Physics B 602 (2001) 23–38 25

If lepton number was spontaneously broken by this vev [11], the so-called triplet Majoron(i.e., the resulting massless Goldstone boson) coupling to the Z boson would be predicted.

This scenario is now ruled out by the known invisible Z width [12]. Moreover, such modelsdo not explain the present observed baryon asymmetry of the Universe. A new scenariowas then proposed in which lepton number is broken explicitly at a very high energyscale [5]. The triplet Higgs scalar would then be extremely heavy. However, it acquires avery tiny vev through its lepton-number violating trilinear coupling to the standard-modelHiggs doublet, which can then give a small Majorana mass to the neutrinos. The decaysof the triplet Higgs scalars also generate a lepton asymmetry of the Universe, which getsconverted to a baryon asymmetry of the Universe before the electroweak phase transition.

To implement the triplet Higgs mechanism in a supersymmetric model, we need to

extend the supersymmetric standard model to include two triplet Higgs superfields. Sincewe want these fields to be very heavy, supersymmetry should be unbroken at that stage andthe generation of the lepton asymmetry will not depend on the supersymmetry-breakingmechanism. We also assume that R-parity is not violated, so that there is no other source of lepton-number violation except for the Yukawa couplings of the triplet Higgs superfields.We introduce one triplet ξ̂ 1([ξ̂ ++1 , ξ̂ +1 , ξ̂ 01 ] ≡ [1, 3, 1] under SU (3)c×SU (2)L×U (1)Y ) andanother triplet ξ̂ 2([ξ̂ 02 , ξ̂ −2 , ξ̂ −−2 ] ≡ [1, 3,−1]) so that a mass term M ξ̂ 1ξ̂ 2 may appear in thesuperpotential. However, CP violation is not possible with just these two Higgs triplets. Forthat, we need two of each type of the above Higgs triplets. So, if heavy triplet superfieldsare used to generate neutrino masses as well as a lepton asymmetry of the Universe,there should be at least four: ξ̂ a1 ([ξ̂ a++1 , ξ̂ a+1 , ξ̂ a01 ] ≡ [1, 3, 1]) and ξ̂ a2 ([ξ̂ a02 , ξ̂ a−2 , ξ̂ a−−2 ] ≡[1, 3,−1]), where a = 1, 2 corresponds to the two scalar superfields, whose mixing givesCP violation for generating the lepton asymmetry of the Universe.

The essential part of the superpotential for the interactions of these scalar superfieldswith the lepton superfields L̂i ≡ (νLi e−Li ) ≡ [1, 2,−1/2] and the standard Higgs doubletsH 1([φ01, φ−1 ] ≡ [1, 2,−1/2]) and H 2([φ+2 , φ02 ] ≡ [1, 2, 1/2]) is given by

(1)W = M ab ξ̂ a1 ξ̂ b2 + f aij L̂i L̂j ξ̂ a1 + ha1

H 1

H 1ξ̂

a1 + ha2

H 2

H 2ξ̂

a2 + µ

H 1

H 2 + · · · ,

where i=

1, 2, 3 is the generation index. The first term gives masses to the triplets. The

condition for leptogenesis and neutrino masses would determine this scale M . The nextterm gives the Yukawa couplings of the triplet Higgs scalar superfield with the left-handedlepton chiral superfields of the three generations. When the scalars ξ a1 acquire vacuumexpectation value (vevs), this term gives Majorana masses to the neutrinos. The next twoterms give small vevs to the triplet Higgs scalars.

The scalars ξ a1 couple to two leptons, to two higgsinos H 1, to two scalars H 2 and to

a H 1H 2 pair. The scalars ξ a2 couple to two higgsinos H 2, to two sleptons, to two scalars

H 1 and to a H 1H 2 pair. This simultaneous decay of the triplets to products with differentlepton numbers breaks lepton number explicitly. Thus the scale of lepton-number violation

is the same as the mass of the triplet Higgs scalars, which is very heavy, say of the order∼ O(109–1014) GeV. However, since SU (2)L is unbroken at this scale, these fields do notacquire any vev. Only after the electroweak symmetry breaking is there an induced tinyvev for these scalars and the neutrinos would acquire mass.

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The vevs of the triplet Higgs scalars are obtained from the vanishing of the F -terms,which corresponds to the minima of the potential. From the conditions F ξ a1 = F ξ a2 = 0, andassuming that R-parity is conserved (so that the sneutrinos do not acquire any vev), we get

F ξ a1 = M ab ξ b2 + f aij L̃i L̃j + ha1H 1H 1 = 0⇒ ξ b2 = ub2 =−M −1ba ha1H 12 =−M −1ba ha1v21,

F ξ a2 = M ab ξ b1 + ha2H 2H 2 = 0

(2)⇒ ξ b1 = ub1 =−M −1ba ha2H 22 =−M −1ba ha2v22 .Since the masses of the triplet scalar fields are several orders of magnitude higher thanthe electroweak symmetry breaking scale v, the effective vev of the triplet Higgs fieldsare several orders of magnitude smaller than v

=246 GeV. 1 Since these vevs give masses

to the neutrinos, the smallness of the neutrino mass is now directly related to the largelepton-number violating scale.

The vevs of the triplet scalars will give a mass to the neutrinos given by

(3)(mν )ij =

a

2f aij ua1 =

a,b

−2f aij M −1ab hb2v22 .

Since the leptons do not couple with the other triplet scalar ξ 2, there is no contribution tothe neutrino mass from ua2 . Since the lepton number is now broken at a very large scaleexplicitly, there is no Majoron in this scenario. There is one would-be Majoron, which

becomes too heavy to affect any low-energy phenomenology. This makes it consistent withthe measured invisible Z width from LEP (Large Electron Positron Collider) at CERN.The decay of these scalars to two leptons or two higgsinos can be read off from the

F -terms in the superpotential. The decays of these scalars into two sleptons and thestandard-model Higgs doublets can be read off from the relevant part of the scalar potential,

V = |M ab ξ b2 + f aij L̃i L̃j + ha1H 1H 1|2 + |M ab ξ b1 + ha2 H 2H 2|2

(4)+ |2ha1H 1ξ a1 + µH 2 + · · · |2 + |2ha2H 2ξ a2 + µH 1 + · · · |2 + · · · .The various decay modes of the scalar and fermionic components of the triplet scalar

superfields are listed below and shown in Figs. 1 and 2. The decay modes of the ξ̂ a

1 (i.e.,the scalars ξ a++1 and the fermions ξ̃ a++1 ) are

(5)ξ a1++→

L+i L

+j (L=−2),

H +2 H +2 (L= 0),H +1 H +1 (L= 0)

and

(6)ξ̃ a1++

→

L̃+i L+j (L=−2),

H +2 H +2 (L= 0),H +1 H +1 (L= 0)1 The smallness of these vevs makes this triplet model perfectly consistent with the usual constraints on

additional triplets coming from the measurement of the ρ parameter at LEP.

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while the decay modes of ξ̂ a2 are

(7)ξ a2++→ ˜

L+i˜

L+j (L=−

2),H +2 H +2 (L= 0),H +1 H

+1 (L= 0)

and

(8)ξ̃ a2++→

L̃+i L

+j (L=−2),H +2 H +2 (L= 0),

H +1

H +1 (L= 0).

The couplings entering in the various decay modes can be read off from the superpotential.

Note that we do not consider the decays proportional to µ2 (i.e., to a H 1 H 2 pair) which arenegligible. If there is CP violation and the decays satisfy the out-of-equilibrium condition,then these decays can generate a lepton asymmetry of the Universe [13,14]. This leptonasymmetry can then get converted to a baryon asymmetry of the Universe [15].

When this lepton asymmetry is generated, the B + L violating (but B − L conserving)sphaleron transitions are taking place at a very fast rate [16]. In fact, during the period

1012 GeV > T > 102 GeV

the anomalous B + L violating sphaleron processes remain in equilibrium. Duringthis period, any lepton asymmetry of the Universe would be equivalent to the B − Lasymmetry. The sphaleron interactions would then convert this lepton asymmetry to abaryon asymmetry of the Universe within this period [17].

3. CP asymmetry in triplet Higgs decay

The various decay modes of ξ a1 and ξ̃ a1 are given in Fig. 1 and the decay modes of

ξ a2 and ξ̃ a2 are given in Fig. 2. The simultaneous decay of the triplet Higgs scalars or the

triplet higgsinos into states with lepton number 0 (two scalar Higgs doublets or higgsinos)

and with lepton number 2 (two leptons or sleptons) implies lepton-number violation. ForCP violation, the tree-level diagrams by themselves are not enough. Even if the couplingsare complex, the probability will be positive definite and hence there will not be any CPviolation. However, if there are one-loop diagrams, which interfere with these tree-leveldiagrams, then the interference may be complex, which gives the CP violation.

In the present case there are one-loop diagrams which are given in Fig. 3 (for ξ a1 andξ̃ a1 decays) and in Fig. 4 (for ξ

a2 and

ξ̃ a2 decays). As in the nonsupersymmetric case,although some of the tree-level diagrams appear similar to the right-handed neutrino decaydiagrams [18–20], there are no one-loop diagrams which are similar to the vertex diagrams

of the right-handed neutrino decays. From this point of view, leptogenesis with the tripletHiggs scalars have this unique feature that CP violation comes only from the self-energydiagrams, which has the interpretation of oscillations of the scalars before they decay [20].Moreover, it was pointed out that the CP violation coming from the self-energy diagrams

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Fig. 1. Tree level diagrams for the decay of ξ̂ 1.

has an interesting feature of resonant oscillation. Thus the amount of lepton asymmetry canget highly enhanced when the masses of the triplet Higgs superfields are almost degenerate

[20].

In none of the loop diagrams of Figs. 3, 4 is there any interference between ξ a1 andξ a2 . So, with one each of ξ

a1 and ξ

a2 , there cannot be any CP violation. In this case, the

relative phases between various couplings can be chosen to be real. Only when there are

at least two ξ a1 or ξ a2 , there can be CP violation. In this case, decays of both ξ

a1 and ξ

a2

will contribute to the amount of CP violation. The relative phases between the couplingsof the ξ a1 to the leptons of different generations cannot generate a lepton asymmetry of the

Universe, because they all correspond to final states of the same lepton number. Among theloop diagrams, figures (c) and (d) are supersymmetric counterparts of figures (a) and (b), so

supersymmetry ensures that the contributions from the first two diagrams are the same asthat of the last two diagrams. In the following we will consider explicitly only the decays

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Fig. 2. Tree level diagrams for the decay of ξ̂ 2.

of the scalar triplets keeping in mind that the decays of their fermionic superpartners givethe same lepton asymmetry.

We shall now calculate the amount of CP violation generated from the interference of thetree-level processes and the one-loop diagrams. In the mass-matrix formalism, it is possibleto give a physical interpretation to this CP violation. A triplet scalar superfield oscillatinginto another type before it decays, has a different decay rate compared to its conjugatestates. Although the total decay rates are equal by CPT, the partial decay rates now differ,which give rise to CP violation. This CP violation will then lead to a lepton asymmetry dueto the fact that (1) the partial decay products do not all have the same lepton number and

(2) the interaction rate is not much faster than the expansion rate of the Universe.Without loss of generality, we shall assume that the mass matrix for the triplet Higgsscalars starts out as real and diagonal,

M ab = M a δab ,

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Fig. 3. One loop diagrams contributing to CP violation in decays of ξ̂ 1.with M a real. However, in the presence of interactions, they will no longer remain real.Including the interactions, the mass matrix for the left and right chiral superfields getsdifferent contributions from the interference of the tree and loop diagrams. The physicalstates of the left and right chiral superfields will evolve in a different way and their decaysinto leptons and antileptons would generate the lepton asymmetry of the Universe. Inthe following we denote by φ̂m1+ and φ̂m2+ with m = 1, 2 the physical states which arecombinations of the left chiral superfields ξ̂ a1 and ξ̂ a2 , respectively, and by φ̂m1− and φ̂m2− the

physical states which are combinations of the conjugates of these superfields ξ̂ a

∗1 and ξ̂ a

∗2respectively (which are the right chiral superfields).The effective scalar triplet mass matrix we obtain at one loop is given by

(9)ξ a†1M

21

ab

ξ b1 + ξ a†2M

22

ab

ξ b2 ,

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Fig. 4. One loop diagrams contributing to CP violation in decays of ξ̂ 2.

where, for a given value of the squared momentum p2ξ of the incoming or outgoing particle:

(10)M2k =

M 21 − iΓ k11M 1 −iΓ k12M 2−iΓ k21M 1 M 22 − iΓ k22M 2

,

with Γ kab M b = (Γ kba )∗M a and

Γ 1ab M b = 18π

i,j

f aij ∗

f bij p2ξ + ha∗1 hb1p2ξ +M a M bha2 hb∗2 ,

and

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Γ 2ab M b = 1

8π

M aM b

i,j f aij f

b∗ij + ha∗2 hb2p2ξ +M a M bha1hb∗1

.

The decay widths Γ φak± of the tree scalars in the triplet φak± are given by Γ φak± = Γ kaa ≡

Γ φak

. Neglecting terms of order [Γ ij M j /(M 21 − M 22 )]2 the two mass matrices have theeigenvalues M φak± = M a and the eigenvectors are

(11)φ1k+ = ξ 1k − iΓ k12M 2

M 21 −M 22ξ 2k ,

(12)φ2k+ = iΓ k∗12 M 2

M 21

−M 22

ξ 1k + ξ 2k ,

(13)φ1k− = ξ 1∗k − iΓ k

∗12 M 2

M 21 −M 22ξ 2∗k ,

(14)φ2k− = iΓ k12M 2

M 21 −M 22ξ 1∗k + ξ 2∗k .

Similarly we have

(15)ξ 1k = φ1k++ iΓ k12M 2

M 21 −M 22φ2k+,

(16)ξ 2k = −iΓ k∗12 M 2

M 21 −M 22φ1k+ + φ2k+,

(17)ξ 1∗k = φ1k−+ iΓ k∗12 M 2

M 21 −M 22φ2k−,

(18)ξ 2∗k =−iΓ k12M 2

M 21 −M 22φ1k− + φ2k−.

Note that, due to CP violation, the φik− are not Hermitian conjugates of the φik+ but the

orthonormality relations φik+|φ

j

k− = φik−|φ

j

k+ = δij between the in and out states aresatisfied (as they should be) when diagonalizing a non-Hermitian mass matrix (see, e.g.,Refs. [21,22]). The resulting lepton asymmetries εmk induced by the decay of the scalartriplet φ ak± are given by

(19)εa1 = 2Γ (φa1−→ ll )− Γ (φa1+→ lclc)

Γ φa1− + Γ φa1+,

(20)εa2 = 2Γ (φa2+→ ll )− Γ (φa2−→ lclc)

Γ φa2+ + Γ φa2−.

Putting Eqs. (15)–(18) in Eqs. (1) and (4) we obtain

(21)εa1 1

2π(M 21 −M 22 )

i,j

M 2a Im

h21h

11∗

f 1ij f 2∗

ij

+ ImM 2M 1h2∗2 h12f 1ij f 2∗ij i,j |f aij |2 + |ha1|2 + |ha2|2

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and similarly we have

ε

a

2 1

2π(M 22 −M 21 )M 1M 2

M 2a

(22)×

i,j

M 2a Im

h22h

1∗2 f

1∗ij f

2ij

+ Im M 1M 2h2∗1 h11f 1∗ij f 2ij i,j |f aij |2 + |ha2|2 + |ha1|2

.

As expected the asymmetries come from the interference of the leptonic sector (throughthe f aij ’s) and the non-leptonic sector (through the h

ak ’s). Such asymmetries are obtained

from the decay of each one of the tree states in each scalar triplet. Equal asymmetries arealso obtained from the decay of the tree fermionic partners of the scalar triplets. Note thatfor M 1 close to M 2, εa1 ∼ εa2 .

When the mass difference between the two Higgs scalars is very small and is comparableto the decay width, there is a resonance in the amount of CP asymmetry, hence in theamount of lepton asymmetry [20]. Our present method fails in the limit when the decaywidth is larger than the mass differences. In this case it is necessary to apply a resummationof the self-energies as it has been done in Ref. [19]. Note that at the resonance (M 1−M 2 ∼Γ k12/2) and with maximal CP violating phase the asymmetries of Eqs. (21) and (22) canbe as large as of order one. However as we did already in Eqs. (11)–(18) we will restrictourselves to a region where the mass squared difference can be small but still much largerthan the decay widths, so that the formalism we consider can be used safely.

4. Boltzmann equations

We shall now check if the out-of-equilibrium condition is satisfied in this scenarioand can generate the required amount of baryon asymmetry of the Universe. The naiveconsideration for the out-of-equilibrium condition that the decay rates of the triplet Higgsscalars to be less than the expansion rate of the universe is satisfied for a wide range of parameters. This out-of-equilibrium condition reads,

(23)Kφak =Γ

φ

a

k

H (M a)< 1

where the Hubble constant H (T ) at the temperature T is given by

(24)H ( T ) =

4π 3g∗45

T 2

M P ,

with g∗ ∼ 100 the number of massless degrees of freedom and M P ∼ 1019 GeV isthe Planck scale. Given any particular temperature, the out-of-equilibrium conditionconstrains the various coupling constants. If this condition is satisfied and if the various

damping terms due to scatterings are negligible, the total amount of lepton asymmetry percomoving volume XL ≡ nL/s = (nl − nl̄ )/s that will be generated through the decays of the four triplet Higgs superfields will be given by

k 6(ε

k1 + εk2)nγ /(2s) =

k 6(ε

k1 +

εk2)45/(2g∗π4) where the entropy s and the photon number density nγ are given by

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(25)s = g∗2π 2

45 T 3,

(26)nγ = 2T 3

π 2 .

For the out-of-equilibrium condition of φa1,2 to be satisfied, we get a bound on theparameters

(27)

i,j |f aij |2 + |ha1 |2 + |ha2|2

M a<

4π 3g∗

45

8π

M P ∼ 4× 10−17 GeV−1.

It is interesting to compare this condition with the condition that a neutrino mass of order

∼10−3 eV is generated from Eq. (3),

(28)−

a

f aij ha2

M a= (mν )ij

2v22∼ 10−17 GeV−1,

where v2 has to be of order v = 246 GeV. A neutrino mass of order 10−3 eV cantherefore be obtained while the out-of-equilibrium condition is satisfied for any value of M 1 and M 2 provided the couplings f

aij and h

a1,2 have the appropriate values.

2 This is ingeneral achieved if ha2 together with at least one of the f

aij for a = 1 or 2 are of order

∼ [(10−17 GeV−1)M 1,2]1/2 (with all other couplings taking smaller values). For M 1,2 ∼1014 GeV this requires f a

ij ∼ha

2 ∼10−2–10−1 while for M 1,2

∼109 GeV this requires

f aij ∼ ha2 ∼ 10−3–10−4. Assuming a maximal CP violating phase, the lepton asymmetryobtained from Eqs. (21), (22) is then typically of order XL ∼ 10−5–10−6 in the formercase and XL ∼ 10−10–10−11 in the latter case. A smaller asymmetry can be generated if for example this CP violating phase is not maximal or if in general larger values of the f ’sand the h’s are taken in such a way that Eq. (28) is satisfied but not Eq. (27). In the lattercase the damping term of the inverse decay process will suppress the asymmetry. A largerasymmetry can be obtained if M 1 and M 2 are more degenerate. For M 1,2

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or U (1)Y gauge bosons, as obtained from the kinetic term of the scalar triplets. This givesa suppression in the generation of the lepton asymmetry of the Universe and implies that

the mass of the triplets cannot be too small (except if the two triplets are almost degenerateas shown below). The presence of this damping term requires the explicit calculation of the evolution of the asymmetry using the Boltzmann equations.

Defining the variable z ≡ M 1/T and the various number densities per comoving volumeXi ≡ ni /s, the Boltzmann equations are:

(29)

dXφak

dz=−zKφak

K1(z)

K2(z)

M φak

M 1

2Xφak

−Xeqφak

+ z 1

sH(M 1)

1−

X2φak

Xeq2φa

k

γ ascatt.,

(30)dXLdz

=a,k

zKφakK1(z)K2(z)

M φakM 1

2εak Xφak −Xeqφak − 12 Xeq

φak

Xγ XL.

In Eqs. (29)–(30) the equilibrium distributions of the number densities are given by theMaxwell–Boltzmann statistics:

(31)nφak = gφakM 2

φak

2π 2 T K2(M φak /T),

where gφak = 1 are the numbers of degrees of freedom of the φak and K1,2 are the usual

modified Bessel functions. The reaction density for the scattering process ξ †a +

ξ a →G1 +G2 is given by

(32)γ ascatt. = T

64π 4

∞ 4M 2a

ds σ̂ a (s)√

s K1(√

s/T),

where σ̂ is the reduced cross section which is given by 2(s − 4M 2a )σ a (s). Note that aprecise result would require an explicit calculation of all scattering processes involvinggauge interactions in all channels. 3 However it can be checked that the dependence of thegenerated lepton asymmetry on the magnitude of the scattering is much slower than linear.Therefore, considering also the fact that the model allows some freedom in the range of parameters used, this explicit calculation will not add much to our understanding in anycase. We will thus make the following estimate:

(33)σ a = 1π√

s

1 s − 4M 2a

g4,

where g is the SU (2)L coupling (which at tree level is given by the relation m2W = g2v2/4).Putting Eq. (33) in Eqs. (32) and (29), it turns out that the scattering term has a smalleffect on the evolution of the lepton asymmetry for values of M 1,2 above 1011–1012 GeV.

For smaller values of M 1,2 the suppression can be very strong due to the fact that the

3 There are more than 20 different physical processes of the type ξ †a + ξ a →G1 +G2 . There is also scatteringof the type ξ †a + ξ a → l + l̄ with an intermediate gauge boson which is of the same order.

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last term of Eq. (29) increases when M 1 decreases and T ∼ M 1. This will suppress theasymmetry which at some point becomes much smaller than ∼ 10−10 except if M 1 andM 2 are sufficiently degenerate. Note that the suppression due to these scattering processesis the most effective when the triplet starts decaying. At lower temperatures, the scatteringeffect is suppressed by the Boltzmann factor due to the higher threshold in Eq. (32). Takingfor example f a22, f

a23, f

a32 and f

a33 (for both a = 1 and 2) equal to the same value f with all

other f aij equal to zero, i.e., assuming negligible all the f a

ij with i = 1 and/or j = 1 (whichconstitutes one of the possible structures leading to a maximal mixing between the secondand third generation of neutrinos) and taking all hak couplings equal to the same value h,four typical sets of parameters which give an asymmetry of order ∼ 10−10 together with aneutrino mass of order 10−3–10−2 eV are shown below:

(34)M 1 = 1013 GeV, M 2 = 3.0× 1013 GeV, h = 1× 10−3, f = 3× 10−2,(35)M 1 = 1012 GeV, M 2 = 3.0× 1012 GeV, h = 1× 10−2, f = 5× 10−4,(36)M 1 = 1011 GeV, M 2 = 2.0× 1011 GeV, h = 8× 10−4, f = 2× 10−3,(37)M 1 = 1010 GeV, M 2 = 1.1× 1010 GeV, h = 1× 10−3, f = 5× 10−4.

A maximal CP-violating phase has been assumed. Note that the degree of degeneracywhich is required for M 1,2 ∼ 1010 GeV is relatively small. Note also that smaller valuesof M 1,2 are possible if they are even more degenerate. As M 1,2 decreases, the degreeof degeneracy required becomes however very high, due to the damping effects of thescattering processes. 4

5. Gravitino problem

So far we have not taken into account the gravitino problem. The main constraintcomes from the fact that the lepton asymmetry has to be generated after inflation, whichis very important in supersymmetric models [8,23]. The thermal production of massivegravitinos restricts the beginning of the radiation-dominated era following inflation. The

reheating temperature after inflation is constrained by requiring gravitino production tobe suppressed so that it will not overpopulate t

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