small cosmological constant in the seesaw mechanism with broken supersymmetry

Small cosmological constant in the seesaw mechanism with broken supersymmetry

V. V. Kiselev1,2 and S. A. Timofeev2

1Russian State Research Center ‘‘Institute for High Energy Physics,’’ Pobeda 1, Protvino, Moscow Region, 142281, Russia2Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow Region, 141700, Russia

(Received 16 October 2007; published 18 March 2008)

The observed value of the cosmological constant can be naturally related to the scale of breaking downthe supersymmetry in consistency with the schemes of gauge and gravity mediation in particle physics, aswe argue in the framework of the static description of the phenomenological scalar field with assumedproperties of its potential producing the barrier separating the flat vacuum from the anti-de Sitter state,which leads to tuned domain wall fluctuations.

DOI: 10.1103/PhysRevD.77.063518 PACS numbers: 98.80.�k

I. INTRODUCTION

Recent precise observations in cosmology [1–5] preferthe model of the flat universe, which has the energy densitycomposed of three dominant components: baryons, darkmatter, and dark energy with fractions of energy approxi-mately given by �b � 0:04, �dm � 0:21, and �de � 0:75,respectively. The dark energy is dynamically fitted by aquintessence [6], that is a slowly evolving scalar field,whose potential energy imitates1 the cosmological con-stant. The introduction of quintessence seems to be rea-sonable, since the cosmological constant itself [8] shouldgive the energy density

�� 4�; at �� 0:25� 10�11 GeV; (1)

which leads to the artificially small scale in particle phys-ics. The quintessence serves to produce such a scale due tothe evolution of potential energy from natural values to thepresentday point. However, the quintessence does not an-swer the question on the vacuum energy itself. Since thequintessence mechanism suggests that the vacuum energyis exactly equal to zero, that could be heavy to interpret, ifone takes into account huge contributions coming fromknown condensates in the quantum chromodynamics, forexample, and other sources such as the Higgs potential inthe standard model (SM) and so on. Such contributionswould not appear once there are some kinds of cancella-tion, which could take place if the scalar field of quintes-sence would be naturally connected to the fields of thestandard model, for instance. In contrast, one usually as-sumes that the quintessence has nothing in common withthe ordinary matter, and it is actually the extra field. Thenthe quintessence does not solve the cosmological constantproblem from this point of view. Moreover, it considers thedynamical field, whose properties somehow naturallyevolve closely to the true cosmological constant, that isset to zero. In this respect one could distinguish the dy-namical and static aspects of the cosmological constantproblem. The quintessence does not provide us with anyprogress in the static aspect.

The dynamic and static features of the cosmologicalconstant problem were consistently treated in the frame-work of renormalization group running of quantum fieldparameters in the curved spacetime as developed in [9–14]and reviewed in [15].

There is an alternative way to show that the small valueof �� is not artificial but natural in the static aspect of theproblem. Indeed, fluctuations between two vacuum stateswith exact and broken down supersymmetry can result insmall mixing and the appearance of the stationary vacuumlevel with the small cosmological constant. Thus, thecosmological constant could indicate the scale of super-symmetry breaking.

In Sec. II of the present paper we assign the cosmologi-cal constant to the energy density of the vacuum (zero-point) modes. If supersymmetry (SUSY) is exact the vac-uum is flat, while breaking down SUSY results in a nega-tive density of energy determined by the scale of SUSYbreaking�x, and the vacuum state is given by anti-de Sitterspacetime (AdS) [16]. We argue for the two vacua corre-late. The decay of the flat vacuum to the AdS one [17] isforbidden due to the gravity effects [18], introducing acritical density of the AdS state unreachable in supergrav-ity [19]. Therefore, two vacuum levels can get mixing, notthe decay. Note that the SUSY breaking terms do notviolate the common picture, since their contribution tothe vacuum energy has the same order of magnitude, butit cannot change the sign of vacuum energy [16] thatpreserves the AdS state.

In Sec. III we consider the static spherically symmetricaction of gravity and scalar field interpolating betweentwo of its positions in the minima of potential withzero and negative values of energy density. Such a con-figuration describes the bubble of the AdS vacuumseparated from the flat vacuum by the domain wall. Weshow that the domain wall does not propagate toinfinity. Contrary, it has a finite size. We comparethe situation with the case of gravity switched off aswell as with the calculation of static energy describingthe decay of the flat vacuum if not forbidden. Weestimate the size of bubble fluctuations responsible forthe mixing.1See the review of quintessence phenomenology in [7].

PHYSICAL REVIEW D 77, 063518 (2008)

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http://dx.doi.org/10.1103/PhysRevD.77.063518

The mixing of two stationary vacuum levels is studied inSec. IV in cases of both thin and thick domain walls. Thesuppression of the mixing matrix element leads to a seesawmechanism with a small mixing angle [20], so that theobserved small value of the cosmological constant is natu-rally derived in terms of the SUSY breaking scale �x andPlanck mass. Note that at scales greater than �x, thedynamics is supersymmetric and, hence, its contributionto the cosmological constant is equal to zero, since it doesnot disturb the flat supersymmetric vacuum, while at scalesmuch less than �x, contributions of other nonsupersym-metric effects, like the gluon condensate in quantum chro-modynamics with the energy density �QCD ��4

QCD at�QCD � 1 GeV or the minimum of the Higgs potential inthe standard model with the admissible energy density�SM ��4

SM at �SM � 102 GeV, etc., are negligiblysmall in comparison with the supesymmetry breaking ef-fect because �x � �QCD; SM. Hence, the primary vacuaare not sensitive to the low-energy contributions. Themixing due to the domain wall depends on the vacuapositions and the height of potential barrier, which is fixedby the potential model. Then, we actually perform a kind ofreasonable adjustment of the vacuum mixture. Therefore,the mechanism of generating the cosmological constant issafe with respect to the static effects of low-energy physics,while the loop corrections do not break the main texturedue to the supersymmetric invariance of the action. Thus,the low-energy contributions appear as additional terms inthe energy density of the AdS vacuum with SUSY brokendown, which are numerically negligible, while the actualcosmological constant is generated due to the mixing thatforms true stationary vacuum states.

The estimates in Sec. V show that thin domain wallscorrespond to the low scale of SUSY breaking about �x �104 GeV, while thick domain walls give high scales of theorder of �x � 1012�13 GeV.

In Sec. VI we formulate a model of superpotential,which allows us to demonstrate that thin domain wallscorrespond to gauge-mediated SUSY breaking as well asthick domain walls do to gravity-mediated SUSY breaking.We stress that the potential introduced in Sec. VI serves asan illustration of the principal opportunity to get propertiessuitable for the offered mechanism to work. Then, weevaluate the mixing angle in Sec. VII.

A connection of the vacuum superposition to the prob-lem of generations in the SM of particle interactions isdiscussed in Sec. VIII, wherein we qualitatively map theway for the origin of three generations.

In the conclusion we summarize our results and focus onsome further questions.

II. VACUUM MODES AND COSMOLOGICALCONSTANT

The quantization of free bosonic and fermionic fieldsgive Hamiltonians in terms of creation and annihilation

operators

EB �Z d3k

�2��31

2fayB�k�aB�k� aB�k�a

yB�k�g!B�k�;

EF �Z d3k

�2��31

2fayF�k�aF�k� � aF�k�a

yF �k�g!F�k�;

(2)

respectively, for each mode with !�k� ��m2 k2p

.The commutation and anticommutation relations for

bosons and fermions

aB�k�; ayB�k

0�� faF�k�; ayF�k

0�g � �2��3��k� k0�; (3)

involve the delta function at zero if k � k0. It is related tothe spatial volume

�2��3��k�jk�0 �Z

d3r � eir�kjk�0 � volume:

Then, the energy of single field mode is given by theexpression

E �Z d3k

�2��3ay�k�a�k� �!�k� ��1�F� � volume; (4)

where F � f0; 1g denotes the fermion number for the bo-sonic or fermionic mode, correspondingly, while the en-ergy density of the zero-point mode � equals

� �1

2

Z d3k

�2��3!�k�: (5)

The vacuum energy has the density2

� �X

modes

��1�F�: (6)

At !> 0, the exact supersymmetry guarantees the fol-lowing:

(i) The number of bosonic modes is equal to the numberof fermionic ones

IW �X

modes

��1�F � 0:

(ii) Masses of superpartners are equal to each other

mB � mF;) !B�k� � !F�k�:

Therefore, the supersymmetric vacuum state j�Si has zeroenergy density �S � 0 due to the contribution by thevacuum zero-point modes. The Witten’s index IW [21]counting for all physical modes would differ from zero inthe supersymmetric theory [16], if one introduces different

2Other procedures of quantization differ from the acceptedway by an introduction of the arbitrary renormalization ofvacuum energy, that should involve some physical reasons. Wedo not see such reasons for the subtractions.

V. V. KISELEV AND S. A. TIMOFEEV PHYSICAL REVIEW D 77, 063518 (2008)

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numbers of bosonic and fermionic modes with zero energy! � 0, but such a situation would correspond to the case,when, due to the conservation law for the number ofunpaired zero-energy modes, the supersymmetry cannotbe spontaneously broken in evident contradiction withobservations.

A loss of balance between the modes produces a nonzerocosmological constant. The balance could be lost becauseof essential deviations from dispersion laws of free parti-cles that can appear due to a strong-field dynamics beyondthe asymptotically free region. Then, SUSY is brokendown.

In ordinary schemes the SUSY breaking down is de-scribed by a generation of different masses for superpart-ners at scales below �x, the characteristic energy of SUSYbreaking. For instance, in the gauge-mediated scenario ofSUSY breaking the superpartners of fields in the SMacquire masses of the order3

m��g4�

�x �x;

while the number of modes in the matter sector of theory ispreserved, and the masses satisfy a rule of splitting

Xmatter modes

��1�F � 0;X

matter modes

��1�Fm2 � 0: (7)

Effectively at scales below �x we put the dispersion law!�k� �

��k2 m2p

. SUSY is restored at scales higher than�x. Then, the integration in the energy density of singlevacuum mode is actually cut off by �x because of exactcancelling by the superpartner contribution,4 and we easilyget

� �1

2

Z �x

0

k2dk

�2��3��k2 m2

p Zd�

�2

�16��2m4�sinh4y� 4y�; (8)

where

y � arcsinh�x

m� ln

��x

m

��2

x

m2 1

s �:

At �x

m � 1 the leading contribution to the vacuum en-ergy in the observable matter sector is about

Xmatter modes

��1�F��X

matter modes

��1�Fm4 ln�x

m; (9)

since terms of the form�4x are cancelled due to the balance

between the superpartner modes, i.e. Witten’s index isequal to zero, while terms of the form m2�2

x nullify dueto the sum rule for the mass splitting (7). The superchargerelation with the Hamiltonian ensures the positivity of

matter contribution to the vacuum energy (9), i.e. up tofine effects in higher orders of small ratiom=�x one shouldexpect the following sum rule

Xmatter modes

��1�Fm4 ln�x

m< 0:

However, the direct breaking down SUSY at tree level inthe minimal extension of the SM contradicts with obser-vations, since the mass sum rules (7) introduce too light ofsuperpartners for the particles of the observable sector[16]. So, SUSY is broken in a hidden sector, which cancarry zero or nonzero quantum numbers of SM, and theparticles of the observable sector acquire masses due toloops with particles from the hidden sector, that plays therole of messenger. The first scenario with messengerscarrying nonzero SM charges refers to the gauge-mediatedSUSY breaking, while the second possibility of sterilemessengers does to the gravity-mediated one. The massesof messengers are of the order of the SUSY breaking scale,m��x. Hence, the contribution of the hidden sector to thedensity of vacuum energy is dominant, ��4

x. The signcan be certainly fixed, if one takes into account the resultby W. Nahm, who algebraically found [23] that SUSYrealization is forbidden in four-dimensional (4D) space-time with a positive density of vacuum energy, while it ispermitted in 4D spacetime with a negative density ofvacuum energy.

In the gravity sector, the SUSY breaking leads to twomassless modes of graviton with spirality �2 as well as totwo massive modes of graviton superpartner, the gravitinowith spirality � 3

2 , while in addition the goldstino withspirality � 1

2 becomes massive and it complements higherspirality modes of the gravitino to the full set f� 3

2 ;�12g.

Therefore, the goldstino breaks the balance between thenumber of bosonic and fermionic modes in the gravitysector. Hence, the vacuum energy could gain the largenegative contribution of two goldstino modes

Xgravity

��1�F��X

goldstino

��1

8�2 �4x: (10)

However, the goldstino is a composition of hidden sectorspinor fields, i.e. its two modes are superpartners for thebosonic modes from the nongravity sector. Therefore, thetrue value of vacuum energy is determined by the wholehidden sector as it has been matched above.

Thus, the vacuum modes in supergravity with SUSYbroken below �x give the negative cosmological termthat corresponds to the anti-de Sitter spacetime. We denotesuch a state by j�xi, which has the negative energy density� � ��x ��

4x.

Such nature of vacuum energy assumes that two statesj�Si and j�xi correlate, i.e. they are not completely inde-pendent, since the vacuum modes with momenta greaterthan�x are common for both states. In other words, we canintroduce the correlation length determined by the scale of

3See details in Weinberg’s textbook [16].4See notes on the scheme of regularization in [22].

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SUSY breaking �x � 1=�x, so that dynamical processes atcharacteristic distances less than �x involve the correlationof two vacuum states with zero and negative cosmologicalconstants. The transitions between two states can have astatus of whether we get the decay of the unstable state intothe stable one or mixing that leads to two stationary levels.The overlapping of two vacua is associated with the do-main wall separating the bubble of the lower-energy AdS-vacuum from the exterior of the higher-energy flat vacuum.The process of decay is described in terms of bounce, thesolution of 4D Euclidean spherically symmetric field equa-tions for a scalar field interpolating between local minimaof its potential in the region of the domain wall. Thebounce determines the quasiclassical exponent of penetra-tion between two levels of vacuum. Coleman and DeLuccia [18] show that the bounce is essentially modifiedby gravity that introduces a critical surface tension of thedomain wall, while S. Weinberg [19] found that the realsurface density of energy exceeds the critical one in super-gravity. Thus, the decay does not take place.5 Therefore,we focus on stationary 3D spherically symmetric fluctua-tions of the scalar field that provide the mixing of twovacuum states, if the domain wall cannot evolve to spatialinfinity.

III. STATIC ENERGY AND DOMAIN WALL

For fields independent of time, the action is converted tothe static potential Ustat multiplied by the factor of totaltime

S �Z

L��gp

d4x � Sstat � �UstatZ

dt; (11)

since the metric could be also written in the static form,too. In the case of spherical symmetry we get the metric

d s2 � ~B�r�dt2 �1

B�r�dr2 � r2�d#2 � sin2#d’2�; (12)

so that��gp

� r2 sin#��~B=B

q, while in the Lagrangian of

the real scalar field ��r� dependent of the radius

L F �12g��@��@�� V��;

the gradient term survives in the form

g��@��@�� 0�2B; (13)

where the prime denotes the derivative with respect to thedistance r. Then the field equation reads as follows:

�00 u0�0 2

r�0 �

1

B@V@�

(14)

with u � 12 ln�~BB�. The field equation allows the treatment

in terms of Newtonian mechanics by the assignment of �00

to the ‘‘acceleration’’ of ‘‘coordinate’’ �, so that the forcecontains the ‘‘potential term’’ @V=@� with ‘‘external pa-rameter’’ B and the ‘‘friction’’ proportional to the ‘‘veloc-ity’’ �0. The friction coefficient 2=r enters because of thespatial dimension equal to 3, while the gravitation resultsin the friction if u0 is positive, otherwise the gravitationcauses the enlarging acceleration.

The energy-momentum tensor T�� @��@��g��LF is composed by diagonal elements

Ttt � 12��

0�2B V; Trr � �12��

0�2B V; (15)

and T## � T’’ � Ttt , which enter the Einstein equation

R�� 12g��R � 8�GT��:

Hence, due to the relation of the scalar curvature with thetrace of an energy-momentum tensor

R � �8�GT;

the Lagrangian of general relativity equals to

L GR � �R

16�G;

and the static field Lagrangian equals to

L F � �Ttt :

We get the stationary energy depending on the size ofsphere rA inside of which the matter has a nonzero energy,

Ustat�rA� � �4�Z rA

0V��

��~BB

sr2dr: (16)

The static potential equals zero if the scalar field isglobal, and is positioned at a local minimum of its potentialwith V � 0. If the local minimum at the constant field ispositioned at negative V � ��x, then we arrive to anti-deSitter spacetime with

~B AdS � BAdS � 1r2

‘2 ;1

‘2 �8�G

3�x; (17)

and the positive static potential6

UstatAdS �

4�3r3A�x: (18)

Let��r� be the solution, which interpolates between twolocal minima of potential with zero energy and negativeV � ��x. To the moment, we restrict ourselves by theconsideration of a thin domain wall, so that the field isessentially changing in a narrow layer of width �r near thesphere of radius rA and �r rA. Then, the stationary

5See some further arguments in [24].

6From (17) we conclude that the gravitational potential in AdSspacetime is given by ’AdS � r2=2‘2 � 4�G�xr

2=3, and it isattractive in contrast to the naive expectation for a dust cloudwith negative energy. The reason is the large negative pressure inthe AdS vacuum p � ��, so the pressure makes a work, i.e. itproduces the positive energy, which gravitates, too.


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potential is composed of two summands with integration inlimits 0; rA� and rA; rA �r�, respectively,

Ustat�rA� �4�3r3A�x � 4�r2

AWA; (19)

where WA determines the surface energy per unit area

WA�rA� �1

r2A

Z rA�r

rAV��

��~BB

sr2dr; (20)

and it is positive if the local minima are separated bysufficiently high potential barrier.

At �r rA ‘ we can safely neglect the contributionof friction in the field equation (14), since by the order ofmagnitude �00 � ��=��r�2, while the spatial term is at thelevel of �0=r� ��=��r�2 � �r=rA �00, and the metricelements ~B, B are infinitely close to the unit, so that u0�0 �r2A=‘

2 � 1=�r � ��=�r �00. Therefore, in this limit thefield equation does not involve any scale parameter exter-nal with respect to the potential V, and it reproduces the‘‘kink’’ solution with the small value of rA and the width �rdetermined by a mass parameter in V, since the fieldequation yields 1=��r�2 � �V=��2 � @2V=@�2. Notethat the gradient contribution to the energy density Tttequals the potential term [18]. The kink sets the distribu-tion of matter determining the behavior of the metric. Thus,the thin domain wall can be established in the limit of asmall bubble.

At �r rA � ‘ the gravitational contribution to thefield equation has two regimes. At the inner surface ofthe domain wall, i.e. at the edge of AdS spacetime, themetric elements ~B, B are about unit and u0 > 0 at u0 �rA=‘2 � 1=rA, so that one could neglect its contribution aswell as the friction term. Inside the wall the metric ele-ments ~B, B can rapidly fall to unit and u0 < 0 at u0 � 1=�r,so that u0�0 ��00 and the gravity term accelerates theevolution of field from the negative minimum to positiveone, if the field evolves from a small value to a larger one.Therefore, the surface tension WA can depend on thebubble size, but the width of the domain wall still remainsat the same order as it was at small rA, that preserves themagnitude of WA, too. In this region of bubble size thegradient term in the energy density is comparable to thepotential.

We can evaluate the surface tensionWA by setting ~B� Band V � ��0�2, so that WA �

R ��Vp

�0dr�R ��

Vp

d�, whilein the supersymmetric theory with the chiral superfield thepotential is determined by the superpotential f as V �j@f=@�j2, hence, WA � jf0j, where f0 is the superpoten-tial value at the vacuum.7 In supergravity the negativevacuum energy at the extremal of the superpotential isassigned to the superpotential itself in the linear order in

the Newtonian constant G

�x � 24�Gjf0j2; (21)

that yields

WA �mPl�2x; (22)

where mPl � 1=��Gp� 1019 GeV is the Planck mass.

At rA � ‘ the metric elements at the edge of AdSspacetime become large ~B� B� r2

A=‘2 � 1, and the

gravity term in the left-hand side of the field equation (14)can still be essential, since at �u� 1 we estimate u0�0 ��=��r�2 ��00, while condition B� 1 leads to the sup-pression of the gradient term in the energy as well as to amore thick domain wall because of the approximation B ��00 � @V=@�, hence, ��0�2 V and 1=��r�2 �@2V=@�2 � ‘2=r2

A. Note that the width of the domain wallessentially exceeds its ‘‘natural’’ value �r0 determined bythe parameters of the potential V at small rA, and it linearlygrows with rA like �r� �r0 � rA=‘. Switching the regimesinWA versus rA depends on the parameters of the potential.The simple example with WA � W0

A1rAb‘ �1 tanhfrAb‘�

b0g�� at b0 � 1 allows us to draw a conclusion on thecritical behavior of WA versus the scale of switch rA �bb0‘, as it is depicted in Fig. 1, that shows the staticpotential Ustat. Moreover, at large rA � ‘ the domainwall could disintegrate at all.

We assume that the critical scale is large enough in orderto provide the materialization of the bubble with zero staticpotential. Then, the bubble can arise in the vacuum withzero density of energy. The characteristic size of such abubble is given by solving Ustat � 0, that gives

rA �3WA

�x� ‘: (23)

The materialization of a bubble in the flat vacuum results in

rA

0

Ustat

FIG. 1 (color online). The static potential of the bubble withthe domain wall versus the bubble radius at a different behaviorof surface tension: naively constant W0

A (long-dashed curve),with a large scale of switching the regime (solid curve) and a lowscale of switching (dotted curve). The low scale of switching isnot realistic, since it should mean the opportunity of the domain-wall motion to infinity, i.e. the decay, that is forbidden (see text).

7The derivation closely follows the original study by S.Weinberg in [19].


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the instability, since it takes place at the size of rA, that isnot positioned at the local minimum of the static potential:the domain wall begins to move to the bubble center (seeFig. 1). Furthermore, the zero size of the bubble is alsounstable: the flat vacuum suffers from fluctuations due tothe bubbles of the AdS vacuum.

This situation is opposite to the case of switching off thegravity. Indeed, the elimination of gravitational actionresults in the static potential of the bubble

Ustat0 � �

4�3r3A�x 4�r2

A~WA;

where

~W A�rA� �1

r2A

Z rA�r

rA

�V��

1

2��0�2

�r2dr:

This static potential formally has the opposite sign incomparison with (19). Therefore, the domain wall canmaterialize after the penetration through the potential bar-rier, but it will move to spatial infinity; that means thedecay of flat vacuum to the AdS one. The description ofpenetration in the presence of gravity was considered byColeman and De Luccia [18], involving the Euclideanaction and spherical symmetry. So, the critical surfacetension was found, and in fact [19] the decay is forbidden,since the tension exceeds the critical value.8

Indeed, at the weak gravitational field, i.e. atG! 0, onecan easily evaluate the static energy Estat by summing up

(i) the energy of the AdS-vacuum bubble

MB � �43�r

3A�x;

(ii) the energy of the domain wall Mdw � 4�r2A

~WA,(iii) the gravitational potential of the wall-bubble inter-

action

’AdSMdw � ’AdS4�r2A

~WA �163�

2Gr2A�x

~WA;

(iv) the gravitation of the thin domain wall itself

Z’dwdMdw � �

GrA

ZMdwdMdw

� �8�2Gr4A

~W2A;

that yields

Estat � �4�3r3A�x 4�r2

A~WA

�1

1

2

r2A

‘2

�� 8�2Gr3

A~W2A:

Beyond the weak-field approximation in [19] S. Weinbergfound

Estat � �4�3r3A�x 4�r2

A~WA

��1

r2A

‘2

s� 8�2Gr3

A~W2A;

where the only modification of the wall-bubble term isrelated to the strict definition of the thin domain-walldensity of energy in terms of the Dirac delta function

�dw �~WA��Bp ��r� rA�

with B � BAdS, that preserves the invariance under repar-ametrizations of the radius. Such static energy is the massdetermining the Schwarzschild metric beyond the bubbleand domain wall, so it has nothing with the static value ofaction, Ustat � Estat. It is an easy task to find that Estat

nullifies at

rA �r0A

1� �r0A

2‘�2; at r0

A �3 ~WA

�x:

Therefore, r0A < 2‘ and the critical density is given by

�cx � 6�G ~W2A. S. Weinberg showed that the surface ten-

sion is constrained by the superpotential as ~WA > 2jf0j.Thus, �cx > 24�Gjf0j

2 � �x, and the flat vacuum cannotdecay to the AdS one. We have to stress two points. First,the above conclusions on the behavior of Estat is made atexactly constant surface tension ~WA. Second, at arbitrary~WA, nullifying the static energy Estat describes the materi-

alization of the bubble, which is strictly considered in [18]in terms of Euclidean 4D-symmetric action, so that onegets the standard quasiclassical calculation of bounce.Contrary, the static action corresponds to unstable fluctua-tions usually called sphalerons,9 which are considered in3D space. Such bounce and sphaleron are generally differ-ent classical solutions, so certainly Estat � Ustat.

As we have just shown the gravity induces the materi-alization of the bubble not propagating to infinity, thatmeans the mixing of two levels, but not the decay.

Thus, due to the unstable bubbles the vacua are noteigenstates of a true Hamiltonian.

IV. TWO-LEVEL SYSTEM

Consider the quantum system of two stationary vacuumlevels within the domain wall, which is described by theHamiltonian density H � Hvac=volume,

H � ��xj�xih�xj �Sj�Sih�Sj

~�fj�xih�Sj j�Sih�xjg; (24)

where �x ��4x in the AdS vacuum with broken SUSY,

while in the supersymmetric vacuum �S � 0. We defineglobal complex phases of states, so that the quantity ~�takes a real positive value. The transition is associated with

8This fact supports our previous assumption on the large scaleof switching the regimes in the surface tension WA.

9More strictly, sphalerons actualize a minimal value of thepotential barrier.


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fluctuations described by the domain wall corresponding tothe overlapping region of states. The bubble of the AdSvacuum has the size rA � ‘, the domain wall has a width�r. Let us, first, evaluate the width of the domain wall �r invarious cases and, second, estimate the mixing matrixelement ~� � h�SjHj�xi.

A. The thin domain wall

If the domain wall is thin, its mass is given by theexpression Mdw � 4�r2

AWA � 4�‘2�rV0, where V0 isthe characteristic height of the potential barrier inside thewall. This mass is compensated by the negative mass of thebubble MB � �4�r3

A�x=3��4x‘3, so that under ‘�

mPl=�2x we get

�r � V0 � ‘�x �mPl�2x: (25)

Furthermore, for the chiral superfield, the potential isdefined by V � j@f=@�j2, where in the linear order in Gthe superpotential f0 at the stationary point is related to thenegative density of vacuum energy by (21), that gives

f0 �mPl�2x ) V0 �

f20

��2�m2

Pl�4x

��2; (26)

where �� is the characteristic change of field in thedomain wall, i.e. the ‘‘distance’’ between two extremalpoints of the field. Hence, we evaluate the width of thedomain wall in terms of the evolution change of the field,

�r��2

mPl�2x

: (27)

Putting �r rA � ‘, we find

�� mPl: (28)

Therefore, the domain wall is thin, if the field dynamics isessentially sub-Planckian.

For instance, we get

��x ) �r�1

mPl; (29)

��mPl�xp

) �r� �x �1

�x: (30)

The case of �r� �x looks like the most natural situation,since the domain wall has the size of correlation length oftwo vacua. At

��mPl�xp

�� mPl the domain wallbecomes thick with respect to the correlation length �x.This case requires special consideration.

The correlation energy of two states can be estimated interms of mixing density of energy multiplied by the vol-ume of the bubble,

Ecorr � ~� � ‘3: (31)

On the other hand, it is determined by the energy in theoverlapping region restricted by the correlation length �x,

i.e. in the element of the thin domain wall with the area ofthe order of �2

x. Hence, Ecorr is given by the surface tensionWA � �r � V0 in the area of correlation

Ecorr �WA � �2x: (32)

Value (32) gives the energy of the domain wall in thebeginning of materialization at rA�x.

Therefore, under WA � f0 �mPl�2x we get the estimate

~��2

x

‘2 ��6

x

m2Pl

; (33)

implying ~� �x.At

��mPl�xp

�� mPl the correlation energy is de-termined by the height of potential barrier within thecorrelation volume Ecorr � V0 � �

3x, that yields ~�

�6x=m2

Pl satisfying the same condition ~� �x as above.

B. The thick domain wall

The mass of the thick domain wall is estimated in termsof characteristic height of the barrier Mdw � ��r�3V0, thatis opposite to the mass of the bubble with size rA � ‘,where the energy density is negative. So,

��r�3 � V0 � ‘3�x; (34)

that leads to

��r�3 �mPl��

2

�6x

: (35)

Putting �r� ‘, we get

�� mPl; (36)

and the dynamics of the thick domain wall is related tosuper-Planckian fields.

The correlation energy is determined by the dominantvolume of the thick domain wall

Ethickcorr � ~� � ��r�3; (37)

which is equal to the characteristic energy inside the wallwithin the correlation volume

Ethickcorr � V0 � �

3X: (38)

Therefore, we get

~��mPl�

7x

��4; (39)

and again ~� �x due to (36) and �x mPl.

C. Seesaw mechanism

We have just drawn the conclusion that the matrix of thetwo-level Hamiltonian of vacuum has the form

H ��x ~�

~� 0

� �at ~� �x; (40)


063518-7

so that such a texture is well known in the particle phe-nomenology as the ‘‘seesaw mechanism’’ for describingthe mixing of charged currents, for instance [20]. Someapplications of the seesaw mechanism to the cosmologicalconstant problem have been recently considered in [25],while the small scale in the quintessence dynamics gen-erated due to the seesaw has been studied in [26].

The eigenvalues of (40) are equal to

�� 12��x �

��2

x 4~�2q

�; (41)

and due to ~� �x they are reduced to

�dS� �

~�2

�x; �AdS

� � ��x; (42)

that corresponds to the expanding de Sitter (dS) universeand collapsing AdS universe. Both vacua are stationarylevels with no mixing or decay. We are certainly living inthe universe with the dS vacuum.

The eigenstates are described by the superposition ofinitial nonstationary vacua

jvaci � cosKj�Si sinKj�xi;

jvac0i � cosKj�xi � sinKj�Si;(43)

with the mixing angle10 equal to

tan2K �2~��x; (44)

well approximated by

sinK �~��x 1: (45)

Thus, we arrive at the analysis of the cosmological constantin different schemes of fluctuations in the region of over-lapping the two initial vacuum states, i.e. the domain wall.

V. ESTIMATES

The thin domain wall determines

�dS� �

�8x

m4Pl

; (46)

and due to �� 4� we get the estimate11

�x ��mPl��p

� 104 GeV: (47)

Thus, the thin domain wall is relevant to the low scale ofSUSY breaking.

For the thick domain walls we arrive at the estimate

�dS� �

m2Pl�

10x

��8: (48)

Then, the comparison with the observed cosmologicalconstant gives rough estimates at various evolution changeof field, for example,

��m2

Pl

�x) �x � 1012 GeV;

��m2

Pl

�x

��mPl

�x

s) �x � 1013 GeV:

(49)

Therefore, thick domain walls are relevant to the high scaleof SUSY breaking.

The relation of the SUSY breaking scenario with differ-ent regimes of domain-wall fluctuations can be clarified byconsidering some typical properties of scalar-fieldpotential.

VI. MODEL POTENTIAL

For simplicity, consider the real scalar field and AdS-vacuum density modeled by a single fermionic mode offormula (8)

�x � �: (50)

Introduce the field M defined as the bottom boundary ofintegration versus the vacuum modes in the energy density,

��M� �Z �x

M

k2dk

�2��2��k2 m2

p: (51)

This field should be physical, since it describes the gen-eration of SUSY breaking. At M � �x, SUSY is exact,while at M � 0 we get �x � ��0� and SUSY is brokendown.

The field M is constrained by the limits M 2 0; �x�.In addition, the above definition can involve noncanonickinetic energy. Therefore, M is actually expressed interms of canonic scalar field �, i.e. M �M��.

Let us assign the superpotential12 of� by the supergrav-ity relation

f2�� 1

24�G��M� �m2

Pl�4x: (52)

Then, the potential is given by the expression13

V�� @f@�

��2; (53)

10The subscript ‘‘K’’ is the abbreviation of Russian ‘‘kachely’’translated as ‘‘seesaw.’’

11Estimate (47) was obtained by T. Banks in [27] in anotherway of physical argumentation for the mechanism of SUSYbreaking.

12It is important to emphasize that we deal with the low-energyeffective potential of the scalar field, that should be considered asthe correction to a true superpotential safely neglected at suchvalues of the field, where the introduced correction is essential.

13Remember, we deal with the real field.


063518-8

which is calculated as the derivative of the compositefunction. This fact causes three important points.

First, at M! �x the vacuum density of energy nullifies��4

x �M4 at m �x or ��3x �M3 at m� �x,

while actually m��x, so that any way the superpotentialbehaves like

f�

��1�

M�x

s;

and there is the singularity

@f@M

�1��

1�M�x

q :

The simplest way to avoid the singularity is to postulate anappropriate behavior of derivative for M with respect to�like

dM

d�� 1�

M

�x: (54)

Then, the potential will be regular at its local minimumcorresponding to the flat vacuum with �S � 0. The solu-tion of (54) is given by the exponential potential. In a moregeneral form, we put

�M

�x

�� 1� exp

��2

~m2 1 C��; (55)

where ~m is a scale, � is an integer, while C�� is a poly-nomial function, introducing corrections to the quadraticdependence of the exponent argument versus the field. Thequadratic behavior is introduced in order to preserve thelimits of M as well as the invariance under �$ ��, forthe sake of simplicity.

Second, at M! 0 the vacuum density tends to its AdSvalue as ��4

x �m0 �M3, so that the superpotential

acquires the dependence in the form14

f� 1� ~bM3

�3x

: (56)

At this point, SUSY is broken, hence @f=@� � 0, that canbe easily satisfied if

M 3 ��! 0: (57)

This condition is provided by the ansatz (55) at � � 6,since C ! 0 at �! 0.

Third, the vacuum energy in the scalar sector given by Vat �! 0 is modified by supergravity [16]

�!��@f@�

��2�24�Gf2�0� �

��@f@��2��x: (58)

To preserve the AdS spacetime we should require

��@f@��2

& 24�Gf2; at �! 0;

or approximately

m2Pl

~m2�4

x & �4x;

that can be satisfied by putting ~m � ~mthin, where

~m thin �mPl

; (59)

so that 2 ��x=m� 1 withm being the mass in the singlevacuum density of energy (51), and such a value of provides the correct expectation V�0� ��4

x that is appro-priate for thin domain walls as we will see below, since itprovides the sub-Planckian changes of field in the domainwall.

At V�0� �4x, one could expect that V�0� is suppressed

by the gravitational constant G, and hence,

~m thick �mn1

Pl

�nx� mPl; at integer n > 0; (60)

that is appropriate for thick domain walls with super-Planckian changes of field.

So, the potential model in (55) is almost defined. Theonly uncertainty is entered through the integer n andfunction C��, whose properties are related to the dynam-ics of SUSY breaking down.

A. Gauge-mediated SUSY breaking

The correction function could look like the expansion ininverse �g ��x determined by a strong-field interactionin the gauge sector, so that to the leading order one couldexpect

C ��2

�2g: (61)

The complete potential energy of the field, includinglinear G-corrections from supergravity, has the form15

U�� V�� 24�G�f��

�3

@f@�

�2

16�

3G�2

�@f@�

�2: (62)

14The relation between the superpotential and density of vac-uum energy in general involves higher orders in the Newtonianconstant, so that subleading terms can induce a linear correctionto the cubic dependence as

M3

�3x

�bM

�x

��2

m2Pl

;

that slowly modify the potential behavior at �! 0, which is notimportant for our purposes. 15See, for instance, [16].


063518-9

The characteristic behavior of quantity (62) under (61) isshown in Fig. 2.

It is clear that U�� starts to rapidly grow from U�0� at��g ��x, where C�� effectively starts to dominatewith respect to the unit. The potential begins to fall at ��

~m�gp

��mPl�xp

� �x. Then, the characteristic changeof field between two minima of potential16 is about ��mPl�xp

, which corresponds to the thin domain wall.Thus, the thin domain wall is relevant to the gauge-

mediated SUSY breaking at low scales �x � 104 GeV.

B. Gravity-mediated SUSY breaking

If the gravity is responsible for the transition of SUSYbreaking to the observed matter sector, the expansion of Cis composed versus powers of the Newtonian constant, i.e.in the inverse Planck mass. Therefore, to the leading orderone expects

C ��2�2

m2Pl

; at �� 1: (63)

The leading term depends on the mass scale ~mthick,which can be estimated by

�2

~m2thick

��2�2

GR

m4Pl

; (64)

where �GR �x denotes the characteristic scale of ob-served fields or superpartner masses, which is composed bythe breaking scale �x, and it includes powers of inversePlanck mass, too. Therefore,

~m thick �m2

Pl

�GR;

while

��~mthickmPl

p:

For instance, at �GR ��2x=mPl �

��Gp

�2X we find the

distance between fields fitted to the minima of the potential

��m2

Pl

�x;

while �GR ��3x=m

2Pl �G�

3x corresponds to

��m2

Pl

�x

��mPl

�x

s:

Both the above cases of�GR represent two known versionsof the standard scenario for the gravity-mediated SUSYbreaking [16].

Since the field is exposed to super-Planckian changes,we deal with the thick domain walls in the gravity-mediated SUSY breaking at high scales about1012�13 GeV.

To the end of this section, we especially emphasize thatat super-Planckian changes of field in thick domain walls,the height of potential barrier takes the values much lessthan the energy density of the AdS vacuum, V0 �4

x.Therefore, one should control the dimensionless parame-ters like � in order to get positive values of the actualpotential (62) within the wall. In this respect, one can seethe role of the presented potential as a toy model, thatserves to demonstrate some general features of scale de-pendence in the problem. In practice, the form of truepotential strongly depends on the field contents in thetheory. Moreover, remember that we have accented theattention on the nonperturbative low-energy contributionand neglected a tree potential.

VII. ANGLE �K

The mixing angle of two levels K takes different valuesdepending on the scenario of SUSY breaking.

For the thin domain wall we get

K �~��x��2

x

m2Pl

��

mPl: (65)

Therefore, its value is certainly fixed by presentday data onthe cosmological constant, K � 10�30.

In contrast, for thick domain walls we write down

K �

��~�2

�2x

s�

��dS

�

�x

s��2

�

�2x

; (66)

where �x depends on the scheme of gravity-mediatedSUSY breaking. In the above examples we roughly getthe estimate K � 10��46�48�.

ΜX

ΓΜXmPl

ΓΦ

X4

mPl ΜX 3

U Φ

FIG. 2 (color online). The potential of the scalar field U�� inthe gauge-mediated scheme of SUSY breaking at �g � �x=and ~m�mPl=. The upper curve shows the potential with nosupergravity corrections.

16The method of potential reconstruction in the model does notallow us to make certain conclusions about an actual potentialbehavior at infinity �! 1 because it can be not related with theenergy of vacuum modes. Therefore, the true form of potentialfar away from local minima are not shown in Fig. 2.


063518-10

VIII. GENERATION PROBLEM

The vacuum states j�Si and j�xi are determined byclassical values of the scalar field in the local minima ofits potential. So, the quantization of dynamical fields in thevicinity of such vacua are straightforwardly standard. Thequestion is how can we quantize the fields over the truevacuum being the superposition of two such states inaccordance with (43)?

First, we can determine the field masses in vacua j�Siand j�xi, respectively, in an ordinary way. Say, let mS andmx be the masses of the fermion field as given by such aprocedure. Hence, the masses correspond to the cases ofexact and broken SUSY.

Second, the superposition of vacuum states is equiva-lently described by the 2D vector or column

jvaci�cosK

sinK

� �: (67)

Therefore, the mass term of the fermion field should begiven by a 2� 2 matrix of general form

M �mx �m�m mS

� �: (68)

It is clear that such construction is responsible for twogenerations of the same field.

Thus, the vacuum structure in the form of superpositioncan be the origin of generations observed in the standardmodel. Then, one should suggest the superposition of threevacuum levels, at least. Probably, one could prefer for thesituation with two flat vacua and single AdS vacuum as itdepicted in Fig. 2. Then, the Hamiltonian of the vacuumcontains the mixing of the AdS level with each flat statej�Si and j�Si� at positive and negative values of flatminima, while the eigenstate relevant to our universe takesthe form of the superposition

jvaci3G �1��2p fjvaci jvaci�g; (69)

which is represented as a 3D vector

jvaci3G �1��2p

2 sinK

cosK

cosK

0@ 1A; (70)

in the basis of states fj�xi; j�Si; j�Si�g, that could beactual for 3 generations, probably, with some realistictextures of mass matrices of matter fields.

We finalize at this point, since the consideration ofspectroscopy is beyond the scope of the present paper.The problem is reduced to the calculation of nondiagonal‘‘masses’’ a la �m in (68).

IX. CONCLUSION

In this paper we have offered the mechanism for thedynamical generation of the small cosmological constantdue to the seesaw mixing of two initial vacuum statesdescribing the phases of the exact and broken supersym-metry. The current value of the cosmological constant canbe consistent with the estimates obtained in the frameworkof our model with phenomenological entries of the SUSYbroken scale in particle physics.

The mechanism works due to fluctuations formed bybubbles of the AdS vacuum separated by domain wallsfrom the flat vacuum. We have classified the cases of thinand thick domain walls related to gauge or gravity-mediated SUSY breaking, respectively. The mixing resultsin the superposition of the initial vacua that could set theorigin of three generations of fermions in the standardmodel.

Further studies of such a mechanism have to answerimportant questions on the spectroscopy of matter andsuperpartners as well as on a role of the mixing angle K

and methods of its direct measurement. In addition, oneshould clarify why we are living in the vacuum we havegot. An answer to this question could disfavor the schemewith two flat vacua as presented in Sec. VIII. Then, aninverse picture with two AdS vacua and a single flatvacuum could be more realistic. This possibility will beinvestigated elsewhere [28]. Nevertheless, basic features ofscale dependence found in the present paper should remainvalid with no changes. The dynamical aspect of the cos-mological constant problem related to the universe evolu-tion is not investigated in the present paper.

To finalize, we list some drawbacks of our approach tothe cosmological constant problem:

(i) We do not get full control on the quantum effects oflow-energy physics and its parameters nor on thephase transitions during the universe evolution.

(ii) The vacuum mixing is described phenomenologi-cally through the scalar field with some suitablepotential, that does not give a complete dynamicalexplanation of the mixing.

(iii) We do not consider any fundamental origin of thefield and its interpretation within a primaryframework.

(iv) We do not prove that the field can possess theproperties we have ascribed to it, since no calcu-lations from the first principles have been done.

(v) The final result we have inherently depends on theassumptions and adjustments made, that producesthe limitations for the further development.

ACKNOWLEDGMENTS

The work of V. V. K. is partially supported by theRussian Foundation for Basic Research, Grant No. 07-02-00417.


063518-11

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small cosmological constant in the seesaw mechanism with broken supersymmetry

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