Cosmo 2000 Proceedings

|2 -\— •. However, these breaking terms are negligible in t h e present analysis as long as \ip\ < m _ 1 .

(13)

138

since eK does not contain (p. For rj, \X\ < 0(1), we can rewrite the potential as V(v,
+ V2) + m2\X\2.

(15)

Thus, the term proportional to ip2 becomes dominant and the chaotic inflation takes place if the initial value ip > 1. During the chaotic inflation, the effective mass squared of r/, m 2 , becomes m2 ~ m V 2 - 6K 2 ,

(16)

where H(~ A=m
(17) (18)

where the dot represents the time derivative. Also, here and hereafter, we assume that X is real and positive. But, one should note that these classical equations of motion are valid if and only if quantum fluctuations are unimportant for both ip and X fields. Therefore, we need to clarify when the classical description is feasible. For this purpose, first, we compare quantum fluctuations with classical changes for the field

(is)

On the other hand, the amplitude of quantum fluctuations Sipq ~ H/(2TT). Thus, the above classical equation of motion for

Otherwise, the universe is in a self-reproduction stage of eternal

inflation. Therefore, we consider only the region ip
TO-1/2.

Next, in order

If we take t h e higher order term £,\X\A with 5 < —9/8 in t h e Kahler potential, the effective mass squared of X becomes larger than 9 H 2 / 4 so that X rapidly oscillates around the origin and its amplitude goes to zero.

139 to estimate the amplitude of quantum fluctuations of X, we use the FokkerPlanck equation for the statistical distribution function of X, P[X,t],

which is obtained through the Langevin equation based on eq.(18) with use of the stochastic inflation method of Starobinsky. Taking p> as a time variable by virtue of eq.(17), we find that ( ( A X ) \ asymptotically approaches

On the other hand, the classical value of X, Xc is at most (p/(pi — V/my>/v47*'V/6- Thus, during the chaotic inflation, ((AX)2)

< Xc ~ V^ip < 1 « (p

(22)

because m < l a s shown later. Thus, for X, quantum fluctuations are smaller than the classical value, and moreover our approximation that both rj and X are much smaller than unity is consistent throughout the chaotic inflation. The density fluctuations produced by this chaotic inflation is estimated as

^ - 4 = - ^ W + * 2 ).

(23)

Since X -C p, the amplitude of the density fluctuations is determined only by the ip field and the normalization at the COBE scale (Sp/p — 2 x 10~ 5 for VCOBE - 14) gives

m ~ 10 13 GeV.

(24)

The spectral index ns becomes ns ~ 0.96 for <^COBE — 14. After the inflation ends, an inflaton field (p begins to oscillate and its successive decays cause reheating of the universe. In the present model the reheating takes place efficiently if we introduce the following superpotential: W = XXHH,

(25)

where H and H are a pair of Higgs doublets whose .R-charge are assumed to be zero and A is a constant. Then, we have the coupling of the inflaton ip to the Higgs doublets as L ~ XrrvpHH,

(26)

140

which gives the reheating temperature

— 8 - v (T^)(lo^v)" 2 -

<">

In order to avoid the overproduction of gravitinos, the reheating temperature TR must be lower than 109GeV, which requires the small coupling A < 10~ 5 . The small coupling A is naturally understood in 't Hooft's sense provided that HH is even under the Z2 symmetry in eq.(7). 3

N e w inflation in supergravity with a chaotic initial condition

In this section, we propose new inflation with a chaotic initial condition as an alternative scenario because it predicts low reheating temperature straightforwardly and can easily generate density fluctuations with a tilted spectrum. 3,1

model

We introduce the inflaton chiral superfield $(2:, 0) and assume that the Kahler potential, K{<&, $*, • • •), is a function of $ + 4>* by imposing the NambuGoldstone like shift symmetry. If this Nambu-Goldstone-like symmetry is exact, however, x (= v 2 Im$) cannot have any potential and chaotic inflation would be impossible no matter how large an initial value it could take. Hence this global symmetry must be explicitly broken. For the purpose, we introduce the following superpotential which breaks the shift symmetry with a dimensionless coupling parameter g', W = v2X - g'X<$>2 = v2X{\ - g<S>2),

(28)

where we have introduced another chiral superfield X(x, 9). Here v is a scale generated dynamically, and g = g'v~2. The above superpotential W is invariant under U(1)R X ZI symmetry. Under U(\)R symmetry, X{0) -> e-2iaX(eeia), $((9) -» $(<9eiQ).

(29) (30)

Also, X is even and $ is odd under Zi symmetry. Although this superpotential W is not invariant under the Nambu-Goldstone-like shift symmetry, it is natural in the sense of 't Hooft. For the symmetry is recovered if the small parameter g1 is set to be zero. Since the correction to the Kahler potential from this breaking term is negligible if g -C 1, we consider the Kahler potential which is invariant under the Nambu-Goldstone-like shift symmetry and

141

U(1)R

x Z2 symmetry, «"(*, **, X, X*) = tf[($ + $*) 2 , XX*}.

(31)

Below we take a minimal Kahler potential, for which scalar kinetic terms are canonical, K = h$

+ §*)2 + XX*,

(32)

and extend it later to incorporate higher-order effects. 3.2

Dynamics

The scalar Lagrangian density L($, X) is given by L($, X) = d^d"**

+ d^Xd^X* - V{$, X),

(33)

where the scalar components of the superfields are denoted by the same symbols as the corresponding superfields. The scalar potential is given by V = v4eK [ |l - g$2\2 (1 - \X\2 + |X| 4 ) + | X | 2 | - 2 p $ + ($ + $ * ) ( l - < 7 $ 2 ) |2 ] .

(34)

Now, we decompose the scalar field $ into real and imaginary components, $=-L(y,

+

i X ).

(35)

Then, the Lagrangian density L(ip, x, X) is given by L{ip,x,X)

=

l

-d^d^+

^xd^x

+ d^Xd^X* -V(v,x,X),

(36)

with the potential V(ip,x, X) given by V(v,x,X)=v4exp(\X\2 x

2

+ ¥>

)

{(1 - f ^ 2 ) 2 + 9X2(l + \f2

+ f X 2 )} (1 - \X\2 + |X| 4 )

+ \X\2{2g2x2 + 2{g - 1) V + 2g{g + 1 W +2 f f ( f f -lV + 2^ V + X 2 ) 2 } |2_1_,„2

(37)

While , x can take a value much larger than unity without costing exponentially large potential energy. Then,

142

the amplitudes of both ip and X soon become smaller than unity due to this steep potential and the exponential factor can be Taylor expanded around the origin. This is the situation we deal with hereafter. Then the scalar potential is dominated by V ^ \\x\

(38)

with A = g2vA. Thus chaotic inflation can set out around the Planck epoch. chaotic inflation, the mass squared of
H^±X\

During

(39)

where H is the Hubble parameter in this era. Then,

end ~ //end ~ A 1 / 2 =

9V\

On the other hand, the mass squared of X, m2^-, is dominated by, m\ c 2«7V X 2 ~ -2E\

(40)

X

which is much smaller than the Hubble parameter in the early stage of chaotic inflation, when X moves towards the origin only slowly. Below we set X to be real and positive making use of the freedom of the phase choice. In this regime classical equations of motion for X and x a r e given, respectively, by 3H X ~ -m2xX, 3tf

3

X

- -Ax ,

(41) (42)

from which we find, X cc x2-

(43)

This relation holds actually if and only if quantum fluctuations are unimportant for both x a n d X. As for x, the amplitude of quantum fluctuations acquired in one expansion time is larger than the magnitude of classical evolution in the same period if x •> A - 1 / 6 , when the universe is in a self-reproduction stage of eternal inflation. Hence let us consider the regime x < A^ 1 / 6 and (42) holds. Then we can estimate the root-mean-square (RMS) fluctuation in

143

X using the Fokker-Planck equation and found that ((AX) approaches /

\

\ asymptotically

A2/3

which is much less than unity because A must be a tiny number as will be shown later. From (43) and (44) the amplitude of X becomes much smaller than unity by the time Y ~ A/24- Thereafter (41) no longer holds and X oscillates around the origin rapidly and its amplitude decreases even more. Thus our approximation that both if and X are much smaller than unity is consistent throughout the chaotic inflation regime. As x becomes of order of unity, chaotic inflation ends and the field oscillates coherently with the mass squared m^ ~ 2gv4. Since g must take a value slightly larger than unity as shown later, the energy density of this oscillation becomes less than the vacuum energy density ~ vA soon. At this stage

-(3-l)+ff(l + |)x2, (45) m2 which becomes negative only after the square amplitude of x becomes smaller than Xc = 2(<; — l ) / 3 <§C 1. Thus, the second inflation takes place, which is driven by a false vacuum energy, before if becomes unstable. Once the inflation occurs, the amplitude of the oscillation rapidly goes to zero because the amplitude damps in proportional to a - 3 / 2 . Then, the potential energy with x — 0 reads, V ( ^ ) ~ i > 4 e x p ( ^ 2 + |X| 2 )

i - f y 2 ) 2 ( i - l ^ l 2 + |X|4) +|X|2 J2(
v*

C

..4._2

2^V,

(for^,|X|«l) (46)

with c = 2(g — 1). Thus, if g > 1, ip rolls down slowly toward the vacuum expectation value 77 = \/2/g and new inflation takes place. Here and hereafter we set ip to be positive by use of Z2 symmetry( <-> —$). The initial value of if in the beginning of new inflation is set by the amplitude of quantum fluctuation and is of order of H then. The dynamics of

144

if is also governed by quantum fluctuations until the classical motion during the Hubble time, Acp ~ \(p\H~l, becomes larger than the quantum fluctuations, H/(2TT), acquired during the same period. In our case, this condition is equivalent to ^

^

=

^

(4?)

^ -

Since c < 1 as shown later, quantum fluctuations dominate the dynamics initially. Therefore, the universe enters the self-regenerating stage so that the imprint of chaotic inflation is washed away and current horizon scale must be contained in one of a new inflation domain in which ip got larger than ipc and the classical description of the dynamics of ip with the potential Eq.(46) became possible. The slow-roll condition for the inflaton ip is satisfied for 0 < c < 1 and 0 < ip < 1. The Hubble parameter during new inflation is given by H ~ v2/\/3. Then, the number of e-folds acquired for ip > (p^ is given by hi

V

c

\Vf

)

c

where the prime represents the derivative with respect to ip and
*H{N) ^ AT—,

(49)

ctpN where / — 3/5 (2/3) in the matter (radiation) domination. Since the COBE normalization requires $#(AT) ~ 3 x 1 0 - 5 at N ~ 60, the scale v is given by 2V3TT

v ~ 2.3 x 1 0 - 2 v / c e - ^ | w = 6 0 - 1-8 x 10~ 3 - 3.6 x 10" 4 ,

(50)

for 0.02 < c < 0.1. The spectral index ns of the density fluctuations is given by ns ~ 1 - 2c.

(51)

The data of anisotropics of the cosmic microwave background radiation (CMB) by the COBE satellite also implies ns = 1.0 ±0.2 so that 0 < c < 0.1, which leads to 1.00 < g < 1.05.

145

After new inflation, (p oscillates around the minimum tp = 77 and the universe is dominated by a coherent scalar-field oscillation of a =

ev

= e " 2 (l + 2rja + • • • ) ,

a has linear interactions of gravitational strength with all scalar and spinor fields in the theory including those in the minimal supersymmetric standard model (MSSM). Let us consider, for example, the term W = yiDiHSi in the superpotential in MSSM, where Dit Si are doublet (singlet) superfields, H represents Higgs superfields, and j/j are Yukawa coupling constants. Then, the relevant interaction Lagrangian reads, Ant ~ vhaDlSl

+ • • •,

(52)

which leads to the reheating temperature TR, Tfi~ Kr10-lCr8,

(53)

for 0.02 < c < 0.1, which is low enough to avoid the overproduction of gravitinos in a wide range of the gravitino mass. Finally we comment on higher order terms in the Kahler potential. Fourth order terms in the Kahler potential such as AK = ||X| 2 (<1> + $*) 2 + | | X | 4 + | ( $ + d>*)4,

(54)

roughly generate three minor changes associated with ki. The first change is as follows; during chaotic inflation, the mass squared mx of X acquired the additional terms, 2

<-, 2 4

2

mx ~ 2g v*x ~ -3k2H2.

"-2

2 4 4

j P T (55)

Thus, if &2 < — f, X oscillates rapidly so that the amplitude of the oscillation damped into zero. Thus, we can safely set \X\ to be zero in the whole analysis of the present model. The second one is the change of c = 2(g — 1) into c = 2(g + k\ — 1). % is almost irrelevant for the dynamics of ip field and only changes its vacuum expectation value r/ due to the redefinition of

i) = E;4f($+$*) 2 |V>i| 2 . Then, the interaction Lagrangian density is given by £i n t = EiAit/^d^jC^V*: which yields the similar reheating temperature.

146

4

Discussion and Conclusions

We have shown that a chaotic inflation naturally takes place if we assume that the Kahler potential has the Nambu-Goldstone-like shift symmetry of the inflaton chiral multiplet $ and introduce a small breaking term of the shift symmetry in the superpotential. Unlike other inflation models the chaotic inflation model has no initial value problem and hence it is the most attractive. However, it had been difficult to construct a natural chaotic inflation model in the framework of supergravity because the supergravity potential generally becomes very steep beyond the Planck scale. Therefore, the existence of a natural chaotic inflation model may open a new branch of inflationmodel building in supergravity. Furthermore, the chaotic inflation is known to produce gravitational waves ( tensor metric perturbations ) which might be detectable in future astrophysical observations. We have also proposed new inflation with a chaotic initial condition. Chaotic inflation takes place around the Planck scale so that the universe can live long enough. In this regime, the inflaton responsible for new inflation dynamically relaxes toward zero so that new inflation sets in. The reheating temperature is low enough to avoid overproduction of gravitinos in a wide range of the gravitino mass. Furthermore, our model generally predicts a tilted spectrum with the spectral index ns < 1, which may naturally explain the recent observation of anisotropies of CMB by the BOOMERANG experiment and the MAXIMA experiment. In the present model, the initial value of the inflaton for new inflation may be so close to the local maximum of the potential that the universe enters a self-regenerating stage. Therefore all the scales observable today left the Hubble radius during the last inflation and we cannot verify the chaotic inflation stage directly because the minimum of Kahler potential during chaotic inflation coincides with the local maximum of the potential for new inflation. By appropriately shifting the local minimum of the new inflaton's potential during chaotic inflation one can construct a model in which duration of new inflation is short enough that the trace of chaotic inflation is observable on the large-scale structure. References 1. M. Kawasaki, M. Yamaguchi, and T. Yanagida, Phys. Rev. Lett. 85, 3572 (2000). 2. M. Yamaguchi and J. Yokoyama, hep-ph/0007021, to appear in Phys. Rev. D. 3. See also references in the above two references.

A N T H R O P I C SELECTION EWAN D. STEWART Department of Physics, KAIST, Taejon 305-701, South Korea E-mail: [email protected] I discuss anthropic selection and related topics. 1

What is the Anthropic Principle?

There are two versions: 1.1

The Strong Anthropic Principle

The Fundamental Theory should be such that it gives rise to life. This is more religion than science. 1.2

The Weak Anthropic Principle

This is just a selection effect. What we observe is biased by the fact that we are not external observers but are part of the universe and live in particular places. For example, we see an oxygen atmosphere, liquid water, etc. although these are known to be rare in the universe as a whole. /Probability of \ I observable 1 = \ being observed J

/Probability of^ observable in the Fundamental \ Theory J

/Probability

of\

I observer I being there y t o observe it J

(1)

One cannot deny the Weak Anthropic Principle. The real question is for which observables is it an important selection effect. 2

The Structure of the Fundamental Theory

Again, there are two versions: 2.1

Unique vacuum

The Fundamental Theory uniquely predicts the observed low energy physics. This would seem to require the Strong Anthropic Principle, though most physicists who favor a unique vacuum vehemently oppose the Strong Anthropic Principle! 2.2

Many vacua

The Fundamental Theory has many vacua and associated low energy laws of physics. The observed low energy physics is selected by a combination of anthropic selection, random chance, and possibly other factors. This is the weak anthropic approach. 147

148 3

Conjectured Structure of String Theory

Many (1010 , oo?) discrete non-supersymmetric vacua (this includes vacua with low energy supersymmetry breaking) and many continuous families of supersymmetric vacua. This fits well with the Weak Anthropic Principle. 4

Structure of the Eternally Inflating Universe

There are many types of inflation that are natural from the particle physics point of view: 1. False vacuum inflation

Figure 1. False vacuum inflation

This inflates eternally. 2. Rolling scalar field inflation

Figure 2. Rolling scalar field inflation

Inflates eternally at the maximum if m2 < 6Vo! in units where 8nG = 1. 3. Thermal inflation Even this inflates eternally if ^Hawking ~ ^critical4-1

Discrete Eternal Inflation

Slow-roll inflation is motivated by observations requiring the spectral index of the density perturbations to be n ~ 1 and is not necessary for, and is probably not relevant for, eternal inflation. Instead, one expects eternal inflation to be dominated by the more generic types of inflation listed above, and so to occur at the discrete points in field space corresponding to false vacua and maxima, with quantum tunneling between these

149

Figure 3. Thermal inflation

points. This will populate all the eternally inflating points in field space, somewhere in the eternally inflating multiverse, allowing all the (connected) vacua and their associated low energy laws of physics to be realized.

Figure 4. Discrete eternal inflation

However, the infinite expansion factors of eternal inflation make it impossible to give a probability to any given final state vacuum. Cosmology doesn't seem to help much with vacuum selection. 5

The Accelerating Universe

Observations of distant supernovae have shown that the expansion of the universe is accelerating. The simplest explanation of this is that the energy density of the universe is dominated by (positive) vacuum energy. Such vacuum energy would have magnitude pA = (2.2 x 1(T 3 eV) 4 = 2 x 1(T 59 TeV4

(2)

This is about 10 60 times smaller than particle physicists can understand. Furthermore, to understand the coincidence that it is just beginning to dominate now seems to require anthropic arguments.

150 Table 1. Current composition of the universe Vacuum energy Cold dark matter Ordinary matter (baryons) Stars Neutrinos Cosmic microwave background radiation Spatial curvature

6

65% ± 10% 30% ± 10% 5% 0.5% 0.3% x ( m „ / 0 . 1 e V ) 0.006% 0% ± 10%

Anthropic Selection Rules

For anthropically selected fine-tuning to be a consistent explanation we require: 1. Enough freedom in the Fundamental Theory. For anthropic selection to select a vacuum, the vacuum must exist in the first place. For example, > 1010 vacua are needed for accidental cancellations to produce a vacuum with a sufficiently small cosmological constant. 2. There should be no better solution. For example, before the discovery of inflation it was legitimate to use anthropically selected fine-tuning to explain the small value of the spatial curvature. However, it is no longer legitimate. Any anthropic selection mechanism would simply choose inflation rather than fine-tuning. In this respect, supersymmetric vacua are a challenge to the anthropically selected fine-tuning explanation of the value of the cosmological constant. Either supersymmetric vacua must be extraordinarily rare, at least 1010 times rarer than non-supersymmetric vacua, or supersymmetric vacua must somehow be incompatible with life. My guess is the latter, perhaps because matter would be unstable to bosonization and subsequent collapse because of the loss of Fermi exclusion. 3. Small changes in the observed value should have a significant effect. For example, current bounds on the spatial curvature disfavor an anthropic explanation for its small value. Small changes in the observed value of the cosmological constant have a significant effect on galaxy formation making an anthropic explanation plausible. 7

Anthropically Selected Fine-Tuned Cosmological Constant versus a Solution to the Cosmological Constant Problem

There are two classes of theories for why the cosmological constant is so small: 1. There is some symmetry or other mechanism which can set the cosmological constant to zero.

151 2. There is no mechanism (compatible with life - see comments above on supersymmetric vacua) which can set the cosmological constant to zero and anthropic selection selects a vacuum, which has a cosmological constant sufficiently small for life due to accidental cancellations, from a large number of vacua. The first predicts the cosmological constant should be zero. If the symmetry is broken (this includes quintessence scenarios) it predicts an a priori roughly speaking logarithmic type probability distribution which, when combined with the anthropic bound, would predict a value essentially no different from zero. Detailed calculations of the second predict a probability distribution for the cosmological constant with expected values of order or a small factor larger than the current matter energy density. Observations have confirmed the latter prediction which is highly unlikely in the former scenario. Note that this observational evidence strongly suggests that it is a waste of time looking for a 'solution' to the cosmological constant problem. It is also important to note that the case of the cosmological constant is different from that of many other parameters that appear to have an anthropically finetuned value. Firstly, it is cleaner, but more importantly the observed value is on the extreme edge of the anthropically allowed region (on a roughly speaking logarithmic type scale). This would be extremely unlikely if a symmetry were at work. However, for most other apparently anthropically fine-tuned parameters the observed value seems to take a typical value within the anthropically allowed region (again on a roughly speaking logarithmic type scale). In these cases it would be quite consistent and even likely for anthropic selection to select a symmetry or other mechanism to obtain an anthropically allowed value rather than merely using brute-force fine-tuning. Thus in these cases it is important to look for (broken) symmetries or other mechanisms to explain the observed value even if the value seems to be anthropically fine-tuned. 8

Outlook String Theory —> Map of the world

(3)

The field space of String Theory is a map of the local laws of physics, in particular of vacua and the associated low energy laws of physics. It is string theorists' job to draw this map. Cosmology + Particle Phenomenology —> You are here

(4)

Cosmological and particle physics observations will determine where we are in the field space of String Theory. Cosmology + Anthropic Principle —> Why we are here

(5)

We will need to use cosmology and the Weak Anthropic Principle to understand why we are here and how we got here.

152 Acknowledgments This work was supported by the Brain Korea 21 Project and Korea Research Foundation Grant (2000-015-DP0080).

A T M O S P H E R I C A N D SOLAR N E U T R I N O MASSES A N D A B E L I A N FLAVOR S Y M M E T R Y KIWOON CHOI Department

of Physics,

Korea Advanced Institute of Science Taejon 305-701, Korea

and

Technology

Recent atmospheric and solar neutrino experiments suggest that neutrinos have small but nonzero masses. They further suggest that mass eigenvalues have certain degree of hierarchical structures, and also some mixing angles are near-maximal while the others are small. We first survey possible explanations for the smallness of neutrino masses. We then discuss some models in which the hierarchical pattern of neutrino masses and mixing angles arises as a consequence of U(l) flavor symmetries which would explain also the hierarchical quark and charged lepton masses.

1

Introduction

Atmospheric and solar neutrino experiments have suggested for a long time that neutrinos oscillate into different flavors, thereby have nonzero masses x. In particular, the recent Super-Kamiokande data strongly indicates that the observed deficit of atmospheric muon neutrinos is due to the near-maximal ^M —*• vr oscillation 2 . Solar neutrino results including those of SuperKamiokande, Homestake, SAGE and GALLEX provide also strong observational basis for ve —>• v^ or vT oscillation 3 . The minimal framework to accomodate the atmospheric and solar neutrino anomalies is to introduce small but nonzero masses of the three known neutrino species. In the basis in which the charged current weak interactions are flavor-diagonal, the relevant piece of low energy effective lagrangian is given by eZMeeR

+ gW-feZlnVL

+ {vLyMvvL

,

(1)

where the 3 x 3 mass matrices Me and M" are not diagonal in general. Diagonalizing Me and M", (Ue)^MeVe

{U")TMVVV

= De = d i a g ( m e , T O M , T O r ) ,

= Dv = diag ( m i , m 2 , m 3 ) ,

(2)

one finds the effective lagrangian written in terms of the mass eigenstates eZDeeR + W-'eZ^UvL 153

+ TPCfDvvL

,

(3)

154

where the MNS lepton mixing matrix is given by U = {Ue)]Uv.

(4)

Upon ignoring CP-violating phases, U can be parametrized as U =

/I 0 \0

0 C23 -S23

0 \ S 23 C23/

/ C13 0 \-Si3

0 1 0

C13C12 -S12C23 S23S12 -

(

512 C12 0

0' 0 1

S13

\

S23S13S12

s23ci3

Sl3\ 0 C13J

Ci2 -aia \ 0

S12C13

S23S13C12 S13C23C12

C23C12 -S23C12 -

S13S12C23

(5)

C23C13/

where Cjj = cosOij and Sij = sinOij. Within this parameterization, the masssquare differences for atmospheric and solar neutrino oscillations can be chosen to be A

™atm = ml - m\,

Ams2ol = m\ - m?,

(6)

while the mixing angles are given by #atm = #23,

#sol=#12,

#rea = #13,

(?)

where 9Tec describes for instance the neutrino oscillation z/M —» ve in reactor experiments. The atmospheric neutrino data sugget near-maximal v^ —>• z/r oscillation 2 with Am2tm~3xlO-3eV2, sin2 20 a t m ~ 1.

(8)

As for the solar neutrino anomaly, four different oscillation scenarios are possible 3 though the large mixing angle (LMA) MSW oscillation is favored by the recent Super-Kamiokande data: SMA MSW : Am 2 ol ~ 5 x 1 0 - 6 eV 2 ,

sin2 26>soi ~ 5 x 10~ 3 ,

LMA MSW : Am 2 ol ~ 2 x 1CT5 eV 2 , sin2 26»soi ~ 0.8, LOW MSW : Am 2 ol ~ 10~ 7 eV 2 , sin2 2<9soi ~ 1, LMA VAC : Am 2 ol ~ 10~ 10 eV 2 , sin2 2<9soi ~ 0.7.

(9)

There is in fact an important constraint from reactor experiments, e.g. CHOOZ 4 , indicating no v^ oscillation into ve, thereby leading to 4C/23 «sin 2 26>i 3 < 0.15

(10)

155 P u t t i n g the atmospheric and solar neutrino d a t a together while taking into account the CHOOZ constraint, one can consider the following three p a t t e r n s of neutrino masses and mixing angles: I. Bi-maximal mixing with LMA MSW solar neutrino oscillation: 777,2/7773 ~ A or A ,

(M, M, M ) ~ (7f>7f>Afc)>

(n)

II. Bi-maximal mixing with LOW MSW or LMA VAC solar neutrino oscillation: 7772/7773 ~ A4 or A5 ,

( M . M . M ) ~ (^, ^ , A * ) ,

(12)

III. Single-maximal mixing with SMA MSW solar neutrino oscillation: 7772/7713 ~ A , ( | « 2 3 | , |«i2|, | S l 3 | ) ~ ( ^ , A 2 , A * ) ,

(13)

where A = s i n # c ~ 0.2 for the Cabbibo angle 8c and 7773 ~ 5 x 1 0 " 2 e V ,

k>

1

(14)

in all cases. These neutrino results can be compared with the following quark and charged lepton masses a n d mixing angles: (mt,mc,mu) (mb,ms,md) {mT,mll,me)

~ 1 8 0 ( 1 , A 4 , A8) 2

4

~ 4(1, A , A ) 2

GeV, GeV,

5

~ 1.8(1, A , A ) 2

GeV,

( s i n ^ 2 3 , s i n 0 1 2 , s i n ^ 1 3 ) ~ (A , A, A 3 ),

(15)

where
156

H

• L

H

'»

MN 4 * N

• N

i

4 L

Figure 1. Small neutrino mass from the exchange of superheavy singlet neutrino

keeping 623 and 912 near maximal. Similarly, for the scenario III, one can ask what would be the flavor structure yielding small 7712/7713 ~ A2 and #12 ~ A2, but near maximal 623- In this talk, we first survey possible explanations for the smallness of neutrino masses, and then discuss some models in which the hierarchical patterns of neutrino masses and mixing angles arise as a consequence U(\) flavor symmetries 5 , e which would explain also the hierarchical quark and charged lepton masses. 2

W h y neutrinos are so light?

Here we discuss four possible mechanisms which would suppress the resulting neutrino mass. These mechanisms are not orthogonal to each other, so one can take more than one mechanism in order to make the neutrino mass small enough. A.

Seesaw-type mechanism

In seesaw-type model, neutrinos are light since their masses are induced by the exchange of superheavy particles. At low energies, the effects of such heavy particles are described by the operator -77LLHH (16) M where L and H are the lepton and Higgs doublets, respectively, and M denotes the mass scale of the exchanged heavy particle. This gives a neutrino mass

™.~^~.x,o-.(i2^v).v

157

H

H * *

* *

T

*

:

L

L

Figure 2. Small neutrino mass from the exchange of superheavy triplet Higgs

which can be as small as the atmospheric neutrino mass for M ~ 1015 GeV. There are two different ways to generate the above d = 5 operator. One is the exchange of superheavy singlet neutrino 7 (Fig. 1) which corresponds to the conventional seesaw mechanism, and the other is the exchange of superheavy triplet Higgs boson 8 (Fig. 2). For the case of singlet neutrino exchange, the underlying lagrangian includes hHLN + MNNN

+ h.c,

(18)

where A'' is a singlet neutrino with huge Major ana mass MJV. Integrating out N then yields the operator (16) with M = M^/K1. For the case of triplet Higgs exchange 8 ' 9 , one starts from hTLL - \-MlTT*

- M'TTH*H* + h.c,

(19)

where T is a Higgs triplet with huge mass MT- Again integrating out T leads to (16) with M = MT/hMT. The seesaw-type mechanism is perhaps the simplest way to get small neutrino mass. However it would be rather difficult to probe other effects of the involved superheavy particles than generating the neutrino mass. B.

Frogatt-Nielsen mechanism

Neutrino mass can be small if the couplings which are responsible for neutrino mass are suppressed by a spontaneously broken (gauge) symmetry by means of the Frogatt-Nielsen mechanism 10 . As an example, consider again a model with singlet neutrino N but now with weak scale M^ ~ 102 GeV. Suppose that the operator HLN carries a nonzero integer charge n

158

of some discrete or continuous (gauge) symmetry G of the model and G is spontanesly broken by the VE V of a standard model singlet <j> which has charge — 1. Then HLN in the bare action is forbiden, however there can be higherdimensional coupling
' HLN = enHLN,

(20)

and so the neutrino mass <2n{H){H) mv

n

2

2

0 GeV\ 5 x 1 0 -2((\3- x^e10-rJ V f \ /(\11-^^) eV. M )

MN

N

(21) y '

By choosing appropriate values of n and e, one can easily accomodate the atmospheric neutrino mass even when the singlet neutrino has an weak scale mass. In supersymmetric models, one can implement the Frogatt-Nielsen mechanism for small neutrino mass without introducing any singlet neutrino. As an example, consider a supersymmetric model with U(l) flavor symmetry whose symmetry breaking order parameter e ~ A (Cabbibo angle). The (7(1) charges of H\H2 and LH2 are assumed to be —1 and —n, respectively, where Hx 2 and L denote the two Higgs doublets and the lepton doublet superfields in the MSSM. Then the supergravity Kahler potential can contain

K=

w*H* + Kfc) LH»

(22)

while the holomorphy and t/(l) do not allow the supergravity superpotential contain a term like 4>mH\HmLHi- After the spontaneous breaking of supersymmetry and also of U(l), this Kahler potential gives rise to the yu-type terms in the effective superpotential: Weff=tiH1H2

+

^LH2,

n

where fi ~ e*m3/2 ~ Am 3 / 2 and / / ~ e* m3/2 ~ Xnm3/2. Here the first term in Weff is just the conventional /x-term and the second corresponds to the bilinear _R-parity violating term. As is well known, the bilinear /^-parity violation leads to the neutrino mass n »'2(H2)(H2) l**M1/2

mv ~ ^ - ^ 9 ^ ~ A 2 ^ " 1 ^ ^ *

(23)

where the gaugino mass Mx/2 and fi are assumed to have the weak scale value Mweak- This neutrino mass can be of order the atmospheric neutrino mass if

159

n = 9, which would be obtained for instance if the U(l) charges of Hi, H2, L are 4, —5, —4, respectively. In fact, supersymmetric models always contain an intrincically small symmetry breaking parameter, i.e. m3/2/Mpianc): describing the size of SUSY breaking. If mv is suppressed by a symmetry G which is broken by the SUSY breaking dynamics, small m,vjMwea^ and m3/-2IMpiancit have a common dynamical origin. They are then related to each other by the Frogatt-Nielsen mechanism of G, e.g ^ ^ ~ e * , Mpianck

7 ^ - ' ,

(24)

Mweak

where e is the symmetry breaking order parameter of G, and k and I are model-dependent integers. If k/l — 4/3 and m.3/2 ~ Mweak, one obtains Tnv/Mweak ~ 10 - 1 2 which is the correct value for the atmospheric neutrino mass. m3/2/Mpianck and rnv/Mweak is always related to each other when G is a discrete /^-symmetry 12 which appears quite often in compactified string theory. It is also possible to relate mv/Mweak with m3/2/'Mpianck by means of other type of symmetry l o . C.

Radiative generation of neutrino mass

Even when the neutrino mass is zero at tree level, if the lepton number symmetry is softly broken, there can be small finite radiative corrections to neutrino mass 14 . The resulting neutrino mass is suppressed by the loop factor as well as the (potentially) small Yukawa couplings which are involved in the loop. The most typical example is the Zee model with £ = }HLEC + f'S+LL

- AS+HH'

+ ...,

where H and H' are 5f/(2)-doublet Higgs fields, 5+ is a charged SU{2)singlet Higgs field, and L and Ec stand for the conventional lepton doublet and anti-lepton singlet. Here we will ignore the flavor indices of couplings for simplicity. It is then easy to see that a nonzero neutrino mass is generated at one-loop (Fig. 3), yielding

m

" ~ 1^S{H){H'}>

(25)

where ms is the mass of S+. It is also possible to construct a model in which mv is generated at higher loop order 14 . A typical example is given by £ = JHLEC + f'S+LL

+ f"S—EcEc

- AS+S+S~-

+ ...,

160

H'

S+ >' •

/

L

r

L

Ec

f

L

H Figure 3. One-loop neutrino mass in Zee-type model

*\s+

s+y 5 * i

*

• %

r L

Ec

f

*

i

>

•

L

l

/"

H

* I

f

<—

L

H

Figure 4. Two-loop neutrino mass in the variant of Zee model

where S+ and S are charged 5f/(2)-singlet Higgs fields. In this model, nonzero m„ appears at two-loop (Fig. 4), yielding

Af2fr2f" (H)(H). 2 2

(26)

(167r ) m|

Note that even when S+ and S have weak scale masses, the resulting neutrino mass can be as small as the atmospheric neutrino mass if the Yukawa

161

couplings/,/',/'~1(T2. D.

Localizing singlet neutrino on the hidden brane

Recently it has been noted by Randall and Sundrum (RS) that the large hierarchy between Mpianck and Mweak can be achieved by localizing the gravity on a hidden brane 15 . In the RS model, the spacetime is given by a slice of d — 5 AdS space with two boundaries. A flat 3-brane with positive tension is sitting on one of these boundaries at y = 0, while a negative tension 3-brane is on the other boundary at y — b. Massless d = 4 graviton mode is localized on the positive tension brane (the hidden brane) while the observable standard model fields are confined in the negative tension brane (the visible brane). Since d = 4 gravity is localized on the hidden brane, matter fields on the visible brane naturally have very weak gravitational coupling, so a large disparity between Mweak and Mpiancf.. Attempts have been made to incorporate small neutrino mass in the RS model 16 . The model contains a bulk fermion \fr and a bulk real scalar $ with the following orbifold boundary condition: * ( - ! / ) = 75*(i/), H-v) = -*(»)> (27) in addition to the bulk gravition and the standard model fields. The action is given by - A B - ^1ADA*

S = j d^xdy^^^M^R^ + / dAxyf^)[-Kv Jy=b

++ I

+

+ K'H^CL

KHVL


- / $ # * - m * c * + ...] + -£-LLHH M*

+ ...}

...],

(28)

Jy=0

where M* is the 5-dimensional Planck mass, R^ is the d = 5 Ricci scalar, tyc is the charge-conjugation of the d = 5 spinor $ , and H and L are the Higgs and lepton doublets confined in the visible brane. Here A„ and A^ denote the visible brane tension and the hidden brane tension, respectively, and we write explicitly also the terms for neutrino mass in the visible brane action. If the bulk and brane cosmological constants satisfy k

=

~AB

=

A h

=

K

(9Q\

l \] 6Ms 6M3 6M.3 ' the model admits the following form of d = 4 Poincare invariant spacetime and the corresponding massless d = 4 graviton mode:

ds1 = e- 2 *l»l(rv + hll„{x))dxlidxv

- dy1.

(0
b).

(30)

162

Obviously /iM„ is localized around y = 0, so its coupling to the energy momentum tensor at y — b is exponentially suppressed by e~2kb. Equivalently, all dimensionful quantities on the visible brane are rescaled by e~2kb. This results in the standard model mass parameter M2ieak ~ e~2kbM2, while the d = 4 Planck mass is given by Mplanck = M^(l-e~kb)/k, so an exponentially small ratio Mweak ~M Mplanck

-kb e

/on

•

(31)

The small ratio mv/Mweak can be similarly obtained in the model of (28) by localizing the zero mode of \P on the hidden brane. To implement this mechanism, we need first the lepton number violating couplings (both in bulk and on branes) to be suppressed enough, for instance h, K' and m/M, should be less than 10~ 12 in order for mu < 0.1 eV. This can be easily achieved by imposing a discrete symmetry under which * -> e2ni/N*S>,

L -> e27ri/NL,

(32)

which would result in h = K' = m = 0.

(33)

Then the Dirac equation for the zero mode of VP is given by d g

2/c + i 7 5 / ( $ ) J * 0 = 0 ,

(34)

leading to the following solution g-3%1/2^

=

e-(2/(*>-*)|»|/2^(a:)

( 3 5 )

where r] denotes the canonically normalized (in d = 4 sense) singlet neutrino mode. On the parameter region with k < 2 / ( $ ) , this mode is localized on the hidden brane. As a result, rj has an exponentially small Yukawa coupling with H and L on the visible brane, so an exponentially small Dirac neutrino mass. After the proper rescaling of the involved fields, one finds - ^ -

~ «e-(2/<*>-*>»/2

(36)

M-weak

which can be small as 10~ 12 to provide the atmospheric neutrino mass. Note that the small neutrino mass obtained by localizing singlet neutrino on the hidden brane is a Dirac mass, however the current neutrino oscillation experiments do not distinguish the Dirac mass from the Majorana mass.

163

3

Models with abelian flavor symmetry

Here we discuss some models in which the hierarchical patterns of the atmospheric and solar neutrino masses and mixing angles are obtained by means of U(l) flavor symmetries. Our discussion will be limited to a specific example for bi-maximal mixing with LMA MSW solar neutrino oscillation and another example for near-maximal atmospheric neutrino oscillation and SMA MSW solar neutrino oscillation. A. A model for bi-maximal mixing The neutrino masses and mixing angles for bi-maximal mixing with LMA MSW solar neutrino oscillation are given by 7712/7713 ~ A o r A 2 ,

(\s23\,]Sl2l\s13\)~(-j=,-j=,\k)

(*>1).

(37)

One issue for this pattern of neutrino masses and mixing angles is how could one obtain small #13 and m-ijm-i while keeping #23 and #12 near maximal. Comparing the MNS mixing matrix U = U^U" with the parametrization (5), one easily finds that U automatically has a small #13 with bi-maximal #23 and 0i2 if Ue has only one large mixing by #23 and also U" has only one large mixing by #12 • A form of charged lepton mass matrix which would lead to such Ue is Me ~mrl

1

(38)

where a C 1. For the neutrino mass matrix, we can consider two different forms leading to such V. One is of pseudo-Dirac type:

(

61 a

a b2

c\ d\

(39)

c d 1/ with bi
a2

a3

d

,

(40)

where all a, are of the same order and c, d
164

Here we assume that Mv is induced by the conventional seesaw mechanism in supersymmetric model and explore the possibility that the above mass matrix textures are obtained as a consequence of U{\) flavor symmetries. It is not so trivial to obtain the mass matrix textures (38), (39), (40) from (7(1) flavor symmetries since Mv needs to have same order of magnitudes for the 1st and 2nd generations while Me needs different ones. This difficulty becomes more severe if we want to obtain a smaller value of #13. Among the models of U{\) flavor symmetries, the simplest one would be the case of single anomalous U(l) whose breaking is described by a single order parameter e = (<j>)/Mt ~ A. Unfortunately, it turns out that the desired textures can not be obtained in this simplest case. The next simple model would be the case of single non-anomalous U(l) which has two symmetry breaking parameters with opposite £7(1) charges:

_ (4>+)

?=

(-)

M,' M» ' where <j>± has the £7(1) charge ± 1 . Note that if the scale of £7(1) breaking is much higher than the scale of supersymmetry breaking, vanishing £7(1) Z?-term assures |e| = |e|. One can also consider the case of two £7(1)'s in which one £7(1) is anomalous while the other is non-anomalous. One plausible symmetry breaking pattern in this case is that e and e have the £7(1) x £7(1) charges (—1,-1) and (0,1), respectively. In this case, vanishing D-term of anomalous £7(1) leads to |e| ~ A, while that of non-anomalous £7(1) gives |e| = |e|. In the below, we will present a simple example for each case which gives rise to the mass matrix textures for (37). Let us first consider the case of single £7(1) with two symmetry breaking parameters e and e. We will assume that |e| = |e| ~ A. The light neutrino mass matrix which is obtained by the seesaw mechanism is given by M" = MD(MMy1(MD)T, D

(41) M

where M is the 3 x 3 Dirac mass matrix and M is the 3 x 3 Majorana mass matrix of superheavy singlet neutrinos. Let small letters denote the £7(1) charges of the capital lettered superfields, e.g. U for the lepton doublets Li, ei for the anti-lepton singlets E\, m for the superheavy singlet neutrinos Ni. Then for the charge assignments of fj = ( 2 , - 2 , 0 ) ,

n,- = ( 2 , - 2 , 0 ) ,

one finds the mass matrices /A Mv=m3\A

\X2

4

A A4 A2

17

e* = (1,5,5),

h = h2 = 0,

(42)

which are are given by A2\ A2 , 1/

/A7 Me = (H1) A3 \AS

A7 A3 A5

A3\ A Xj

(43)

165 where A is of order one, but does not exceed 1. This mass matrix provides the near bi-maximal #23 and #12, and also 013~A2,

Am2tm~(l-.42)m2,

Ams2ol ~ A 4 .4 2 m 2 ,

(44)

which can accomodate all experimental data for reasonable values of A and m3. The desired forms of mass matrices can be obtained for the case of U(l) x f/(l) also. If one U(l) is anomalous while the other is non-anomalous, which is the case that appears quite often in compactified string theory, it is quite plausible that U(l) x U(l) are broken by the two symmetry breaking parameters e and e with the U(l) x U(l) charges (—1, —1) and (0,1). In this case, |e| = |e| and they are naturally of order the Cabbibo angle A. Then for the charge assignment m=(0,-l),

n 2 = (0,l),

Ji = ( 0 , - 1 ) ,

^2 = (1,2),

n 3 = (0,0), f3=(0,0),

one easily finds M" ~ m 3

/A2 A VA

A A\ 0 0

(45)

0 1)

which yields 913 ~ A,

Am52ol ~ A 2 Am 2 tm -

(46)

B. A model for large atmospheric and small solar neutrino mixings The neutrino masses and mixing angles for large atmospheric and small solar neutrino mixings are given by m2/m3

~ A2,

( M . M . M ) ~(^,A2,A*)

(*>1).

(47)

The issues for this pattern would be how could one obtain small m2/m 3 even when #23 is near maximal, and also what would be the reason for small #12 and #13. Here we present a supersymmetric model with U(l) flavor symmetry in which such pattern of neutrino masses and mixing angles arises naturally. The model under consideration is the MSSM with i?-parity breaking couplings which are suppressed by an anomalous U(l) flavor symmetry with e ~ A 18 . The most general SU(3)C x SU(2)L X U(1)Y-invariant superpotential of

166

the MSSM superfields includes the following lepton number (L) and /^-parity violating terms: AW = XijkLtLjEi

+ \'ijkLiQjD%

(48)

where (Li,E£) and (Qt, [/?, Df) denote the lepton and the quark superfields, respectively. Another L and ft-parity violating term fi^LiH-2 in the superpotential can always be rotated away by a unitary rotation of superfields. Soft SUSY breaking terms also contain the L and .R-parity violating terms: AV80ft = mliHl

LtHl + BiLiH.2 + C^LiLjE^

+ C'^LiQjDi,

(49)

where now all field variables denote the scalar components of the corresponding superfields. In the basis in which /XjLj//2 in the superpotential are rotated away, non-vanishing Bi and m2LH result in the tree-level neutrino mass n

wv)T

* g °M )(i>;) '

(50)

1/2

where Ml/% denote the SU(2)i x £/(l)y gaugino masses and the sneutrino VEV's are given by ,~n 2Mz(m2L.HicosP + BiSin0) {Vi) { * m? + ! M i c o s 2 ^ ' ' where tan/? = (#2)/'(Hi), Mz is the Z-boson mass, and m;~ is the slepton soft mass which is assumed to be (approximately) flavor-independent. There are also additional neutrino masses arising from various one-loop graphs involving the squark or slepton exchange 19 . Let the small letters qi,Ui, e.t.c. denote the 1/(1) charges of the superfields Qi,Uf, e.t.c. If all L and /^-parity violating couplings are suppressed by some powers of A as is determined by the U(l) charges of the corresponding operators, the resulting neutrino mass matrix takes the form: A 2 , 1 Mn A'»+' a M 1 2 \ A^Mis /

v

(M )ij

= m3

A' l s + , 2 Mi2 \2l**A22 A'23^

\ll3Al3\ A''M 2 3 , ^33 /

(52)

whereTO3is the largest mass eigenvalue, Uz = h — h, and all Ay are of order unity. It is then straightforward to see that the near maximal #23 requires l2 = l3j while the small 0 12 ~ A2 requires h = h + 2. This eventually leads to the MNS mixing matrix of the form: / 1 A2 A2' U ~ A2 1 1 I . I A2 1 1

(53)

167

which gives #13 ~ A2. So far, we could get the mixing angle pattern #23 ~ 1, #12 ~ #13 ~ ^ 2 J u s t by assuming the U(\) charge relations: h — 2 = l-i = I3. Still we need to get the mass hierarchies 7712/7713 ~ 4 x 10~2 and also m3/Mweak ~ 10~ 12 . In the model under consideration, U(l) flavor symmetry assures that ^-parity violating couplings are all suppressed by A' i+/l2 or \li~hl compared to their i?-parity conserving counterparts. As a result, 7773 from ft-parity violation obeys roughly m3/Mweak

~ xW'+k')

or

A2^-^.

So if the U(l) charges are arranged to have h +h,2 =h - hi =7

or 8,

the resulting 777.3 naturally fits into the atmospheric neutrino mass scale. One may wonder how 7772/7773 can be as small as 4 x 1 0 - 2 even when the 2nd and 3rd neutrinos mix maximally. The neutrino mass matrix (52) from .R-parity violation automatically realizes such unusual scenario since it is dominated by the tree mass (50) which is a rank one matrix. Then the largest mass rn.3 is from the tree contribution while 7772 is from the loops, so 7712/7773 ~ LOOP/TREE independently of the value of #23- In fact, we need to make this loop to tree ratio a bit bigger than the generic value in order to get the correct mass ratio 777,2/7773 « 4 x 1 0 - 2 . This is difficult to be achieved within the framework of high scale SUSY breaking, e.g. gravity-mediated SUSY breaking models, while it can be easily done in gauge-mediated SUSY breaking models with relatively low messenger scale. In gauge-mediated SUSY breaking models 2 0 , Bi and ni\.H can be simultaneously rotated away as n't at the messenger scale Mm, i.e. Bi(Mm) = 777'i iHl (M m ) = 0 in the basis of \Ji = 0, and their low energy values at Mweak are determined by the RG evolution. Then the tree mass (50) which is determined by these RG-induced Bt and m\.H can be made smaller by taking a lower value of the messenger scale. Soft parameters in gauge-mediated models 20 typically satisfy: Ma/aa w 777^/0:3 ^ 777^/0:^2 at Mm where Mn, rriq, and mj denote the gaugino, squark and slepton masses, respectively, and aa = dtl^ for the standard model gauge coupling constants. The size of the bilinear term BH1H2 in the scalar potential depends upon how fi is generated. An attractive possibility is B(Mm) = 0 for which all CP-violating phases in soft parameters at

168

Mweak are automatically small enough to avoid a too large electric dipole moment 21 . In this case, the RG-induced low energy value of B yields a large tan/3 w (m2Hl + m'2H2 + 2n'2)/B{Mz) = 40 ~ 60. Analyzing the neutrino masses from i?-parity violating couplings which are determined by the RG evolution with the boundary conditions that trilinear soft scalar couplings, B, Bt and rn2LH are all vanishing at Mm, and also Ma/aa ~ rriq/az « mj/ai^ at Mm, one finds 18 ff\tree - , , n - 1 . 41, ..

{M")^

/ M "^Z

* lO" ^; \^f\

,

(54)

where at = J/bA^33 ~ \li~hi for the 6-quark Yukawa coupling yi and t = ln(M m /mj)/ln(10 3 ). The loop mass is given by 18 ( M " ) ^ * 10- 2 i 2 y^?^3 3 (feA i 33 + ^ 3 A i3 3) ( ^ ^

J .

(55)

where yT is the r-lepton Yukawa coupling and the smaller contributions are ignored. These tree and loop masses then give the following mass hierarchies: mz/Mweak m2/m3

« [Ui3{M")t^UjZ}/Mweak

~ 10- 1 A 2 (' 3 -' l l >,

« [^2(M*,)J5opl/;,-2]/m3 ~ (LOOP/TREE),

m i / m j « [t/„ (M^U^/m? 2

~ A4 ,

(56) 3

2

where (LOOP/TREE) = 1 0 - ( A 2 3 3 / A 2 3 3 ) ( l n l 0 / l n ^ ) , and we have used tan/3 ~ 50, m;- « 300GeV and \x « 2m;- which has been suggested to be the best parameter range for correct electroweak symmetry breaking 21 . To summarize, in this model, small m^lvriz is due to the loop to tree mass ratio, while the other small mass ratios m i / m 2 and mz/Mweak are from U(l) flavor symmetry. Acknowledgments: I thank E. J. Chun and K. Hwang for useful discussions, and also Y. Kim for drawing the figures. This work is supported by the BK21 project of the Ministry of Education, KRF Grant No. 2000-015DP0080, KOSEF Grant No. 2000-1-11100-001-1, and KOSEF through the CHEP of KNU. References 1. For a recent global analysis, see M. C. Gonzalez-Garcia et al, hepph/0009350.

169

2. Y. Fukuda et al, Phys. Rev. Lett. 82 2644 (1999); Phys. Lett. B467, 185 (1999). 3. J. Bahcall, P. Krastev and A. Smirnov, hep-ph/0006078. 4. CHOOZ Collaboration, Phys. Lett. B420, 397 (1998); Phys. Lett. B466, 415 (1999). 5. M. Leurer, Y. Nir and N. Seiberg, Nucl. Phys. B398, 319 (1993); ibid, B420, 468 (1994); L. Ib'anez and G. G. Ross, Phys. Lett. B332, 100 (1993); P. Binetruy and P. Ramond, Phys. Lett. B350, 49 (1995); V. Jain and R. Shrock, Phys. Lett. B352, 83 (1995); E. Dudas, S. Pokorski and C. A. Savoy, Phys. Lett. B356, 45 (1995); P. Binetruy, S. Lavignac and P. Ramond, Nucl. Phys. B477, 353 (1996). E. J. Chun and A. Lukas, Phys. Lett. B 387, 99 (1996); K. Choi, E. J. Chun, and H. Kim, Phys. Lett. B 394, 89 (1997); Z. Berezhiani and Z. Tavartkiladze, Phys. Lett. B396, 150 (1997). 6. N. Irges, S. Lavignac and P. Ramond, Phys. Rev. D58, 035003 (1998); P. Binetruy, E. Dudas, S. Lavignac and S. A. Savoy, Phys. Lett. B422, 171 (1998); J. Ellis, S. Lola and G. G. Ross, Nucl. Phys. B526, 115 (1998); C. D. Frogatt, M. Gibson and H. B. Nielson, hep-ph/9811265; Y. Nir and Y. Shadmi, JHEP 9905, 023 (1999); S. F. King, Nucl. Phys. B562, 57 (1999); Nucl. Phys. B576, 85 (2000); J. Feng, Y. Nir and Y. Shadmi, Phys. Rev. D61, 113005 (2000); M. Berger and K. Siyeon, hep-ph/0010245; Q. Shafi and Z. Tavartkiladze, hep-ph/9904249; hepph/9905202;hep-ph/0002150; 7. M. Gell-Mann, P. Ramond and R. Slansky, in Supergravity (ed. P. van Nieuwenhuizen and D. Freedman, North-Holland, Amsterdam, 1979) p. 315; T. Yanagida, in Proceedings of the Workshop on the Unified Theory and the Baryon Number in the Universe (ed. O. Sawada and A. Sugamoto, KEK Report No. 79-18, Tsukuba, Japan, 1979). 8. E. Ma and Utpal Sarkar, Phys. Rev. Lett. 80, 5716 (1998); E. Ma, Phys. Rev. Lett. 81, 1171 (1998). 9. K. Choi and A. Santamaria, Phys. Lett. B267, 504 (1991). 10. C. D. Frogatt and H. B. Nielsen, Nucl. Phys. B147, 277 (1979). 11. L. Hall and Suzuki, Nucl. Phys. B231, 419 (1984); I.-H. Lee, Phys. Lett. 138B, 121 (1984). 12. K. Choi, E. J. Chun and H. Kim, Phys. Rev. D55, 7010 (1997). 13. N. Arkani-Hamed et al., hep-ph/0006312; F. Borzumati and Y. Nomura, hep-ph/0007018. 14. A. Zee, Phys. lett. B93, 389 (1980); Phys. Lett. B161, 141 (1985); K. S. Babu, Phys. Lett. B203, 132 (1988); Y. Okamoto and M. Yasue, Phys. Lett. B466, 267 (1999); D. Chang and A. Zee, Phys. Rev. D61,

170

15. 16. 17. 18. 19.

20. 21.

071303 (1999); T. Kitabayashi and M. Yasue, Phys. Lett. B490, 236 (2000). L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999); Phys. Rev. Lett. 83, 4690 (1999). Y. Grossman and M. Neubert, Phys. Lett. B474, 361 (2000). Y. Nir and Y. Shadmi, JHEP 9905, 023 (1999). K. Choi, E. J. Chun and K. Hwang, Phys. Rev. D60, 031301 (1999). For recent works, see E. J. Chun and S. K. Kang, Phys. Rev. D61, 075012 (2000); S. Davidson and M. Losada, JHEP 0005, 021 (2000); O. C. W. Kong , hep-ph/0102113. For a review, see G. F. Giudice and R. Rattazzi, Phys. Rep. 322, 419 (1999). R. Rattazzi and U. Sarid, Nucl.Phys. B501, 297 (1997).

C U R R E N T E X P E R I M E N T A L STATUS OF N E U T R I N O OSCILLATION K. N A K A M U R A High Energy

Accelerator

Research Organization (KEK), Oho, Tsukuba, 305-0801, Japan E-mail: [email protected]

Ibaraki

Compelling evidence for neutrino oscillation was obtained from the SuperKamiokande's observation of atmospheric neutrinos with Am'2 ~ 1 0 - 3 eV 2 . There are two other indications of neutrino oscillation with different A m 2 from solar neutrino observations and the LSND experiment. On the other hand, an important bound was obtained by the CHOOZ reactor neutrino oscillation experiment. The results from these experiments as well as some other neutrino oscillation experiments are reviewed. Particular emphasis is placed on the recent results from Super-Kamiokande, which were updated in June, 2000. In addition, the status of the K2K experiment and the prospects of forthcoming neutrino oscillation experiments are reviewed.

1

Introduction

Neutrino oscillations provide the most sensitive means to explore the finite neutrino mass as well as the mixing in the neutrino sector. At present, there are three evidences of neutrino oscillations with different Am 2 : (i) Am 2 ~ 3 x 10~ 3 eV2 from atmospheric neutrino observations, (ii) Am 2 ~ 10~ 5 eV2 or less from solar neutrino observations, and (iii) Am 2 ~ 1 eV2 from the LSND experiment. Recent neutrino oscillation experiments and planned forthcoming experiments mainly focus on further investigation of these evidences. A simple observation is that these three different Am 2 cannot be accommodated with only three neutrino species. Therefore, if all the three evidences are true, at least one sterile neutrino species, vs, must exist because of the LEP constraint on the number of light, active neutrino species. Alternatively, one of the three evidences may not survive. Among them, the evidence from atmospheric neutrino observations seems compelling. Most of the results were obtained by Super-Kamiokande (hereafter abbreviated as SuperK), 1 , 2 , s with supporting evidence from Kamiokande, 4 1MB, 5 MACRO, 6 ' 7 and Soudan 2. 8 - 9 ' 10 The observed mixing is nearly maximal. The SuperK data favors v^ —>• vT oscillation and disfavors v^ —> vs oscillation as shown in Sect. 2. n Solar neutrinos have been observed by five experiments, Homestake chlorine experiment (^ e 3 7 Cl -> e" 3 7 Ar), 12 SAGE 13 and GALLEX 14 gallium 171

172

10

0.1 0.01 10 "-1

«S-io- 4

>

g 10-5

<

10~ KamLAND day-night asymmetry in IBs

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experiments (^ e 7 1 Ga ->• e _ 7 1 G e ) , and Kamiokande 15 and SuperK 16 z/e scattering experiments. The solar neutrino fluxes measured by all these experiments are significantly lower than the standard solar model (SSM) predic-

173

tions. 1 T Astrophysics alone cannot explain these results. In terms of neutrino oscillations, a global analysis of the data obtained by all the experiments show various allowed regions in the two-flavor (Am 2 , sin22#) plane. Different regions are called the MSW Small Mixing (Am 2 ~ 5 x 10~ 6 eV2 and sin220 ~ 3 x 1(T 3 ), MSW Large Mixing(Am 2 ~ 5 x 10" 5 eV2 and sin226> ~ 1), MSW Low (Am 2 ~ 10~ 7 eV2 and sin226> ~ 1), and Vacuum (or Just-So) (Am 2 ~ 10" 1 0 eV2 and sin22<9 ~ 1) solutions. The LSND experiment at LAMPF reported evidence of neutrino oscillations for both Vp -> i>e 18 a n d i/M -> ve. 19 The allowed (or "favored") (Am 2 , sin2 26) regions are consistent for the two CP-conjugate oscillation channels. However, the Karmen 2 experiment 20 at ISIS, Rutherford Appleton Laboratory, found no evidence though it has similar, but slightly less, sensitivity to LSND. Figure 1 neatly compiles the two-flavor oscillation results from important experiments as well as the sensitivities of some of the forthcoming experiments. 21 In this review, * the atmospheric neutrino results are presented in Sect. 2, and solar neutrino results in Sect. 3. Results from reactor and accelerator neutrino oscillation experiments are briefly reviewed in Sect. 4 and Sect. 5, respectively. The prospects of forthcoming experiments are discussed in Sect. 6, and conclusions are given in Sect. 7. 2

Atmospheric Neutrino Results

The most recent results from SuperK atmospheric neutrino observations correspond to an exposure of 70.4 kton-year or 1144 live days. Fully contained (FC) events are classified into sub-GeV events with Evls < 1.33 GeV and multi-GeV events with £JV;S > 1.33 GeV. Partially contained (PC) events are classified as multi-GeV events, though the minimum track length of about 2.5 m corresponds to muons with > 700 MeV/c momentum. To study the atmospheric neutrino flavor ratio (u^ + i>/J)/(i/e + P e ), the double ratio R = (/i/e)Data/(/Ve)MC is measured, where (fi/e) denotes the ratio of the numbers of fi-like to e-like neutrino interactions observed in the data or predicted by Monte Carlo (MC). The ratio of the data to MC is taken to cancel uncertainties in the neutrino flux and cross sections. The expected value for R is unity if there is agreement between the experiment and the theoretical prediction. *This review is an updated version of the one presented at CIPANP2000. mainly the SuperK results and the K2K status.

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Figure 2 shows the zenith-angle dependence of sub-GeV e-like, sub-GeV /i-like, multi-GeV e-like, and multi-GeV /z-like (FC+PC) events. The dashed histograms show the Monte Carlo prediction for the hypothesis of no oscillation. These data indicate that the /i-like events show a strong deviation from expectation in the shape of the zenith-angle distribution. In particular, multi-GeV ^t-like events show strong up/down asymmetry in contrast to the calculated up/down ratio of near unity. On the other hand, the zenith-angle distribution of e-like events is consistent with the prediction. These results update the published SuperK results. 2

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The fluxes of high-energy atmospheric z/Ms can be measured with underground detectors through the detection of muons produced by the chargedcurrent interactions of v^s in the rock surrounding the detector. The SuperK Collaboration measured both the data of upward through-going muons and

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upward stopping muons that stop in the detector. Typical neutrino energies corresponding to upward through-going and upward stopping muon events in SuperK are 100 GeV and 10 GeV, respectively. Figures 3 (a) and (b) show the SuperK results for the zenith-angle distributions of upward through-going and stopping muons. The observed zenithangle distribution for the upward through-going muons is steeper than the prediction with no neutrino oscillations. Also, the observed flux of the upward stopping muons is significantly smaller than the prediction with no neutrino oscillations. These results update the published SuperK results. 2 ' 3 The upward-going muons were also measured by the MACRO experiment 6 ' 7 at Gran Sasso. Figure 3(c) shows the zenith-angle distribution for the upward through-going muons observed by MACRO. 7 An oscillation analysis of the SuperK atmospheric neutrino data was made by including the upward through-going and stopping muons in addition to the FC and PC events. The obtained 68%, 90%, and 99% CL allowed regions for the hypothesis of v^ —> vr are shown in Fig. 4. In this figure, the 90% CL

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allowed regions reported by Kamiokande, 24 Soudan 2, 10 and MACRO 7 are also shown. The best fit parameters from SuperK are sin2 26 = 1 and Am 2 = 3.2 xlCT 3 eV 2 . The solid histograms in Figs. 2, 3(a), and 3(b) are calculated with these parameter values. The 90% CL allowed intervals for

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the oscillation parameters obtained from the SuperK combined analysis are 1.5 x 10~3 eV2 < Am 2 < 5 x 1(T 3 eV2 and sin2 29 > 0.88.

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There is an alternative scenario that can explain the SuperK FC atmospheric neutrino events in terms of neutrino oscillations. As far as the FC events are concerned, the hypothesis of v^ —» vs gives an equally good fit to the zenith-angle distribution as the standard hypothesis of z/M —¥ vT. These two hypotheses may be discriminated by using the fact that for v ii —* vs (i) fewer neutral-current events should be observed than i/M —> vT because vs does not interact with matter, and (ii) matter effects suppress the oscillation probability 25 while there is no matter effects for v^ —>• vT. The difference of the oscillation probability is appreciable only for > 15 GeV neutrinos traveling through the Earth.

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First, neutral-current enriched sample of events were selected from multiring FC events with Evis > 400 MeV and the most energetic ring being e-like. A Monte Carlo study indicates that the fraction of neutral-current events in this sample is 29% for no oscillations. Figure 5(a) shows the zenith-angle distribution of these events with Monte Carlo predictions. Next, matter effects were studied using PC events with Evis > 5 GeV (for these events, the typical energy of the parent atmospheric neutrino is 20 GeV) and upward through-going muons (the typical energy of the parent neutrino is 100 GeV). Figures 5(b) and (c) show the zenith-angle distributions for the PC events and the upward through-going muons with Monte Carlo predictions. A combined statistical analysis was performed using the SuperK neutralcurrent enriched FC events, high-energy PC events, and upward throughgoing muons, and the two hypotheses, v^ —>• vT and v^ —>• vs were tested. For neutral-current enriched FC events and high-energy PC events, an up/down ratio, up(—1 < cos# < -0.4)/down(0.4 < cos# < 1), is used as a discriminant to cancel some systematic errors. For upward through-going muons, a vertical/horizontal ratio, vertical(—1 < cos# < — 0.4)/horizontal(—0.4 < cos# < 0), is used. Note that the matter effects depend on the sign of Am 2 , so that both positive and negative Am 2 values were tested for v^ —• vs. Figure 6 shows the excluded regions for each hypothesis. In this figure, the parameter regions allowed by the analysis of the SuperK FC data are also shown. For "fi —> "s, the parameter regions allowed for the FC data are excluded at the 99% CL. Therefore, the SuperK results disfavor the hypothesis of v^ —> vs for the atmospheric muon neutrino oscillation. n 3

Solar neutrino results

The latest results of the solar neutrino flux measurements are listed and compared with the prediction of the Bahcall-Pinsonneault (BP98) SSM calculations 26 in Table 1. The SuperK result in Table 1 is obtained from 1117 live days of solar neutrino observation, and updates the previous SuperK result from 300 live days of observation. 16 Compared to the SSM (standard solar model) prediction of BP98, 26 Data/SSM(BP98) = 0.465 ± 0.005j^lnlThe SuperK solar neutrino observations produce not only the average flux but also other interesting data. Among them, the day-night flux difference and the recoil electron energy spectrum have important bearing on the discrimination of various possibilities to solve the solar neutrino problems. The day-night flux difference, if any, would be caused by the regeneration of ves by the Earth. Should the MSW Large-Mixing solution be correct, an observable day-night flux difference would be expected. However,

180 Table 1. Results of the solar-neutrino flux measurements compared with the BP98 2 6 SSM calculation. The Homestake, GALLEX, and SAGE results and the corresponding calculations are given in terms of neutrino capture rates in units of SNU. The Kamiokande and SuperK results and the corresponding calculated results are given in terms of the 8 B solar-neutrino flux in units of 1 0 6 c m _ 2 s - 1 . Experiment

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the SuperK results on the day and night fluxes are (2.35 ± 0.04t°;o7) x ! ° 6 cm" 2 sec" 1 and (2.43 ± Q.Oltom) x 106 c m - 2 sec" 1 , respectively, and (Night-Day)/i(Night+Day) = 0.034 ± 0.022±g;g}|. The nighttime data is subdivided into 5 bins according to the zenith angle 6Z of the sun; Nl (0 < cos0z < 0.2), N2 (0.2 < cosl9z < 0.4), ... , N5 (0.8 < cos9z < 1.0). The flux for each subdivision is shown in Fig. 7, where fluxes predicted for ve —\ V^,{VT) oscillation with some (sin22f9, Am 2 ) values are also shown for comparison. The SuperK recoil-electron energy spectrum normalized to the prediction of BP98 SSM calculation 26 is shown in Fig. 8 and compared with predictions for ve —>• vti{vT') oscillation with some (sin22f?, Am 2 ) values. Although the

182

Figure 9. T h e light shaded areas are the excluded regions at 95% CL for v& —> v^v-r) (left) and Vf,, —»• u, (right) from the day and night spectra analysis. The dark shaded regions show allowed regions at 95% CL from a global fit to the fluxes measured by the gallium (GALLEX and SAGE), chlorine (Homestake) and SuperK experiments.

data points at high-energy end are slightly higher than the average, the energy spectrum distortion is statistically not significant. Using the recoil-electron spectrum distortion or the day-night flux difference, an oscillation analysis can be done independently from uncertainties in the standard solar model (SSM) flux predictions. The absence of significant spectrum distortion and zenith angle variations in the SuperK data constrains neutrino mixing and Am 2 . Actually, the recoil-electron spectrum for each of Day, Nl, ... , N5 zenith angle (see Fig. 7) are independently taken into account in the x2 calculation for the oscillation analysis. Two neutrino oscillation scenarios, ue -> Ufi(uT) and v^ -> i/s, are tested. The left panel of Fig. 9 shows the excluded regions from this analysis for ve —> v^{uT), overlaid with the allowed

183

regions from a global fit to the solar neutrino fluxes measured by the gallium (GALLEX and SAGE), chlorine (Homestake) and SuperK experiments, assuming the SSM fluxes. 26 It can be seen that the Small Mixing and Vacuum solutions are disfavored. The right panel of Fig. 9 shows the excluded regions for z/e —>• vs. All possible solutions from the global fit are disfavored in this case. 4

Reactor Experiments

Two reactor neutrino oscillation experiments, CHOOZ 27 and Palo Verde 28 investigated the Am 2 range down to 10~ 3 eV2 for ue disappearance, ve ->• i>x, using liquid scintillator detectors. The CHOOZ detector is located in a 300-m water equivalent (mwe) underground laboratory at a distance of about 1 km from the Chooz power station in the Ardennes region of France. The Palo Verde detector is lacated at a distance of 750 - 890 m from three reactors at the Palo Verde Nuclear Generating Station near Phoenix, Arizona, but it has only 32 mwe overburden. Consequently, the Palo Verde experiment suffered from a large background rate, though the detector is subdivided. The 90% CL excluded regions from these experiments are shown in Fig. 1. These results excludes the possibility that the atmospheric neutrino anomaly observed by Kamiokande 4 is due to v^ —>• ue oscillations, thus in concordance with the SuperK result. In the context of three-flavor neutrino oscillations, the CHOOZ result gives a bound to the angle #13, sin 2 0 13 < 5 x 1 0 - 2 . 5 5.1

Accelerator Experiments LSND and Karmen 2

The LSND experiment searched for the appearance signal of the v^ —\ ve oscillation. The 90% "favored" region from this experiment 18 is shown in Fig. 1. In addition, LSND found an evidence for v^ —• ve oscillations. 19 The oscillation probabilities are (3.3 ± 0.9 ± 0.5) x 10~ 3 for z>M —> ve and (2.6 ± 1.0 ± 0.5) x 10~ 3 for v^ —>• ve. The favored regions and oscillation probabilities for these CP-conjugate oscillation channels are consistent. For the i>M —> ve oscillation, the Karmen 2 experiment has a sensitivity that nearly, but not entirely, covers the LSND favored region. However, Karmen 2 found no evidence, and the result of oscillation search 20 is shown in Fig. 1. At 90% CL, the region 0.2 < Am 2 < 1 eV2 of the LSND favored region is not excluded yet.

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5.2

CHORUS and NOMAD

CHORUS 29 and NOMAD 30 experiments at CERN searched for J/M -> vr oscillations by looking for the appearance of T~ , in the context of massive neutrinos which might be a component of mixed dark matter. Their sensitivity goals are similar, sin2 20 ~ 2 x 10~4 for large Am 2 . At present, oscillations are excluded for sin22# > 10~ 3 at 90% CL in the large Am2 limit (see Fig. 1). Also, Am 2 > 1 eV2 is excluded at sin228 = 1 (maximal mixing).

5.3

Status of K2K

The K2K (KEK-to-Kamioka) long baseline neutrino oscillation experiment aims at exploring neutrino oscillations fM —> vx in the Am 2 range of 10~ 2 ~ 1 0 - 3 eV 2 , suggested by the atmospheric neutrino observations. vx may be ve, z/T, or vs, though the SuperK results strongly favor vT. Muon neutrino beams are produced by the KEK 12-GeV Proton Synchrotron, and are shot to the SuperK located at a distance of 250 km from KEK. With a horn-focused neutrino beam with an average energy of 1.3 GeV, v^ -> ve is studied in the appearance mode, while v^ —> vT is studied in the disappearance mode. The goal of the experiment is to observe a few hundred charged-current events (in the case of no oscillations) in the 22.5 kton fiducial volume of SuperK with 1020 protons on the production target. The K2K started data taking in June, 1999, and the preliminary results were oabtained from about 100 days of data taken from June 1999 to June 2000. The total number of protons brought to the target amounted to 2.3 x 1019 for the runs used in the analysis. In SuperK, FC events which were correlated in time with 1.1 /xsec neutrino beam pulses from KEK were searched for, and 27 FC events having the vertex within the 22.5 kton SuperK fiducial volume were observed. On the other hand, the expected number of SuperK FC events with the vertex within the 22.5 kton fiducial volume for the case of no neutrino oscillation was 4 0 . 3 t ^ . This number was estimated from the neutrino flux measured by the near detector and the near/far flux ratio deduced from the measurement of pion kinematics just after the magnetic horn. Based on these numbers, the preliminary K2K data disfavor no neutrino oscillations at the 2a level. It will take about 5 years to accumulate 1020 protons on target. When this goal is achieved, the K2K experiment is expected to be sensitive to Am 2 > 2 x 1CT3 eV2 at the 90% CL.

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6

Forthcoming Experiments

There are a number of forthcoming neutrino oscillation experiments which are at various stages; a new solar neutrino experiment SNO already started observation, some are under construction, and others are yet to be formally approved. The main problems to be addressed by these forthcoming experiments are (i) to confirm the SuperK evidence for atmospheric v^ oscillations and to determine the oscillation mode by long baseline neutrino oscillation experiments, (ii) to solve the solar neutrino problem by identifying the unique solution, and (iii) to test the LSND evidence for v^ -> Pe oscillations. For the confirmation of the SuperK evidence for atmospheric z/M oscillations, K2K already started data-taking. There are three other long baseline neutrino oscillation experiments with the baseline distance of 730 km; MINOS (Fermilab to the Soudan mine, under construction, expected to turn on in 2003), and OPERA and ICANOE (CERN to Gran Sasso, yet to be approved formally, expected to turn on in 2005). MINOS is a magnetized calorimeter experiment and OPERA is an emulsion experiment. The ICANOE detector is a combination of liquid argon TPC and a magnetized calorimeter. Though K2K is a fM —>• vx disappearance experiment, other three experiments have capabilities to measure the appearance of T~ . Figure 10 compares the sensitivities of these experiments for both v^ —\ vT and v^ —) ve. 31 In the solar neutrino observations, the measurements of energy spectrum of the solar neutrinos and the day-night flux difference, and the measurement of solar-neutrino flux by utilizing neutral-current reactions are key issues. SuperK continues high-statistics measurements of the day-night flux difference and recoil electron spectrum. SNO, which uses 1,000 tons of heavy water D2O, started data-taking at the end of 1999. SNO measures solar neutrinos through both inverse beta decay [yed —> e~pp) and neutral-current interactions (vxd -> vxpn). In addition, ve scattering events will also be measured. The Borexino experiment with 300 tons of ultra-pure liquid scintillator is under construction at Gran Sasso, and expected to turn on in 2001. The primary purpose of this experiment is the measurement of the 7 Be solar neutrino flux by lowering the detection threshold for the recoil electrons to 250 keV: if the vacuum oscillation is the solution, it causes seasonal variation of the 7 Be solar neutrino flux. KamLAND, which is under construction at Kamioka and will be completed in 2001, is a multi-purpose neutrino experiment with 1,000 tons of ultra-pure liquid scintillator. One of the primary purposes of KamLAND is the search for neutrino oscillations by measuring neutrinos produced by power

186

Figure 10. Comparison of ICANOE, OPERA, and MINOS sensitivities.

reactors. The sensitivity region of KamLAND (shown in Fig. 1) includes the MSW Large Mixing solution. It may be proved or disproved in 5 years of measurements. KamLAND can also observe the 7 Be solar neutrinos if the detection threshold can be lowered to a level similar to that of Borexino. The sensitivities to the solar-neutrino problem of the 7 Be solar neutrino observations are also shown in Fig. 1. Finally, to test the LSND evidence for z?M ->• ve and v^ —> ve oscillations, the MiniBooNE experiment at Fermilab is under construction: it is expected to turn on late in 2001. The MiniBooNE detector will be diluted liquid scintillator (total mass is 807 tons and fiducial mass is 445 tons) and will measure the electron appearance signature in the v^ beam from the Fermilab 8-GeV Booster. The expected sensitivity of MiniBooNE is shown as "BooNE expected" in Fig. 1.

187

7

Conclusions

The evidences for the neutrino oscillation from the SuperK atmospheric neutrino observations and from the solar neutrino experiments are becoming solid. To accommodate yet another evidence from LSND, a sterile neutrino species must be invoked in addition to the three known neutrino species. These evidences will be challenged by the forthcoming neutrino oscillation experiments. They, together with the ongoing experiments, eventually settle the current problems and may bring new surprizes. Once established, the neutrino mass will lead us not only to a new era of neutrino physics, but also to a wealth of new physics beyond the Standard Model. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

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188

Marciano (American Institute of Physics, New York, 2000), p. 164. 23. M. Honda et al, Phys. Rev. D 52, 4985 (1995). 24. S. Hatakeyama et al, Phys. Rev. Lett. 8 1 , 2016 (1998). 25. For example, see P. Lipari and M. Lusignoli, Phys. Rev. D 58, 073005 (1998). 26. J.N. Bahcall, S. Basu, and M.H. Pinsonneault, Phys. Lett. B 433, 1 (1998). 27. M. Apollonio et al, Phys. Lett. B 466, 415 (1999). 28. F. Boehm et al, Phys. Rev. Lett. 84, 3764 (2000). 29. E. Radicioni, Nucl. Phys. B (Proc. Suppl.) 85, 95 (2000). 30. A. Lupi, Nucl. Phys. B (Proc. Suppl.) 85, 91 (2000). 31. Taken from http://pcnometh4.cern.ch7LNGS_nov99.pdf.

THE PAMELA EXPERIMENT: A CLUE TO THE ENIGMA OF ANTIMATTER IN SPACE

SERGIO BARTALUCCI* LNF-INFN, P.O. Box 13, 1-00044 Frascati (ROMA)Jtaly E-mail: Sergio.Bartalucci@lnf. infn. it * on behalf of the PAMELA Collaboration PAMELA is a satellite-borne experiment whose main physics goals are the search for cosmic antimatter of primary origin and the measurement of the spectrum of cosmic ray particles and antiparticles in the energy range from -50 Mev to 200 GeV and beyond. Additional objectives are the continuous monitoring of the cosmic ray solar modulation, the study of the solar flares and the stationary perturbed fluxes in the Earth magnetosphere. The PAMELA apparatus will fly in a low polar sun-synchronous orbit for at least three years. It consists of a magnetic spectrometer with a particle identification system. The scientific objectives, the mission profile and a general description of the instrument are presented.

1

Introduction

The PAMELA experiment is the main part of the Russian Italian Mission (RIM) program that was conceived in the 90 s to study Cosmic rays in space. The RIM program follows an approach consisting of several steps. The first experiment (RIM-0.1, alias Si-Eye-1, Bidoli et al., 1997), has taken data onboard of the russian Space Station MIR starting from October 1995, and a second one (RIM-0.2 experiment, alias Si-Eye-2) was brought to MIR in 1997. These two experiments, based on small silicon telescope detectors, measure the trajectory and the energy lost by high ionizing particles in coincidence with the light flashes seen by astronauts during the flight. The second RIM experiment (called NINA, Bidoli et al., 1999) has been launched in space onboard of the russian satellite Resurs-Arktika n.4 in July 1998. Its main goal is to study the anomalous, galactic and solar components of cosmic rays at low energy (10 100 MeV/n). The PAMELA experiment is expected to be launched at the beginning of year 2003 on board of the russian Resurs-Arktika n.5 satellite, which will fly on a sun-synchronous, polar orbit (986 inclination, average height 690 km) for at least three years.

189

190 The Scientific Case

2.1

The Antimatter Enigma

The beautiful symmetry of microscopic physics laws seems not to hold true in the real macroscopic world and, if the violation of symmetry is very small at the subnuclear scale, it is absolutely prevailing already at the atomic scale. Nevertheless, the existence of antimatter, both as an artificial product from particle accelerators and as a natural product from interaction of cosmic rays with interstellar matter, is strongly suggestive of a possible evolution of the present asymmetric universe from an earlier, symmetric one. Unfortunately, no observation can support such primordial symmetry so far, although the existence of separate domains of matter and antimatter on cosmological scale cannot be absolutely excluded. The early theoretical models and the observational tests of a symmetric Universe were reviewed firstly by Steigman [1976], but since then no significant improvement in either field has been done. The most recent work on this subject is due to Cohen et al. [1998], who explore the possibility of universal (but non-local) matter-antimatter symmetry, with a typical domain size today > 20 Mpc. They find that matter-antimatter encounters at domain boundaries are unavoidable, and try to put constraints on such an Universe arising from two observations, a non-thermal distortion in the Cosmic Background Radiation (CBR) and an enhancement in the spectrum of Cosmic Diffuse Gamma-rays (CDG), red-shifted to the current energy of order 1 MeV. In both cases their HEAT comb. rather conservative estimates far exceed CAPRICE 98 CAPRICE 94 the measured values, so they conclude " Neulr. annihilation • Secondary bkgd that, if such antimatter domains exist, - Signal + bkgd their size is comparable to the Hubble s distance. Perhaps it is worth mentioning that in this case there should be annihilation radiation coming from the interfaces of those large domains, giving rise to detectable anisotropics for the v io2^ ! „ ^ T » C D G in the 100 M e V energy range. Such measurement is well within the io! capabilities of the GLAST based Positron energy (GeV) LargeArea Telescope instrument [GLAST-LAT, 1999]. Figure 1. The experimental and theoretical positron fraction with possible distortion from % annihilation

In the past twenty-five years, several balloon-borne experiments have measured the antimatter/matter ratio as the experimentally most accessible

191 parameter. Some of the most recent results are shown in fig. 1 for e*/(e + e+) ratio, where all experiments mostly suppport the secondary origin of high-energy positrons, although the positron fraction does not appear to fall as rapidly as predicted by the various models. In particular, the HEAT [Barwick et al., 1997] result in the energy range from 7 to 20 GeV might indicate new mechanisms of positron production, but statistics is still poor. A similar remark can be done for the antiproton flux (fig.2): although results in excess of expectations from secondary production were claimed in some early measurements, the next generation of experiment, making use of more sophisticated techniques, clearly showed no evidence for exotic sources of antimatter in the energy range from 200 MeV to, with little statistics, - 50 GeV. -*J ltr 2

to 3

's

1

m = 9 6 4 GeV 10*

• i D X

Mass91 BESS IMAX CAPRICE Secondary total

PAMELA Range

m-»

Figure 2. The antiproton flux with the distortion from % annihilation in a possible MSSM

The basic signature of the existence of significant amount of antimatter in the Universe is certainly the detection of just an antinucleus. There has been speculation that it might be possible for cosmic rays to reach Earth from extragalactic space out to a few hundreds Mpc. Thus, high-energy antinuclei form antimatter domains beyond the gamma-ray limits ( > 10 Mpc) might reach US and be detected, although w i t f j ym]s p r o b a b i l i t y . A n antihelium

could likely be the result of primordial nucleosynthesis, since even the most conservative estimate of the fraction of secondary He produced in collisions of cosmic rays with the interstellar gas gives a He I He of 10"14. An anticarbon would only be produced in an antistar. The present status of search for He is summarized in fig. 3, together with the limit that PAMELA can achieve in three years operation, thanks to its excellent particle identification capabilities. Historically, early in 1967 A. Sakharov suggested that the baryon-asymmetric Universe can evolve from a symmetric one by satisfaction of the three conditions: a) Non-conservation of the total baryon number B b) Both C and CP violation, but CPT invariance c) Baryogenesis in a non-equilibrium environment Condition 1) is quite obvious. Condition 2) means that not only a preference for matter or antimatter must be present (C-violation makes the amplitude of the inverse reaction different), but

192 also CP must not be an exact symmetry, otherwise the rate of a CP-invariant (i.e. time-invariant, because of the CPT theorem) reaction would be equal to the one of its time reversed reaction, thereby determining no change in the net baryon number. 10

J

2 2 io-'

1

RiminrralPT!

JSmoutelal 197.5 Bidhwnrelnl.1978 Ooldenetnl 1997 Bafliin-temctil. 197S

1 2

10 5

10

3

I

G

AMSSTS-91 BESS 93. 94.95.97 and 98 »"<: -mm

10 ''

PAMELA 10"8

Condition 3) is required by the fact that in a thermal equilibrium, when entropy is maximum, the time average of B must be zero, assuming again that the CPT invariance holds true (i.e. masses and total decay widths of particles and antiparticles are exactly equal). The asymmetric Universe is the present reality, but not all the three conditions are supported by experimental evidence. The occurrence of condition 1) was not observed so far: the upper limit on proton lifetime (AB=1 interaction) is very high, around 103" years, implying that the mass scale at which new physics arises is Mx > 1 0 15 GeV. A more stringent test is provided by studies of n

Figure 3. Status and prospects for ^ s e a r c h

present upper limits imply a mass scale M x

_

o s d l l a t i o n

a

n

d n o n -leptonic

modes (AB=2 interaction), ~ 10 GeV.

decay

where

C-nonconservation, as required by condition 2), is maximally observed in weak interactions, but the observed amount of CP violation is definitely too small in the neutral kaon system and doesn t seem large in the B" I B° => fK, decay. While Grand Unification Theories provide a simple and natural environment where Sakharov s conditions are implemented, unfortunately the involved physics is unlikely to be testable in a near or far future. The most widely studied scenario for generating the baryon number of the universe is the electroweak baryogenesis, because it leads to testability at realistic colliders, and therefore to tight constraints. In particular, in the Minimal Supersymmetric Standard Model (MSSM) the behaviour of the electroweak phase transition is dependent on the mass of the lightest Higgs particle and on the mass of the top squark (stop mass) [Riotto and Trodden, 1999 and ref. therein]. Recent LEP results would suggest a candidate Higgs mass > 113 GeV, while electroweak baryogenesis puts an upper bound of 105 GeV in the MSSM. Furthermore, the stop mass should be close to the top mass, -173 GeV, and is presently reachable only at the Tevatron.

193 Only condition 3), which is model-independent, seems quite easily fulfilled, since at early epochs the relevant processes are naturally out of equilibrium, their timescales being slow compared to the expansion rate of the universe. As a last remark, some authors [see Streitmatter, 1996] have suggested the possibility of very large (perhaps on a -100 Mpc scale) domains of matterantimatter, as a consequence of a non-universality of the sign of the CP-violation, what is possible if it arises from spontaneous symmetry breaking.

2.2

The search for exotic Dark Matter

The identification of any excess of primary antiprotons or positrons over the expected flux of secondary antiprotons and positrons may reveal the presence of exotic dark matter, in the form of WIMPs (Weak Interacting Massive Particles). In supersymmetric theories a good candidate for WIMP is the neutralino X< lightest stable supersymmetric particle. These particles could have been accumulated in the halo of galaxies and, through the annihilation with their antiparticles, result in the pair production of antiprotons, positrons and gammas, which can be observed on top of the secondary flux produced in the interaction of cosmic rays with the interstellar medium. In the past, extensive calculations were performed by several authors, mainly looking at possible effects on the p spectrum at low energy (< 2 GeV) , where the spectrum is quite sensitive to solar modulation, however. More recent calculations in the framework of the MSSM point out at the presence of detectable distortions of the high energy spectrum of both positrons [Baltz and Edsj , 1998] and antiprotons [Ullio, 1999]. Some examples are shown in figs 1-2. The particle collection capability of PAMELA, with its large statistics and energy range, will certainly allow one to recognize such signatures within one year of operation, should they exist.

2.3

Summary of the Scientific objectives

The main observational objectives of PAMELA can be summarized as follows: 1. 2.

measurement of the p and p spectrum in the energy range 0.1 150 GeV; measurement of the e~ spectrum in the energy range 0.05 lOOGeV;

search for antinuclei with a sensitivity of about 3¥10 8 in the He I He ratio; measurement of nuclear spectra (from H to C) in the energy range 0.1 200 GeV/n. Moreover, the PAMELA experiment has the following additional objectives:

3. 4.

5.

continuous monitoring of the cosmic rays solar modulation during and after the 23 r maximum of the solar activity

194 6.

study of the time and energy distribution of the energetic particles emitted in solar flares; 7. study of the stationary and disturbed fluxes of high energy particles in the Earth s magnetosphere. These objectives are in the reach of PAMELA because the satellite spends much time in the polar regions, where the cut-off due to the terrestrial magnetic field is low.

2.4

The mission profile

The PAMELA experiment will be installed on the upward side of the ResursArktika n.5 satellite, that will be continuously oriented downward to the Earth during all its mission, in order to fulfill a program of monitoring of ocean-surface, ice and meteorological conditions of Arctic zones. Furthermore the satellite will travel in a quasi-circular, about 700 km high, polar orbit. This is an optimal situation for the observation of cosmic rays, because the polar orbit maximizes the cosmic ray collection rate (important for antinuclei search), and also minimizes the geomagnetic cutoff in a significant portion of the satellite trajectory (important for all scientific issues related to the solar activity and to the terrestrial geomagnetic effects). The upward orientation of the PAMELA telescope on board of the satellite is the required direction for observing cosmic rays without interference with the Earth and keeping far away from the telescope acceptance the showers produced by quasi-horizontal cosmic rays on the terrestrial atmosphere; these showers are responsible of the strong increase of the background at low zenith angles. Finally the high height of the orbit assures a long permanence of the satellite in orbit due to the very small effect of the atmospheric drag; the stabilization of the satellite is obtained by magnetic devices to make the best use of this situation without depending from the possible shortness of fuel.

3

The PAMELA Instrument

A sketch of the telescope is shown in fig. 4. This is composed of: three basic elements: a magnetic spectrometer, an imaging-sampling electromagnetic calorimeter and a transition radiation detector. Also, a time-of-flight/triggering system and an anticoincidence system are implemented. The total height of PAMELA is 105 cm, the mass is 380 kg, the power consumption is 345 W and the geometrical factor is 20.5 cm x sr.

195 3.1

Magnetic spectrometer

The magnetic spectrometer, based on permanent magnet system and silicon microstrip detector, is capable of determining the electric charge with a very high confidence level, and of measuring the particle momentum up to the highest energies, allowing to collect a useful sample of rare particles, according to the PAMELA acceptance and duration of the flight. The general scheme of the magnet system consists of five modules, each 81 mm high, interleaving 6 frames 8 mm high, which accommodate the silicon sensors. The total height of the spectrometer is 445 mm with a rectangular cavity 131x161 mm corresponding to a geometrical factor of 20.5 cm 2 x sr. The magnetic material is a sintered Nd-Fe-B alloy with a large residual magnetic induction (~ 1.3 T). The field within the spectrometer is 0.4 T at the center. Outside the field is shielded by a ferrimagnetic shield. Six tracker planes are inserted in the spectrometer, each composed by 6 silicon sensors, 70x53.3 mm wide and 300 |im thick (The PAMELA collaboration, 1999a). The detectors have been tested with particle beams at PSI (Z rich) and at CERN. The Magnet spatial resolution of a Silicon-Tracker single detector has been found to be 3.0-0.1 u,m in the bending direction and 11.5-0.6 (J.m in the orthogonal direction. The Maximum / Detectable Rigidity (MDR) of the PAMELA Figure 4. The PAMELA Telescope (side view) spectrometer, resulting from the performed tests, is about 800 GV/c and the spectrometer simulation shows that the proton and electron spillover sets a maximum energy limit in the antiproton and positron to about 200 GeV.

3.2

Electromagnetic Imaging Calorimeter

The electromagnetic imaging calorimeter is meant to provide a measurement of the energy lost by the interacting e + , e" and the particle interaction pattern within the

196 calorimeter. The latter allows a clean separation of electromagnetic from hadronic showers ( and from non-interacting particles) with a high level of confidence and efficiency. It is a sampling calorimeter made of silicon sensor planes interleaved with tungsten absorbers [The PAMELA Collaboration, 1999b]. The sensitive area of one detector plane is 240x240 mm' and it consists of a 3x3 matrix of single-sided silicon detectors 80x 80 mm2 wide, which are divided in strips with an implant pitch of 2.4 mm. This high granularity permits a very good path reconstruction of the particle energy released in the calorimetric volume. The thickness of the silicon sensors is 380 |0,m and that of the absorber tungsten layers is 2.6 mm corresponding to 0.7 Xo The whole calorimeter is made of 46 detector layers (23 for the X view and 23 for the Y view) and 22 absorber layers. Several calorimeters of this kind have been already built, tested and used in balloon flight experiments of the WiZard collaboration (TS93, CAPRICE94, CAPRICE98). The performances of these instruments are well studied and the simulation of the SiW calorimeter in the PAMELA configuration gives a resolution in the electron energy measurement better than 5% in the range 20 100 GeV (and ~ 6.5% at 250 GeV) and a rejection power of protons and electrons against positrons and antiprotons of 104 with an overall selection efficiency better than 90%.

3.3

Transition Radiation Detector

The threshold velocity measurement system is based on a Transition Radiation Detector (TRD). It is made of carbon fiber radiators and straw tubes and it helps further discriminate between electrons and hadrons. The Pamela TRD [The PAMELA Collaboration, 1999c] is used to select leptons out of hadrons up to very high energy (-1000 GeV) and it is based on small diameter straw tubes (4 mm) arranged in 9 double layer planes interleaved by carbon fiber radiators. The use of the pulse height measurement technique allows one to perform the particle selection based on energy loss criteria and to track all particles before their entrance in the magnetic spectrometer, cleaning the sample of the particles accepted at its entrance. The PAMELA TRD detector has been tested at CERN using a 3 GeV pion and electron beam, showing pretty good results.

3.4

Time-of-Flight and Anticoincidence Systems

A plastic scintillator hodoscope system is used as time-of-flight (TOF) counter to measure the velocity of particles crossing the apparatus, and gives the trigger signal for data acquisition. It is made up of five layers of detectors, 0.5 cm thick, arranged in three groups, two on top of the instrument, one between the TRD and the magnet, and two just below the magnet. The time resolution is of the order of 100 ps,

197 allowing a separation of ~ 3 std. dev. between protons and electrons up to a momentum of 1.3 GeV/c. The TOF sistem is able to reject albedo (= upward moving) particles at a level better than 1¥ 10. The anti-coincidence system is meant to identify events in which particles may have entered through the mechanical structure of the magnet, or interact in the magnet material and produce secondary particles, or even are produced in interactions in the instrument or spacecraft. It is made up of plastic scintillators covering the top and the sides of the magnet.

4

Conclusions

The PAMELA project is completely defined and each subsystem and detector prototype has been built and tested to measure the detector performances. Moreover, the prototypes have also been tested to probe the resistance against vibrations and shocks during the launch phase, exhibiting a good mechanical behaviour for all sub-detectors. These tests show that the performances meet the requirements and that the PAMELA experiment can fulfil its scientific objectives. Recent (May and July 2000) beam tests at CERN-PS and -SPS with some detectors already in a quasi-final configuration gave very good results, somehow better than the expectations. The PAMELA Qualification Model is under construction and it will be ready for the integration tests and electro-mechanical compatibility with the satellite by mid-2001. Then, according to test results the construction of the Flight Model will start and the integration with the Resurs-Arktika satellite should be completed by the end of 2002. The launch is foreseen at the beginning of 2003 with a mission duration of at least three years. References Baltz, E.A. and J., Edsj, Phys.Rev. D59 (1999),023511. Barwick, S. et al., Nucl. Instr. Meth. 400 (1997),34. Bidoli V., et al, Proc. 25 th ICRC (Durban), 5 (1997), 45. Bidoli V., et al, Nucl. Instr. Meth. A424 (1999), 414. Cohen, A.G. et al, ApJ 495 (1998),539. GLAST-LAT, http://www-glast.stanford.edu/Instrument.html (1999) The PAMELA Collaboration, Proc. 26th ICRC (Salt Lake City), 5 (1999a), 116. The PAMELA Collaboration, Proc. 26th ICRC (Salt Lake City), 5 (1999b),187. The PAMELA Collaboration, Proc. 26th ICRC (Salt Lake City), 5 (1999c), 124. Riotto, A. and M. Trodden, ARN&PS 49 (1999),35. Steigman, G., ARA&A 14 (1976),339. Streitmatter, R., Nuovo Cimento 19C (1996),835. Ullio, P., astro-ph/9904086, Submitted to Astron.Astrophys. (1999).

198

The PAMELA Collaboration 0 . Adriani1 , M. Ambriola2 , G. Barbiellini 3 , L.M. Barbier4 , S. Bartalucci5 , G. Bazilevskaja6, R. Bellotti2 , D. Bergstrom7 , S. Bertazzoni8 , V. Bidoli9 , M. Boezio3, E. Bogomolov 10 , V. Bonvicini3 , M. Boscherini1 , U. Bravar 14 , F. Cafagna2 , P. Carlson7 , M. Casolino8, M. Castellano2 , G. Castellini15, E.R. Christian4 , F. Ciacio2 , M. Circella 2 , R. D'Alessandro 1 ,G. De Cataldo 2 , C.N. De Marzo 2 , M.P. De Pascale9 , N.Finetti1, T. Francke7 , G. Furano 9 , A. Gabbanini 15 , A.M. Galper" , N. Giglietto2, M. Grandi 1 , A. Grigorjeva 6 , M. Hof12, S.V. Koldashov" , M.G. Korotkov" , J.F. Krizmanic 4 , S. Krutkov 10 , B. Marangelli 2 , L. Marino 5 , W. Menn 12 , V.V. Mikhailov 11 , N. Mirizzi 2 , J.W. Mitchell 4 , A.A. Moiseev" , A. Morselli9 ,R. Mukhametshin 6 , J.F. Ormes 4 , J.V. Ozerov 11 , P. Papini 1 , M. Pearce7,A. Perego1 , S. Piccardi1 ,P. Picozza9, M. Ricci5, A. Salsano8, P. Schiavon3 , M. Simon12 , R. Sparvoli9, B. Spataro5,P. Spillantini1 , P. Spinelli2, S.A. Stephens13 , S.J. Stochaj14 , Y. Stozhkov6, R.E. Streitmatter4,F. Taccetti15 , M. Tesi15, A. Vacchi3, E. Vannuccini 1 , G. Vasiljev 10 , V.Vignoli" , S.A. Voronov 11 , N. Weber 7 , Y. Yurkin n ,N. Zampa3 Institutions: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

University and INFN, Firenze (Italy) University and INFN, Bad (Italy) University and INFN, Trieste (Italy) NASA Goddard Space Flight Center, Greenbelt (USA) Laboratori Nazionali INFN, Frascati (Italy) Lebedev Physical Institute (Russia) Royal Institute of Technology, Stockholm (Sweden) Electronic Engineering Department, II University, Roma-Tor Vergata (Italy) II University and INFN, Roma-Tor Vergata (Italy) Ioffe Physical Technical Institute (Russia) Moscow Engineering and Physics Institute, Moscow (Russia) Siegen University, Physics Department, Siegen (Germany) Tata Institute of Fundamental Research, Bombay (India) Particle Astrophysics Laboratory, NMSU, Las Cruces (USA) Istituto di Ricerca Onde Elettromagnetiche CNR, Firenze (Italy)

TUNNELING P H E N O M E N A IN COSMOLOGY: SOME FUNDAMENTAL PROBLEMS

M. Y O S H I M U R A Department

of Physics, E-mail:

Tohoku University Sendai 980-8578 [email protected]

Japan

The fundamental problem of how tunneling in thermal medium is completed is addressed, and a new time scale of order 1/friction for its termination, which is usually much shorter than the Hubble time, is pointed out. Enhanced non-linear resonance is responsible for this short time scale. This phenomenon occurs when the semiclassical periodic motion in a metastable potential well resonates with one of the environment harmonic oscillators coupled to its motion.

The usual picture of how the first order phase transition proceeds in cosmology, for instance the electroweak phase transition which is relevant for the scenario of electroweak baryogenesis 1, is something like this; nucleation of the bubble of the true ground state (of broken symmetry) occurs with some probability in different parts of the universe. If the bubble formation does not take place strongly, the phase transition is terminated only when cosmological evolution changes the form of the potential (or more properly of the free energy) so that the symmetric phase is no longer the local minimum of the potential. This picture relies on the presumption that there is no inherent time scale for termination of the phase transition, or at least even if it exists, it is much larger than the Hubble time scale, which is the time for the potential change. We would like to exmaine this problem by working in a new formalism of the real-time description of the tunneling phenomena 2 , 3 . Clearly, if the new time scale of the phase transition is much shorter than the Hubble time, one must reconsider the usual scenario. This might change the situation on, and even resurect the once failed GUT phase transition for the inflationary universe scenario 4 . Another important ingredient for our consideration is effect of cosmic environment; the order parameter that describes the state of the universe such as the homogeneous part of the Higgs field is necessarily coupled to matter fields that make up the thermal environment. One must then take into accout the presence of thermal medium in discussion of the tunneling rate and its time evolution, and estimate how dissipation due to the environment interaction modifies the basic tunneling rate. There are already many works on the subject of tunneling in medium 5 , and most past works deal with a system in equilibrium as a whole. The 199

200

Euclidean technique such the bounce solution 6 is often used in this context 7 8 , . Our approach here is different, and we attempt to clarify dynamics of time evolution starting from an arbitrary initial state of the tunneling system, which can be either a pure or a mixed state. We find it more illuminating to use a real-time formalism instead of the Euclidean method much employed in the literature. Our formalism is based on separation of a subsystem from thermal environment, and integrating out the environment variable including the interaction with the subsystem. In this picture dissipation seen in the behavior of the subsystem is due to our ignorance of huge environment degrees of freedom. Modeling of the environment and its interaction form with the subsystem is expected to be insensitive to the result one derives in this approach. We use the standard model of environment 9 , 7 . This method is best suited to a (by itself) non-equilibrium system which is immersed in a larger thermal equilibrium state. This methodology has been used by us in a number of problems in cosmology 10 , n . For instance, the relic abundance was calculated n from this approach and the usual estimate of thermally averaged Boltzmann rate was justified at high temperatures. At very low temperatures T <S M (the mass of relic particle) our calculation differs from the usual one; the Boltzmann suppressed number density (MT/2-ir)3/2 e~MlT at the freeze-out is replaced by some temperature power term. (There is some criticism against this calculation 12.) But numerical importance of this effect is presumably minor, although it is a theoretically interesting issue. Here we consider the most basic problem of this kind, one dimensional subsystem described by a potential V{q). This system is put in thermal medium. The potential is assumed to have some local minimum at q = 0, which is separated at q = qs of the barrier top from a global minimum. The environment part is modeled by infinitely many, continuously distributed harmonic oscillators whose coordinates are Q(UJ). Its coupling to the tunneling system is given by a Hamiltonian, /•OO

q /

duc(u)Q(uj).

(1)

J ioc

The coupling strength is c(u>) and uc is some threshold frequency. Needless to say, one may imagine a generalized case in which the system variable q is the order parameter for the first order phase transition, the homogeneous Higgs field, and the environment oscillator Q(co) is a collection of various forms of matter fields coupled to the Higgs field.

201

The basic equation in our problem is m d2Q{io) rco , d-7z2q„ dV f°° . .^. , cFQ(w) , , ,„. 2_. . -£ + — = - I dojc(u)Q(uj), 2f2 + u ; 2 Q M = -c(u;)(7.(2) One may eliminate the environment variable Q(w) to get the Langevin equation 13 ,

^£ + ^+2

J' dsaI(t-S)q(s) = FQ(t),

(3)

where FQ(£) is linear in initial values of environment variables, Qi(w) and

<£j c(w) I Qj(w) cos(wi) + - ^ - ^ sin(wt) j .

(4)

By taking the thermal bath of temperature T = 1/(3 given by the density matrix, \ l/'2

/ Pp(Q,Q')

•n coth(/?c;/2) •exp

W

2 sinh(/3w)

{(Q2 + Q'2) cosh(/3w) - 2QQ')

, (5)

for each environment oscillator, the following correlation formula is obtained;

Z

oo

g^j

dw r(u) cos CJ(T - s) c o t h ( — ) = aR(r - s) (6)

with

The kernel function aj in eq. (3) is given by /*oo

<*/(*) = — /

du>r(oj) sin(wt).

(8)

The combination, afl(t) + iai(t), is a sum of the real-time thermal Green's function for Q(u>)— oscillators, added with the weight c2(w). An often used simplification is the local, Ohmic approximation taking

rM = ^ ,

(9)

7T

with uic = 0, which amounts to a / ( r ) =£W 2 <$(T)+»?<$'(r)-

(10)

202

This gives the local version of Langevin equation, d2q

dV

r

dq

9

,

,

The parameter 5UJ2 is interpreted as a potential renormalization or a mass renormalization in the field theory analogy, since by changing the bare frequency parameter to the renormalized UJ2R the term 5u>2 q is cancelled by a counter term in the potential. On the other hand, r\ is the Ohmic friction coefficient. This local approximation breaks down both at early and at late times 14 , but it is useful in many other cases. Any mixture of quantum states is described by a density matrix, P = ^2wn\n)(n\,

(12)

n

where \n) is a state vector for pure quantum state and 0 < wn < 1. The density matrix obeys the equation of motion; ih^

= [Htot,p],

(13)

where Htot is the total Hamiltonina of the entire system. In configuration space this density matrix is given by its matrix elements, p(q , q'; Q(u) , Q'(u)) = (q , Q(u)\p \q', Q V ) > .

(14)

1

Its Fourier transform with respect to relative coordinates, q—q , Q(ut) — Q'(ui), is called the Wigner function and denoted by fw', fw(q,p;

Q(w),P(w)) = f

P(q +C/2,q-

d e l ] dX(u) exp[-i^P-i

i/2 ; Q(w) + X(u)/2

f

, Q(u) - X(u)/2)

dwX{w)P{u)} .

(15)

It is easy to show from eq.(13) that this quantity obeys dfw

dfw

p—

9t

dg

f , ( r,l \ // * dw , ( I r{<jj)

J

V

®fw

,

2 nx

\

, . + to Q{LO) v ;

3Q(w)

9

fw

. , +

<9P(w)

In one dimensional quantum mechanics the semiclassical approximation is excellent when the potential barrier is large, and we assume that this is also true in the presence of the system-environment interaction. In the semiclassical h —i 0 limit we have 1

/ v ( j . ih

d

\

vf

ihd

Mf

_^dV

dfw

203

The resulting equation, being identical to the classical Liouville equation, has an obvious solution; fw(q,P,Q

,P)=

/ dqtdpi /

dQidPif$(qi,Pi,Qi,Pi)

•6 (q - gci) 6 (P - Pd) 6 (Q - Qd) 5 (P - P cl ) ,

(17)

where ), P(w) integration, fw\q,P;t)= K(q,p,qi,Pi;t)

/ dqidpif^tq{qi,pi)K{q,p,qi,pi;t), = J dQidPi f$iQ(Qi

(18)

, Pi) 6 (q - qcl) S [p - p cl ) . (19)

The problem of great interest is how further one can simplify the kernel function K here. In many situations one is interested in the tunneling probability when the environment temperature is low enough. At low temperatures of T < a typical frequency or curvature scale u>s of the potential V, one has

/c^,jpj(^)

= 0{Vf] « fuT,.

(20)

In the electroweak and the GUT phase transition this corresponds to T < m#(Higgs mass). Expansion of qc\ in terms of Qi(u>), P;(w) is then justified. Thus, we use <* (q - Qd ) = / - ^ exp [i Xq ( q - qcl) ] «

/ ^texp [l A" {q ~q«] " / ^ { QiMQ^i") + p^)ecfV)})

] >(2i)

valid to the first order of Qi(w), Pi(tu). A similar expansion for S {p — pc\) using p^ ,p^\u),p[P(u}), also holds. An alternative justification of the expansion (21) is to keep 0[c(w)] terms consistently in the exponent since both ^ ( w ) and <^P)(u;) are of this order. Gaussian integral for the variables Qi(u), Pi{u)) first and then for Xq , Xp can be done explicitly with eq.(21), when one takes the thermal, hence Gaussian density matrix for the initial environment variable, pQ . The result of

204 this Gaussian integral leads t o an integral transform 2 of the Wigner function, f$ (initial) -> f™ (final), using the kernel function of ui K{q,p,qi,Pi]t)

*\

v/detj = — exp

where the matrix elements of J 1 h i = 1

f00

0{Q ^ 1

Id

>P

Pel ) J

^

(23)

W

1 f°° Bui 1 7 22 = - / ducoth?--\z(Lo,t)\2, /i2 = ^ /"°° dw coth ^ where

z(ui ,t)

(22)

= Uj are given by

3LU 1 du; coth^--\z(u,t)\2,

Jul*

(0) cl

[ (o) (0) \ P-Pcl n — rr,

(24)

- JR ( Z ( w , t)z* (to , t)) = - ^ - ,

= q r , ' (ui ,t) + iui q(ci '' (u ,t),

and

(25)

i ( w , t ) = p£i (ui ,t) +

iujp[P{uj ,t). Quantities t h a t appear in the integral transform are determined by solving differential equations; the homogeneous Langevin equation for q£, and an inhomogeneous linear equation for z{u , t) and z(u ,t),

+ 2 [tdsaI(t-s)q$\S)=0,

— 2+ ( T 0 dt

V dq J 9,.to)

d2z(uj_,t) 2

dt

(26)

y0

ffV

' ( 0 " ) ,«., *<<" • ' ) + 2 { rfs a ' C " s ) ^ •s) = - c(w)e " • (27)

A similar equation as for z(u> ,t) holds for z(u> ,t).

T h e initial condition is

qia)(t = 0) = qi, p^(t = 0)=Pi, z(u,t = 0)=0, z(u,t = 0)=0. T h e physical picture underlying the formula for the integral transform, eq.(18) along with (22), should be evident; the probability at a phase space point (q, p) is dominated by the semi classical trajectory qcl (environment effect of dissipation being included in its determination by eq.(26)) reaching (q ,p) from an initial point (
205

We note that in the exactly solvable model of inverted harmonic oscillator potential the identical form of the integral transform was derived 3 without resort to the semiclassical approximation. Explicit form of the classical and the fluctuation functions (g£j and z{u> ,t)) is given there. To proceed further, one may separate the tunneling potential V(q) at the barrier top location, q = qs- We distinguish two cases of the potential type, depending on the value of V(oo) relative to the local minimum value Vm at q = 0 in the potential well. When V(oo) < Vm, the classical motion in the overbarrier region q > qs is monotonic ending at q = oo, while for V(oo) > Vm the motion is a damped oscillation towards 70 at the true minimum, unless the friction is large. If the friction is larger than a critical value of « 2w* with w« being the curvature of the potential at the global minimum, there occurs the overdamping such that q^ —> qo monotonically. A typical interesting case for V(oo) > Vm is the asymmetric double well as may occur in the first order electroweak phase transition x. We consider here a few problems to illustrate consequences of our general formula of the integral transform. One problem is calculation of the barrier penetration factor; to determine the outgoing flux in the overbarrier region for the type of potential of V(oo) < Vm, assuming an initial energy eigenstate under the potential V. The other problem is the tunneling probability for the asymmetric wine bottle type of potential. Energy eigenstates are special among pure quantum states, since they evolve only with phase factors. Thus, if one takes for |n)(n| in eq.(12) the energy eigenstate of the total Hamiltonian, the density matrix does not change with time. On the other hand, if one takes the energy eigenstate for the subsystem Hamiltonian, then the density matrix changes solely due to the environment interaction. It is thus best to use a pure eigenstate, or its superposition, of the subsystem Hamiltonian when one wishes to determine how the barrier penetration factor is modified in thermal medium. We first discuss the stationary phase approximation for the initial state. Starting from the formula, f${Qi,Pi)

= J

5Z Wn j

dte-**

d£ exp

J2

W

» ^ n f e " fWnfai + f )

£ ipi£ + I n *(
=

£ + -)

(28)

we locate the stationary point by J | = 0, -S- = 0 for each exponent factor. The partial derivative -^- = 0 is taken with the understanding that the rest of pj integration contains smooth functions of pi. This leads to the stationary

206

point at £ = 0, In{x)

Pi = In(qi)

(29)

-i ^*n{x)ipn{x) - tpZ(x)ipn(x) 2 |^ n (a:)| 2

(30)

Here In{x) is the usual flux factor at x for the pure state \n). This gives the result for the initial density matrix, Swili,

Pi) « 5 I

W

n\^n(qi)\'2

27T 5 fa -

(31)

In(qi))

The barrier penetration factor is calculated from the flux formula, i{q,t)=

£

(32)

^pfw'ii'P-'t)

2TT J

Thus, the semiclassical plus the stationary phase approximation gives I(q, t) of the form,

/ ^i^i2vd^exp

2/n

^)+^(?-i0)) (33)

for a pure initial state given by the wave function ip(x), where q* is the turning point in the overbarrier region. In the region of q 2> g» there is always a classical trajectory that reaches the point q of the Gaussian peak from an initial g, in the range qi > g», and the entire region within the Gaussian width ^fl\\ is fully covered by the integral. The factor outside the Gaussian exponent is well approximated by its peak value in the weak coupling case. Thus, one obtains 2 2

2

I(q,t) « |V(*o)| ( - i o ) = mx0)\ p^(x0,t)

,(°h r ^ 1 ^ ^ )

-l

, 04)

where xo (q , t) is determined from q(a\li

= xo ,Pi = / ( x 0 ) , Qi(u) = 0 , Pi(u) = 0-t) = q.

(35)

We now take the WKB wave function in the overbarrier region, 4>{qi) = —r== exp VPw)

f

J x.

dx p{x)

p(x) = J2(E-V(x)),

(36)

207 to get a factorized form of the flux 2 ,

,(„t)-ir(B)lV(,..lB). / - s S ^ p f e ^ ' . O T A feature t h a t characterizes the classical trajectory q^ its initial energy is = 0)a±p(qi)2

Hg(t

is of special interest;

+ V(qi) = E.

(38)

This formula for the flux reproduces our previous result for a specific potential of the inverted harmonic oscillator 3 , V(x) + -| 5ui2 x2 = — | u\ x2 , with UJR the renormalized curvature. T h e present approach actually improves our previous result; 9 x0+gl(x0)

=

_^

9 +ujBg

gl(x0) + guj'2Bx0

uiB(g + ojBg) '

where U>B ~ <JJR — r)/2 is a diagonalized frequency. T h e limiting formula of eq.(39), with I(xo) —> LUBXO as XQ —> oo, is valid in the infinite q limit, as derived in 3 . T h e function
/-oo

#(t) =

H

^

sinh(w B i) + 2 /

«

.,»•.,.,» ^

^

(u2 + u2R)2 +

^

dw H(u) sm(ut)

2

(irr(u)y

>

(40)

(

4 1

)

/»oo

A^2 = 1 - 2

/

C1UJH(LO)U.

(42)

Both at early and late times the factor / « 1, deviating from unity only for the time range of order 1/LUBAccording to the view of 7 the potential V(q) t h a t determines the semiclassical penetration factor |T(£7)| 2 should be expressed in terms of the renormalized parameters, LOR in this case. This explains the bulk of the suppression caused by dissipative environment interaction, as explained in 3 . For discussion of a more general case of finite V(oo) < Vm we use the local, Ohmic approximation, which becomes excellent at late times. A potential t h a t decreases fast as q —> oo is assumed; ~f- —> 0 . T h e acceleration term can then be neglected when the friction satisfies rj1 3> | ^ T - | - This is a slow rolling

208

approximation, and it always holds for q large enough. The classical equation is then solved as Jq a,

dz(—)~1

= -t,

dz

(43)

which gives the factor

'-T-hr

<«>

dqm r]p(co) Thus, the tunneling probability decreases with time along with the decreasing slope of the potential. This result poses a curious question; the tunneling may not be terminated within a limited finite time. We shall encounter a similar situation for the asymmetric wine bottle potential. We next consider the case in which the potential is very steep at both ends; V(±oo) = oo . The tunneling rate, from the inner region at q < qs into the outer region at q > qs, is an important measure of tunneling phenomena and is given by the flux at q — qs; P(t) = — I(qB , t), which is equal to — /

dqidpif$(qi ,pt) (p^ + Yf~{
2/n

(45) On the other hand, the tunneling probability into the overbarrier region at q > x is given by dqidpif${qi,pi)

\ Jx

du

exp V^hi

2/n

(.46)

In both of the quantities, P(t) and P(qB , t) (the tunneling probability into q > qs), it is essential to estimate how q^ and In varies with time. We consider an initial state localized in the potential well so that the dominant contribution in the (qt ,Pi) phase space integration is restricted to q% < <7B- I n the rest of discussion anharmonic terms play important roles, but at first we work out the harmonic approximation;

V(?) + f < ^ V * | w 0 V ,

(47)

near the bottom of the well at q = 0. In the Ohmic approximation the classical solution and the fluctuation is given by qf

= I coscoot + ^-smcoot)

e " ^ 2 Qi + ^

^

e"^2Pi,

(48)

209

Z(U,t)

= —

j

:

W z — UIQ — ILUT]

iut _ u+Qo-iy/2

iaet-nt/i

,

+

u

-

~ Q-o» _~ ir-ll#2/ -

2WQ

.-ia t-t,t/2

e - « w „0* - , t / - 2

^

(4g)

2WQ

using wo = y ^ o — ^ - . In the rest of discussion we assume a small friction, Near w = w>o the fluctuation is approximately 1 C ( 0 J ) . / 0 f t e ^ 0 * - ± smQot) e~^2 . (50) w + wo - "7/2 \ w0 / This formula is valid at t < 0[l/rj]. The appearance of the linear t term is a resonance effect. The resonance roughly contributes to hi{t) by the amount, rj t2 e~r)t x a smooth w integral which is cut off by a physical frequency scale. Thus, the width factor In initially increases with time until the time scale of order 1/rj. The width factor asymptotically behaves as *(w,*) «

/ii(t) = / 1 i(oo) + 0 [ e - ' * / 2 ] ,

(51) 2

/n(oo)«7^- + — TT—T + ^ T . (52) v J ' 2w0 wo e"»/ T - 1 3 w^ The asymptotic value of I n (oo) has the familiar zero point fluctuation of harmonic oscillator and in the last term the dominant finite temperature correction, valid for this Ohmic model at T
*£> + £ - ( * * - * < ? > )"•<>.

(53)

Moreover, the final tunneling probability P(gs , oo) has a finite value, and typically is very small for a large potential barrier. For instance, for the asymmetric wine bottle potential shortly discussed,

p

1

/

W0

<*"°° > * W ^

eo _ 8 v h / u o%

(54)

with V/i the barrier height much smaller than wo. This poses again a curious question; it appears that decay of a prepared metastable state localized in the potential well is never completed. This simple picture is however valid only when one ignores anharmonic terms in the tunneling potential, but they must be there in order to give any realistic tunneling potential. The most important in the following discussion is

210

effect of anharmonic terms in the equation for the fluctuation Z{UJ , t). Presence of anharmonic terms gives a non-trivial periodicity in the coefficient function for (27), assuming a small friction r\ <^i OJQ. c!

The homogeneous part of the Z(UJ ,t) solution which might exhibit the well known parametric resonance 15 is closely related to the behavior for our inhomogeneous z(ui ,t) solution. It is then important to check whether the relevant parameter in the periodic coefficient function f i j-r 1 . falls in the instability or the stability band. As is shown in 16 , unbounded exponential growth of v ^ n does not take place. On the other hand, the power-law growth is observed in numerical computation. Moreover, we find that our z(u ,t) solution belongs to the boundary between stability and instability bands. At the resonance frequency the enhancement factor due to the boundary effect is much larger than what one might expect from the harmonic case (50). We shall interpret this phenomenon as influenced in a subtle way by the parametric resonance, although it is not the parametric resonance itself. It is best to discuss the resonance enhanced tunneling mechanism in concrete examples. We take the asymmetric wine bottle potential, which is described in the well and its vicinity region by V(q) * \{q2 - 2qBqf

.

(55)

The curvature parameters at two externa of q — 0 and q — qs are <X>Q = 2\qB , u% = Xq% , and the barrier height seen from the bottom of the well is Vh, = jq% = Wggg/8. The classical q^ , and the fluctuation z(u ,t) equations in the Ohmic approximation are written using rescaled variables, V = 9ci I IB and r = w0 t/2, v"+y(y-l)(v-2) 2

z"+[4-6(y-^y

)\

+ —v' = 0,, z+ ^

(56) ^

-

^

e

1

^

,

(57)

where ' = d/dr. For a small friction one has approximate forms of solution in terms of the Jacobi's elliptic function;

to- .,« = i + ^-V^ p^\t)

(^7^

+[

-7==i_M)

= 2y/2eVhsxx ( ^ 1 + yfe{r + const),kJ en ( \J 1 + ^ ( r + const), k J

211

1

*

u I 2^

These formulas are valid for wo , 2wo , • • • n&o i' • • > where

u0 = ^J^7e(

, t

Ei


f

,

There are resonances at

dU

V.

(58)

We found so far that these explicit solutions are not very illuminating, and numerically integrated the coupled q^' and z(u, t) equations. In the u) integral (23) for the width factor In the largest contribution is found to come from the fundamental resonance at to = u>o, then contribution from higher harmonics at n CJQ follows. Thus, phenomenon of a non-linear resonance occurs. At a time of 0[1/T?] the width factor In becomes maximal at its value much larger than in the harmonic case. The decrease observed at late times is due to the friction; we explicitly checked that for the zero friction, the fluctuation \z(u> ,t)/c(co)\2 increases in time without a bound, with an averaged time power close to 4 at the resonance. Note that even if the effect of the friction is turned off, there exists an important environment effect here; the environment interaction drives the non-linear resonance oscillation. We refer to our paper 16 for detailed numerical results. We numerically checked 18 the behavior of the most important part given by the product of two competing exponential factors, A = exp

tanh

/3u0 pj + ul q\ LO0

x exp

Zlll

the one for the initial state and the other for the kernel factor in the probability rate. Remarkably, the largest contribution comes, not from the dominant initial component near the zero point energy, rather from the initially suppressed excited component. The first exponent in (59) is in proportion to Ei, while the second one goes roughly like E~2, hence a maximum may appear somewhere away from the lowest energy of Ei = LOQ/2. In the example of rj/ujQ = 0.0025 the maximum product factor is of order 1 0 - 3 at Ei ss 4 x wo/2, 6 orders of magnitudes larger than what one expects from the lowest energy state and also the asymptotic value of order 10 - 3 6 . A large value of the product factor of order unity suggests an interesting possibility of a rapid and violent termination of the tunneling. The reason why one observes enhanced tunneling at resonant frequencies of u) = n WQ is as follows. The semiclassical q—motion in the left potential

212

well has a periodicity 2-K/QQ if one neglects the friction. The environment oscillators act as stochastic fluctuation to this motion in such a way that the effective potential including one environment oscillator, V(q) + qc{u) ( Qi(u) cos{ut) + - ^

sin(wt) j ,

(60)

changes the potential barrier periodically. When the q—particle hits against the potential wall, the potential barrier may become smallest, hence the tunneling rate maximally enhanced, if one of environment oscillators has a period exactly equal to that of the classical motion, u = nuiQ. This is precisely the condition of the non-linear resonance we have been discussing. The resonance enhanced tunneling thus envisaged seems to have little connection to the stochastic resonance 17 much discussed in the literature. In summary, we gave a new mechanism of resonance enhanced tunneling, using the real-time semiclassical formalism. Our approach suggests a new time scale of order I/77, the inverse friction, for completion of the first order phase transition. Evidently much has to be done to apply the idea here to realistic problems. I wish to thank my collaborator, Sh. Matsumoto who shared most of the results presented here. References 1. For a review, A.G. Cohen, D.B. Kaplan, and A.E. Nelson, Annu. Rev. Nucl. Part. Sci. 43, 27(1993). 2. Sh. Matsumoto and M. Yoshimura, "Quantum tunneling in thermal medium", hep-ph/0008025. 3. Sh. Matsumoto and M. Yoshimura, "Dynamics of barrier penetration in thermal medium: exact result for inverted harmonic oscillator", hepph/0006019, and Phys. Rev. A in press. 4. A. Guth, Phys. Rev. D23, 347(1981); K. Sato, Mon. Not. R. Astr. Soc.195, 467(1981). 5. For a review, P. Hanggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys62, 251(1990). 6. S. Coleman, Phys. Rev. D15, 2929(1977); C.G. Callan and S. Coleman, Phys. Rev. D16, 1762(1977); J.S. Langer, Ann. Phys. (N.Y.) 4 1 , 108(1967);

213

7. 8. 9. 10. 11.

12.

13. 14.

15.

16. 17.

M.B. Voloshin, I.Yu. Kobzarev, and L.B. Okun, Sov. J. Nucl. Phys. ,20, 644. A.O. Caldeira and A.J. Leggett, Ann. Phys. (JV.Y.) 149, 374(1983). H. Grabert, P. Schramm, and G-L. Ingold, Phys. Rep. 168, 115(1988). R.P. Feynman and F.L. Vernon, Ann. Phys. (JV.Y.) 24, 118(1963). I. Joichi, Sh. Matsumoto and M. Yoshimura, Phys. Rev. A57, 798(1998); Prog. Theor. Phys. 98, 9(1997); Phys. Rev. D58, 043507-17(1998). Sh. Matsumoto and M. Yoshimura, Phys. Rev. D61, 123508(2000), and hep-ph/991039; Phys. Rev. D61, 123509(2000), and and hepph/9910425. A. Singh and M. Srednicki, Phys. Rev. D61, 023509(2000); M. Srednicki, Phys. Rev. D62, 023505(2000); E. Braaten and Y. Jia, hep-ph/0003135; P. Bucci and M. Pietroni, hep-ph/0009075. G.W. Ford, J.T. Lewis, and R.F. O'Connell, Phys. Rev. A37, 4419(1988). V. Ambegaokar, Ber. Bunsenges. Phys. Chem. 95, 400(1991); I. Joichi, Sh. Matsumoto, and M. Yoshimura, Phys. Rev. A57, 798(1998), and references therein. L. Landau and E. Lifschitz, Mechanics (Pergamon, Oxford, 1960), p80; M. Yoshimura, Prog. Theor. Phys. 94, 873(1995); H. Fujisaki, K. Kumekawa, M. Yamaguchi, and M. Yoshimura, Phys. Rev. D53, 6805(1996); L. Koffman, A. Linde and A. Starobinsky, Phys. Rev. D56, 3258(1997). Sh. Matsumoto and M. Yoshimura, "Resonance Enhanced Tunneling", hep-ph/0009230, and Phys. Lett. B in press. L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev.Mod.Phys70, 223(1998).

T H E COSMOLOGICAL C O N S T A N T A N D T H E B R A N E W O R L D SCENARIO HANS P E T E R NILLES Physikalische

Institut,

Universitat E-mail:

Bonn, Nussallee 12, D-53115 nilles@th,physik.uni-bonn.de

Bonn,

Germany

Although the cosmological constant has primarily cosmological consequences, its smallness poses one of the basic problems in particle physics. Various attempts have been made to explain this mystery, but no satisfactory solution has been found yet. The appearance of extra dimensions in the framework of brane world systems seems to provide some new ideas to address this problem form a different point of view. We shall discuss some of these new approaches and see whether or not they lead to an improvement of the situation. We shall conclude that we are still far from a solution of the problem.

1

Introduction

We know that the cosmological constant is much smaller than one would naively expect. This led to the belief that a natural approach to this problem would be a mechanism that explains a vanishing value of this vacuum energy. While cosmological observations 1 ' 2 seem to be consistent with a nonzero value of the cosmological constant, still the small value obtained lacks a satisfactory explanation other than just being the result of a mere fine-tuning of the parameters. Recently new theoretical ideas in extra dimensions have been put forward to attack this problem. In the present talk I shall elaborate on work done in collaboration with Stefan Forste, Zygmunt Lalak and Stephane Lavignac 3 ' 4 ' 5 , where the problem of fine-tuning has been analyzed in the framework of models with extra dimensions that have attracted some attention recently. One of the most outstanding open problems in quantum field theory is it to find an explanation for the stability of the observed value of the cosmological constant in the presence of radiative corrections. As we will see below (and as has been discussed in several review articles 6 ' 7 ' 8 ) a simple quantum field theoretic estimate provides naturally a cosmological constant which is at least 60 orders of magnitude to large. Quantum fluctuations create a vacuum energy which in turn curves the space much stronger than it is observed. Hence, the classical vacuum energy needs to be adjusted in a very accurate way in order to cancel the contributions from quantum effects. This would require a fine-tuning of the fundamental parameters of the theory to an accuracy of at least 60 digits. From the theoretical point of view we consider this 214

215

as a rather unsatisfactory situation and would like to analyze alternatives leading to the observed cosmological constant in a more natural way. In this talk we will focus on brane world scenarios and how they might modify the above mentioned problem. In brane worlds the observed matter is confined to live on a hypersurface of some higher dimensional space, whereas gravity and possibly also some other fields can propagate in all dimensions. This may give some alternative point of view concerning the cosmological constant since the vacuum energy generated by quantum fluctuations of fields living on the brane may not curve the brane itself but instead the space transverse to it. The idea of brane worlds dates back to 9 ' 1 0 > n . A concrete realization can be found in the context of string theory where matter is naturally confined to live on D-branes 12 or orbifold fixed planes 13 . More recently there has been renewed interest in addressing the problem of the cosmological constant within brane worlds, for an (incomplete) list of references see [14-42] and references therein/thereof. The talk will be organized as follows. First, we will recall the cosmological constant problem as it appears in ordinary four dimensional quantum field theory. We shall then elaborate on some of the past (four-dimensional) attempts to solve the problem. Subsequently the general set-up of brane worlds will be presented. Particular emphasis will be put on a consistency condition (sometimes also called a sum rule) for warped compactifications that has been overlooked in various attempts to address the problem of the cosmological constant and which is a crucial tool to understand the issue of fine-tunings in the brane world scenario. Then we will study how fine-tunings appear in order to achieve a vanishing cosmological constant in the Randall Sundrum model 14>15. We shall argue that a similar fine-tuning is needed in the set-up presented in 21 ' 22 once the singularity is resolved. Finally, we elaborate on the issue of the existence of nearby curved solutions and we will argue that it is this questions that has to be addressed if one wants to understand the small value of the cosmological constant. 2

The problem of the cosmological constant

The observational bound on the cosmological constant is AM|,; < 1CT120

(MPI)4 19

(1)

where MPt is the Planck mass (of about 10 GeV) and the formula has been written in such a way that the quantity appearing on the left hand side corresponds to the vacuum energy density. This is a very small quantity once one admits the possiblilty of the Planck scale as the fundamantal scale of

216

physics. Even in the particle physics standard model of weak, strong and electromagnetic interactions one would expect a tree level contribution to the vacuum energy of order of several hundred GeV taking into account the scalar potential that leads to electroweak symmetry breaking. Moreover, in quantum field theory we expect additional contributions from perturbative corrections, e.g. at one loop \M2Pl = A0Af£ + (UV-cutoff)4 Str (1)

(2)

in addition to Ao the bare (tree level) value of the cosmological constant which can in principle be chosen by hand. The supertrace in (2) is to be taken over degrees of freedom which are light compared to the scale set by the UV-cutoff. Comparison of (1) with (2) shows that one needs to fine-tune 120 digits in Ao Mj,; such that it cancels the one-loop contributions with the necessary accuracy. Supersymmetry could ease this problem of radiative corrections (for a review see 43 ). If one believes that the world is supersymmetric above the TeV scale one would still need to adjust 60 digits. Instead of adjusting input parameters of the theory to such a high accuracy in order to achieve agreement with observations one would prefer to get (1) as a prediction or at least as a natural result of the theory (in which, for example, only a few digits need to be tuned, if at all). This is the situation within the framework of four-dimensional quantum field theories. The above discussion might be modified in a brane world setup which we will discuss in this lecture. We should however mention already at this point that "modification" does not necessarily imply an improvement of the situation. Before we get into this discussion let us first recall some attempts to solve the problem in the four-dimensional framework. 3

Some attempts to solve the problem

A starting point for a natural solution would be a symmetry that forbids a cosmological constant. In fact, symmetries that could achieve this do exist: e.g. supersymmetry and conformal symmetry. Unfortunately these symmetries are badly broken in nature at a level of at least a few hundred GeV and therefore the problem remains. Still one might think that the presence of such a symmetry would be a first step in the right direction. A second possible solution could be a dynamical mechanism to relax the cosmological constant. Such a mechanism could be quite similar to the axion mechanism that relaxes the value of the 9 parameter in quantum chromodynamics (QCD). This mechanism needs a new ingredient, a propagating field that adjusts is vacuum expectation value dynamically. For a review of these

217

questions see 6 ' 4 4 . In string theory the so-called "sliding dilaton" could play this role as has been argued in 45>46. In all these cases, however, one would then expect the existence of an extremely light scalar degree of freedom which would lead to new fifth force that probably should not have escaped our detection. Other attempts to understand the value of the vacuum energy have used the anthropic principle in one of its various forms. For a review see 6 . Given the present situation it is fair to say that we do not have yet a satisfactory solution of the problem of the cosmological constant, at least in the framework of four-dimensional string and quantum field theories. Could this be better in a higher dimensional world? For an alternative way to address the problem in less than four space-time dimensions see 7 . We should keep in mind, however, that the problem of the cosmological constant is just a problem of fine-tuning the parameters of the theory in a very special way. We now want to see whether this can be avoided in a higher dimensional set-up. 4

Extra dimensions as a new hope

In the so-called "brane world scenario" matter fields (quarks and leptons, gauge bosons, Higgs bosons) are supposed to be confined to live on a hyper surface (the brane) in a higher dimensional space, whereas gravity and possibly also some additional fields can propagate also in directions transverse to the brane. Such a picture of the universe is motivated by recent developments in (open) string theory 12 and heterotic M-theory 13>47. Since gravitational interactions are much weaker than the other known interactions, the size of the additional dimension is much less constrained by observations than in usual compactifications. In fact, the size of the additional dimensions might be directly correlated to the strength of four-dimensional gravitational interactions 48 . Looking for example on product compactifications of type I string theory it has been noted that it is possible to push the string scale down to the TeV range when one allows at least two of the compactified dimensions to be "extra large" (i.e. up to a /j.m)53. A first look at the question of the cosmological constant does not look very promising. The naive expectation would be that the cosmological constant in the extra (bulk) dimensions As and that on the brane, the brane tension T, should vanish separately. We would then essentially have the same situation as in the four-dimensional case, with the additional problem to explain why also A B has to vanish. The known mechanism of a sliding field 4 5 ' 4 6 can be carried over to this case 49>50>5i,52^ ^ut ^oes n o t g^j a n y n e w ^ ^ Q n ^e

218

question of the cosmological constant. A closer inspection of the situation reveals the novel possibility to have a flat brane even in the presence of a nonzero tension T. For a consistent picture, however, here one also has to require a non-zero bulk cosmological constant AB that compensates the vacuum energy (tension) of the brane. In some way this corresponds to a picture where the vacuum energy of the brane does not lead to a curvature on the brane itself, but curves transverse space and leaves the brane flat. Curvature of the brane can flow off to the bulk, a mechanism that is sometimes called "self-tuning". For such a mechanism to appear we need to consider so-called warped compactifications where brane and transverse space are not just a direct product. We shall see that in this case we can have flat branes embedded in higher dimensional anti de Sitter space, provided certain consistency conditions have been fulfilled. In the following we will be considering the special case that the brane is 1+3 dimensional and we have one additional direction called y. Then the ansatz for the five dimensional metric is in general (M,N = 0 , . . . , 4 and H, v - 0 , . . . 3) ds2 = GMNdxMdxN

= e2A^g^dx»dx"

+ dy2

(3)

where the brane will be localized at some y. We split (4)


into a vacuum value g^v and fluctuations around it h^. For the vacuum value we will be interested in maximally symmetric spaces, i.e. Minkowski space (M 4 ), de Sitter space (dS4), or anti de Sitter space (adS4). In particular, we chose coordinates such that g

=

' diag (-1,1,1,1) J diag ( - 1 , e2VKt,e2VKt,e2VKt^j 2 1

2 x

2 x

diag Ue ^ *,e ^ *,e ^ *,l)

for: M 4 for: dS4

(5)

for: adS4

That means we are looking for 5d spaces which are foliated with maximally symmetric four dimensional slices. Throughout this talk, the five dimensional action will be of the form

R-\

(dcpf - V () Hv-

Vi) • (6)

We allow for situations where apart from the graviton also a scalar <j> propagates in the bulk. The positions of the branes involved are at «/;. With lower

219

case g we denote the induced metric on the brane which for our ansatz is simply 9nv — GMNS^S^

.

(7)

The corresponding equations of motion read RMN - -ZGMNR

- -zdM + o W ) GMN + TV () GMN

+^E^fe-^^tt = 0 (8) i

dv*^G+ I^M (V^GGMNd <j>) -^-g^Hv-Vi) d4> N

d*fi {) = 0. (9)

After integrating over the fifth coordinate in (6) one obtains a four dimensional effective theory. In particular, the gravity part will be of the form S^grav = M2Pl f

tfx^g

(R - A) ,

(10)

where R is the 4d scalar curvature computed from g. The effective Planck mass is Mpt — J dye2A, (note that we put the five dimensional Planck mass to one). Now, for consistency the ansatz (5) should be a stationary point of (10). This leads to the requirement A = 6A. Finally, the on-shell values of the 4d effective action should be equal to the 5-dimensional one. This results in the consistency condition 4 (see also 24 ), j^~

x

= 6A^,.

(ID

It has to be fulfilled for all consistent solutions of the Einstein equations, independently whether the branes are flat or curved. Especially for foliations with Poincare invariant slices the vacuum energy A should vanish. Curved solutions would require a corresponding nonzero value of A. It is this adjustment of the parameters that replaces the traditional four-dimensional fine-tuning in the brane world picture. 5

A first example: the Randall-Sundrum model

As a warm-up example for a warped compactification we want to study the model presented in 1 4 . There is no bulk scalar in that model. Therefore, we put

) = -kB

, f1=T1

, f2=T2,

(12)

220

where AB, T\ and T2 are constants. There will be two branes: one at y — 0 and a second one at y = y0. Denoting with a prime a derivative with respect to y the yy-component of the Einstein equation gives HA')2

= -^f-

(13)

Following 14 we are looking for solutions being symmetric under y -» —y and periodic under y —> y + 2y0. The solution to (13) is

A

=- w 7 - ¥ >

(i4)

where \y\ denotes the familiar modulus function for — yo < y < j/o a n d the periodic continuation if y is outside that interval. The remaining equation to be solved corresponds to the \w components of the Einstein equation, 3A" = -^6(y)-^S(y-Vo).

(15)

This equation is solved automatically by (14) as long as y is neither 0 nor j/oIntegrating equation (15) from —e to e, relates the brane tension T\ to the bulk cosmological constant A B , 7\ = V - 2 4 A B .

(16)

T2 = - V - 2 4 A B .

(17)

Integrating around yo gives

These relations arise due to A = 0 in the ansatz and can be viewed as finetuning conditions for the effective cosmological constant (A in (10)) 16 . Indeed, one finds that the consistency condition (11) is satisfied only when (14) together with both fine-tuning conditions (16) and (17) are imposed. Since the brane tension T, corresponds to the vacuum energy of matter living in the corresponding brane, the amount of fine-tuning contained in (16), (17) is of the same order as needed in ordinary 4d quantum field theory discussed in the beginning of this talk. Next we have to address the important question: What happens if the fine-tunings do not hold? Does this necessarily lead to disaster or do solutions exist also in this case. Indeed it has been shown in 16 that in that case solutions exist, however with A ^ 0. This closes the argument of interpreting conditions (16) and (17) as fine-tunings of the cosmological constant. It also emphasizes the new problem with the adjustment of the cosmological constant on the brane: how to select the flat solution instead of these "nearby" curved solutions that are continuously connected in parameter (moduli) space.

221

6

Self-Tuning

Thus the generic higher dimensional set-up considers nonzero values of brane tensions and the bulk cosmological constant. A fine tuning is needed to arrive at a flat brane with vanishing cosmological constant. Recently an attempt has been made to study the situation with As = 0. We will focus on a " solution" discussed in 21,22 (solution II of the second reference). In this model there is a bulk scalar without a bulk potential V^)=0.

(18)

In addition we put one brane at y = 0, and a bulk scalar with a very specific coupling to the brane via /o () = T0eH

, with: b = T | .

(19)

Observe that this model already assumes fine-tuned values AB and b which would have to be explained. We now make the same warped ansatz (3) as before. If again we assume A = 0 in (5), the bulk equations seem to be solved by A' = ±±', and 3

y+c

•<»>-{ ±!i ifo s* lJV-C

+ d for: y < 0 + d for: y > 0 '

. K

. '

where d and c are integration constants (they would correspond to the vacuum expectation values of moduli fields in an effective low energy description). Observe that with the logarithm appearing in (20) we are no longer dealing with an exponential warp factor as (3) would suggest. As a result of this we have to worry about possible singularities in the solution under consideration. We shall come back to this point in a moment. Finally, by integrating the equations of motion around y = 0 one obtains the matching condition T0 = 4e±'d.

(21)

This means that the matching condition results in an adjustment of an integration constant rather than a model parameter (like in the previously discussed example). So, there seems to be no fine-tuning involved even though we required A = 0. As long as one can ensure that contributions to the vacuum energy on the brane couple universely to the bulk scalar as given in (19) it looks as if one can adjust the vev of a modulus such that Poincare invariance on the brane is not broken. In fact it seems that a miracle has appeared: "solution" (20) is apparently independent of the brane tension To- So if one would add something to To on the brane, the solution does not change. This would also solve the problem of

222

potential contributions to the brane tension in perturbation theory, as they can be absorbed in T 0 . Is this so-called self-tuning of the vacuum energy a solution to the problem of the cosmological constant? Unfortunately not, since there are some subtleties as we shall discuss now. We first notice that the uniform coupling of the bulk scalar to any contribution to the vacuum energy on the brane may be problematic due to scaling anomalies in the theory living on the brane 4 . Apart from that one would have to worry about the correct strength of gravitational interactions. In order to be in a agreement with four dimensional gravity, the five dimensional gravitational wave equation should have normalizable zero modes in the given background. In other words this means that the effective four dimensional Planck mass should be finite. For the model considered with a single brane at y = 0 and c < 0 this implies that J^° dye2A^ should be finite. However, plugging in the solution (20) one finds that this is not satisfied. Following 21 2 2 ' ' this could be solved by choosing c > 0 and simultaneously cut off the y integration at the singularities at \y\ = | c . This prescription then yields a finite four dimensional Planck mass. With this choice of parameters, however, we are approaching disaster. Checking the consistency condition (11) one finds that it is not satisfied anymore. The explanation for this is simple - the equation of motions are not satisfied at the singularities, and hence for c > 0 (20) is not a solution to the equations of motion. It is the singularities that have created the miracle mentioned above. Of course, it has been often observed that singularities appear in an effective low energy prescription, and that at those points new effects (such as massless particles) appear as a result of an underlying theory to which the effective description breaks down at this point. A celebrated example is N = 2 supersymmetric Yang-Mills theory where singularities in the moduli space are due to monopoles or dyons becoming massless at this point 54 . We might then hope that a similar mechanism (e.g. coming from string theory) may save the solution with c > 0 and provide a solution to the problem of the cosmological constant. It should be clear by now, that this new physics at the singularity would be the solution of the cosmological constant problem, if such a solution does exist at all. In the following, we will investigate such a mechanism and see whether it is connected to a potential fine-tuning of the parameters. Does the new physics at the singularity have to know about the actual value of the tension of the brane at y = 0 or does it lead to a relaxation of the cosmological constant independent of T0? To start this discussion, we first modify the theory in such a way that

223

we obtain a consistent solution in which the equations of motion are satisfied everywhere. This can, for example, be done by adding two more branes, situated at \y\ = | c to the setup. We then choose the coupling of the bulk scalar to these branes as follows, f±()=T±eb±*,

(22)

where the ± index refers to the brane at y = ±§c. These two additional source terms in the action lead to two more matching conditions whose solution is 6 + = 6 _ = 6 = T^

and

T+ = T_ = ~T0.

(23)

It is obvious that here a third fine tuning (apart from A# = 0 and b = = F | ) has to be performed. The amount of fine-tuning implied by these conditions is again determined by the deviation of the vacuum energy on the brane from the observed value. Hence, the situation has worsened with respect to the question of fine-tuning. However, we have learned that the consistency condition (11) is a very important tool to analyze the question of the cosmological constant. A short calculation shows that (23) is essential for the consistency condition (11) to be satisfied. 7

Nearby curved solutions

So far, we focused on a very specific model and there remains the question whether this situation is generic. For the given set of parameters we should then scan the available moduli space of solutions parametrized by the values of the bulk cosmological constant A#, the brane tensions T and the various couplings like 6 of the scalars to the brane. It is quite easy to see that the above observation applies in general (for various explicit examples see 4 ) . The reason is the fact that the amount of energy carried away from the brane by the bulk scalar needs to be absorbed somewhere else. In principle, it could flow off to infinity, but, as we have seen explicitely in the last chapter, this cannot happen since in this case we would not be able it to localize gravity on the brane. In fact, it has been shown in 27 that localization of gravity is possible only if there is either a fine-tuning between bulk and brane parameters as observed in the original Randall-Sundrum model or there are naked singularities as in the models of 21>22. For the latter case the consistency condition (11) requires the exactly fine tuned amount of energy from the singularity to match the contribution from the branes. We have seen this explicitly when studying a simple way of "resolving" the singularities by adding new branes with the appropriate tension. However, any other resolution of the singularities will lead to the same conclusions.

224

So far we have concentrated on solutions that lead to flat branes A = 0. A general discussion, however, should also address the question whether the moduli space of solutions also contains "nearby" curved solutions that are continuously connected to the flat solutions discussed so far. If they exist, the solution of the cosmological constant problem would need to supply arguments why the flat solutions are favoured over these "nearby" curved solutions. A first step in this direction would be to study the response of the system once the fine-tuning (which appears after the singularities are somehow resolved) is relaxed. In various cases it has been shown that there exist also solutions with A ^ 0 23 . Moreover, for any fixed value of A they fulfill the consistency condition (11) for that value of A 4 . This means, that relaxing the fine-tuning to zero cosmological constant will lead to a consistent (curved) nearby solution with a non-vanishing effective cosmological constant A. It has been argued in the literature that for the specific model which we discussed above (6 = ± | ) there do not exist any nearby curved solutions 21>23. This would be a rather remarkable result as it would imply that in some way the solution with A = 0 would be unique and potentially stable. Observe that the option of a smooth deformation of b away from |6| = | is not possible since the |6| ^ | solutions are not smoothly connected to the former ones 22 . The above argumention, however, is only true under the asumption that the bulk cosmological constant A# (or the bulk scalar potential) vanishes exactly. The situation in which the scalar field received a nontrivial bulk potential has been studied in 4 with the result that depending on the value of the bulk potential at zero the effective cosmological constant is constrained to a certain non-zero value. Thus also the flat solution with b = ± | is continuously connected to a nearby curved solution with A ^ 0 and nonvanishing bulk potential. In all the known cases we thus see that the moduli space does not contain isolated flat solutions. This is another way of stating the fact that the problem of the cosmological constant has not been solved. So far we have concentrated on the discussion of classical solutions. As we have argued before there is a second aspect of the cosmological constant problem once we consider quantum corrections as well. Generically we would asume that radiative corrections would destroy any fine-tuning of the classical theory if not forbidden by a symmetry. Supersymmetry is an example, but since it is broken in nature at the TeV scale it is not sufficient to stabilize the vacuum energy to the degree of say 1 0 - 3 eV. The special solution \b\ = | enjoys an additional symmetry, a variant of scale invariance. This symmetry, however, has a quantum anomaly and therefore cannot survive in the full quantum theory. A way out would be to postulate a model with unbroken scale (or conformal) invariance, i.e. a finite theory with vanishing /9-function.

225

But as in the case of supersymmetry we know that this symmetry cannot be valid far below the TeV region and thus cannot be relevant for the stability of the cosmological constant. 8

Outlook

From the above discussion it is clear that the brane world scenario gives a new view on the problem of the cosmological constant. However, the present discussion has not provided a satisfactory solution, since in all the known cases a fine-tuning is needed to achieve agreement with observations. In fact this fine-tuning is of the same order of magnitude as the one needed in ordinary four dimensional field theory. More work needs to be done to clarify the situation. One direction would be to analyze in detail the possible implications of broken (bulk and brane) supersymmetry in the general set-up. In the fourdimensional case we need broken supersymmetry MSUSY to be somewhere in the TeV region and also the value of the cosmological constant is essentially determined by MSUSY • In the brane world scenario one could hope to separate these scales. Some gymnastics in numerology would suggest Mg USY /Mpi ailc k to be relevant for the (small but nonzero) size of the cosmological constant. Unfortunately we have not yet found a satisfactory model where such a relation is realized and the problem of the size of the cosmological constant still has to wait for a solution.

Acknowledgements It is a pleasure to thank Stefan Forste, Zygmunt Lalak and Stephane Lavignac for collaboration on the subjects presented in this talk. The help of Stefan Forste in the preparation of the present manuscript is highly appreciated. This work is supported by the European Commission RTN programs HPRNCT-2000-00131, 00148 and 00152. References 1. S. Perlmutter et al. [Supernova Cosmology Project Collaboration], "Measurements of Omega and Lambda from 42 High-Redshift Supernovae," Astrophys. J. 517, 565 (1999) [astro-ph/9812133]. 2. P. M. Garnavich et al., "Constraints on Cosmological Models from Hubble Space Telescope Observations of High-z Supernovae," Astrophys. J. 493, L53 (1998) [astro-ph/9710123]. 3. S. Forste, Z. Lalak, S. Lavignac and H. P. Nilles, "A comment on self-

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DILUTING GRAVITY W I T H COMPACT H Y P E R B O L O I D S MARK TRODDEN Physics

Department,

Syracuse E-mail:

University, Syracuse NY [email protected]

13244-1130,

USA

I give a brief informal introduction to the idea and tests of large extra dimensions, focusing on the case in which the space-time manifold has a direct product structure. I then describe some attractive implementations in which the internal space comprises a compact hyperbolic manifold. This construction yields an exponential hierarchy between the usual Planck scale and the true fundamental scale of physics by tuning only 0 ( 1 ) coefficients, since the linear size of the internal space remains small. In addition, this allows an early universe cosmology with normal evolution up to substantial temperatures, and completely evades astrophysical constraints.

1

Some Background about Large Extra Dimensions

There are a number of talks in this conference about the idea of large extra dimensions. With this in mind I will begin with a brief overview of the general idea and of the constraints on it. 1.1

Motivations : the Hierarchy Problem

The hierarchy problem, for our purposes, is usually posed in the following way: Why is gravity so much weaker than the other forces? To make this concrete compare GN = M~{2 ~ 1(T 33 GeV" 2

(1)

GF = M^2 ~ 1(T 5 GeV" 2

(2)

with

Phrased in this way, the Planck mass Mp\ is considered to be the fundamental scale of physics and the puzzle is the comparative smallness of M\y, or the comparative strength of the electroweak force. The real problem here is not so much the fact that the two scales differ so drastically, but rather that such an arrangement is doomed to be ruined by radiative corrections in a generic quantum field theory. Thus, theoretical efforts have mostly focused on restoring the technical naturalness of this vast difference is scales via, for example, supersymmetry, in which the offending quadratic divergences are absent. 229

230

Given this suggestive phrasing, a logical alternative statement of the problem becomes clear. If we instead imagine that the fundamental scale of physics M* is close to the weak scale Mw, then the hierarchy problem becomes: Why is Mpi so much larger? When considering this possibility however, one must keep in mind some important facts about physics at these scales: • The Electroweak interactions have been tested up to energies E ~ Mw, or equivalently, down to scales d ~ Mwx = 10~ 16 cm • Gravity, in the form of Newton's law has been tested down to a scale d ~ 10 3 3 /M p l ~ 1cm Thus, we know that there are no quantum gravity effects up to energies E ~ Mw, since, for example, there is no evidence of energy lost to gravitons in e + e~ annihilations. Given these constraints, let us examine how it might be that the Planck mass is a comparatively huge derived quantity in a theory in which the weak scale is fundamental. The fundamental idea behind the new constructions is the following1,2. Imagine that space time has 3 + 1 + d dimensions, and that d of these are compact while the remaining 3 + 1 comprise the familiar space-time in which we appear to live. Now make the following two crucial assumptions: • Standard model particles are confined to the 3 + 1 dimensional submanifold. • Gravity is not confined, and therefore gravitons propagate in the bulk These assumptions will turn out to be crucial, since to obtain the necessary hierarchy we will require large volume extra dimensions, and these are ruled out by the precision tests mentioned above if the weak interactions can take place in the bulk. While such a structure may seem somewhat unnatural from the phenomenological viewpoint taken here, it is important to mention that the appropriate behavior occurs readily in some compactifications of string theory. In particular, in Hofava-Witten theory 3 , precisely this division of interactions occurs. The essential M-theoretic ingredient responsible for this is the existence of D-branes, non-perturbative extended objects, in the string spectrum 4 . Open strings must end on D-branes (with "D"irichlet boundary conditions on their endpoints), while the open strings are free to propagate through the whole larger space. Since the excitations of open strings correspond to standard model-like degrees of freedom, while the closed string

231

excitations describe the geometric (gravitational) degrees of freedom, the resulting structure is one with standard model fields confined to a D-brane, and gravity propagating in the full space. To be concrete, let us describe the original large extra dimension scenario. Imagine that the full space-time manifold is M3+1

x Td ,

(3)

3+1

where Ai describes a flat Minkowski or cosmological space which we observe, and Td is a d-dimensional torus of common linear size R describing the internal space. Consider how gravitational flux spreads out around a point mass m, as in figure 1.

Figure 1. Gravitational flux around a point mass in a direct product space with one extra dimension.

The gravitational acceleration experienced by a test particle at distance r from such a point mass obeys the following G (4+rf) 771 for r < R r2+d

G (4+d)» for r » i? (4) Rdr This is easy to understand in terms of Gauss' law : On small scales r <S R gravity is diluted by spreading out into all the dimensions, whereas on large scales r S> R gravity is smeared out over the internal space and can only spread into the 3 + 1 dimensional space. Now, since we understand gravity extremely well on large scales, we must identify G^G(4+d) d

R

^

M"=M2^Rd.

(5)

232

Thus, the observed Planck mass can be huge, even if the fundamental scale of physics is of order a TeV. All that is needed is that the volume of the extra dimensional space (here Rd) is large enough. In particular, for M* ~ 1 TeV, we require #~10¥-17cm.

(6)

For example, • If d = 1, we need R ~ 10 13 cm, which is obviously excluded by any number of large scale tests of gravity. • If d = 2, we need R ~ mm, a truly large extra dimension. So, in a picture with extra dimensions and standard model particles restricted to the brane, the hierarchy problem can be recast in the interesting guise of a mismatch of spatial scales, rather than one of energies. 1.2

Constraints

The constraints on such a possible structure for space-time and particle physics come from three main sources: the laboratory, astrophysics and cosmology. In the laboratory 0 , the relevant quantity is the amplitude for single graviton emission: (d+2) . (7) / M~ This allows one to write the dimensionless graviton emission rate for a process of energy AE as

7l~G(4+d)AE^,^)d+2

.

(8)

There are two important facts that we shall need from this expression. One is that the constraint becomes weaker the more extra dimensions there are, and the second is that the constraint is stronger for higher values of AE. As it turns out, collider constraints 6 are currently not the strongest ones faced by the large extra dimension picture. This can be seen, for example, by considering the process K —> TT+ graviton, for which the branching ratio is / Bi(K -»• irg) ~ I ^

\d+2 J ~ 1(T 12

"For a nice summary of laboratory constraints see

5

for

d=2.

(9)

233

graviton d

d

Figure 2. A feynman diagram for K —> ir+ graviton.

More important are constraints arising from astrophysics. The basic issue is that the Kaluza-Klein (KK) modes in the model are light MKK > -R -1 > 10~4eV, and numerous NKK - M^/Ml < 10 32 . Thus, although each KK mode is very weakly coupled, of order 1/Mpi, to standard model particles on the brane, there are so many of them that they can be copiously produced by energetic processes on the brane. Therefore, it is possible for astrophysical bodies to lose energy by emitting gravitons into the extra dimensions. The most important ways in which this can occur have been termed gravistrahlung, referring to the production of a KK graviton in heavy nucleon collisions, and the gravi-Primakojf process, in which a KK graviton is produced through a photon scattering from a heavy nucleus (see figure 3). As mentioned earlier,

graviton

graviton

,'WWWJ/VW

"Gravistrahlung"

"Gravi-Primakoff

Figure 3. Two processes through which astrophysical bodies can lose energy to KK gravitons.

the most energetic processes yield the tightest constraints, and so it is useful

234

to consider energy loss from the most energetic astrophysical event yet known, supernova 1987A (SN1987A). A careful analysis7 yields M* > 50 TeV

for d = 2

(10)

(and less important constraints on higher values of d). As is often the case in modern particle physics, some of the most important constraints on this structure arise from cosmological considerations. I will just mention two main issues here. As I have described, extra dimensions provide an alternative mechanism through which astrophysical bodies can cool. This remains true for the universe itself. In the standard cosmology, the universe cools adiabatically as the scale factor a(t) increases, with the temperature-time relationship (in the radiation-dominated era)

t

r^j

T2 = H a ' M*

a —

T

(11)

In the new picture, there is a competing evaporative cooling mechanism due to cosmic gravistrahlung, with temperature-time relationship f-p

rrd+3

f ~ je^

•

(12)

Since the standard cosmology is so successful as a description of the universe at temperatures below that of, for example, primordial nucleosynthesis, we require that the new cooling mechanism be sub-dominant at least at temperatures below that. To this end it has become customary to define the normalcy temperature Tt, to be that temperature below which evaporative cooling is negligible compared to the usual adiabatic mechanism. In the basic large extra dimension scenario that I am reviewing here, this temperature is T* ~ 10 MeV for d = 2 ~ 10 GeV for d = 6 (string theory) .

(13)

The second cosmological constraint arises from the possibility of gravitino overproduction. If the underlying theory is supersymmetric, one expects gravitinos to be thermally produced. If they are produced at temperature T, they have a lifetime Ml If M* is too small, these particles may over-close the universe or distort the 7-ray background. These constraints are most relevant for d = 2, in which

235

case they yield M* > O(TeV) > 110 TeV .

(15)

To summarize the combined constraints from the laboratory, astrophysics and cosmology; d — 1 is experimentally ruled out (easily), d = 2 is quite constrained, and d > 2 is relatively unconstrained. 2

Some Interesting Manifolds

I have spent some time giving an overview of the basic concepts behind and constraints on the large extra dimension model, as originally proposed. I would now like to shift gears a little, and describe how these ideas might be extended to the case when the internal manifold is no longer fiat (merely a torus) 8 . More specifically I will be interested in topologically nontrivial internal spaces, and in particular in the case that the internal space is a compact hyperbolic manifold. Compact hyperbolic manifolds (CHMs) 9 , are obtained from Hd, the universal covering space of hyperbolic geometry (that admitting constant negative curvature), by modding out by an appropriate freely acting discrete subgroup T of the isometry group of Hd. (If V is not freely-acting, then the resulting quotient is a non-flat non-smooth orbifold. Such a structure may be related to the Randall-Sundrum models10) Consider space-times of the form M4 x (Hd/T\{iee)

,

(16)

with MA a Friedmann, Robertson-Walker (FRW) 4-manifold, with metric GudzUz-1

= g$(x)dx»dxv

+ R2cg\f{y)dyidy^.

(17)

In this expression Rc is the physical curvature radius of the CHM, so that gij{y) is the metric on the CHM normalized so that its Ricci scalar is 7Z = —1, and /i = 0 , . . . ,3, i = 1 , . . . ,d. Locally negatively curved spaces exist only for d > 2, and the properties of CHMs are well understood only for d < 3. However, it is known that CHMs in dimensions d > 3 possess the important property of rigidity n . As a result, these manifolds have no massless shape moduli, implying that the volume of the manifold, in units of the curvature radius Rc, cannot be changed while maintaining the homogeneity of the geometry. Therefore, the stabilization of such internal spaces reduces to the problem of stabilizing a single modulus, the curvature length or the "radion".

236

Although CHMs may seem like quite abstract objects, their popularity among mathematicians has led to some useful tools for visualizing their structure. In figure 4 I have used the Geomview package 12 to display two examples of CHMs generated by Jeff Weeks' SnapPea program 13 . These are presented

Figure 4. Two CHMs, one small volume, the other large volume.

in the Poincare metric, and what is important here is that those parts of the manifold that are identified under actions under T are shaded in the same way. I would just like to draw attention to two important features. First, the second example has a much larger volume than the first in units of the curvature radius. In addition, the second example is considerably more topologically complex than the first, as can be seen by the much higher number of identifications under T. These features are interrelated, and are quite general. 2.1

Compact Hyperbolic Spaces and Volume

The central feature of CHMs that I will exploit here is the behavior of their volume as a function of linear size in the manifold. A specific example that it is useful to keep in mind is a 3-sphere of radius r, cut out of an H3 of curvature radius Rc. The volume of such a sphere can be calculated exactly, and is given by Vol(r) =

TTR3C

(18)

237

It is useful to examine this expression in two limits. When r -C Rc, the first term in a Taylor expansion yields Vol(r) ~ ^7rr3 ,

(19)

as one would expect, since the manifold looks fiat on these scales. However, in the opposite limit r ^> Rc, we obtain Vol(r) ~ \Rle*lR*

,

(20)

and therefore the volume of the sphere grows exponentially with linear size for large radius. This result remains true for other compact spaces constructed from H3. In general, the total volume of a smooth compact hyperbolic space in any number of dimensions is Vol(CHM) = Rdc ea ,

(21)

where a is a constant, determined by topology. Since the topological invariant ea characterizes the volume of the CHM, it is also a measure of the largest distance L around the manifold. Although CHMs are globally anisotropic, since the largest linear dimension gives the most significant contribution to the volume, one can employ eq. (18), or its generalizations to d ^ 3, to find an approximate relationship between L and Vol(CHM). For L ~S> Rc/2 the appropriate asymptotic relation, dropping irrelevant angular factors, is

e^expp^l.

(22)

Kc

In a model with such a manifold as an extra dimensional space, this relationship allows us to compute the expression for Mp\ in terms of linear distance in the space, to compare with (5) in the flat case. The relevant expression is Mp2, = Ml+dRdcea

~ Ml+dRdc exp " ( d - 1 ) L l (23) Rc Thus, in strong contrast to the flat case, in which Mv\ has a power law dependence on linear size, with a CHM the relationship is exponential. As I'll mention later, the most reasonable and interesting case is the smallest possible curvature radius, Rc ~ M " 1 , since this is the only scale available in the problem. Taking M* ~ TeV then yields L ~ 35JW71 = l(T 1 5 mm .

(24)

Therefore, one of the most attractive features of a CHM internal space is that to generate an exponential hierarchy between M* ~ TeV, and Mp\ requires only that the linear size L be very mildly tuned.

238

2.2

Eigenmodes and Kaluza-Klein

Excitations

I have tried to convince you that CHMs provide an attractive alternative manifold for implementing large extra dimension ideas. However, if this idea is to be taken seriously it is necessary to examine the constraints and possible experimental signatures. To uncover the relevant physics of these models one must consider the spectrum of small fluctuations h in the metric around the background metric. Writing Gu^Gu+e^huiy).

(25)

one sees 3 different types of KK fluctuations • h^v, the spin-2 piece; • hij, with indices only in the internal directions, giving spin-0 fields for the 4D observer; • /iiM, the mixed case, giving spin-1 4D fields. The 4D KK masses of these states are the eigenvalues of the appropriate internal-space Laplacians acting on h(y). For the spin-2 case the relevant operator is the Laplace-Beltrami operator ALB (the Laplacian on scalar functions in the internal space), defined by A i B 0(i/) = \g(y)\-1/2di

(\g(y)\1/2gijdMy))

-

(26)

Although there are no known analytic expressions for the individual eigenvalues of ALB on a CHM of any dimension, some generic properties are known. First, a variational argument shows that the spectrum of ALB is bounded from below, and the lowest eigenmode is just the constant function on the CHM. This zero mode is the internal space wave-function of the massless spin-2 4D graviton. Second, since the internal space in compact, the spectrum of ALB on a CHM is discrete and has a gap between the zero mode and the first excited KK state. The size of this gap is all important. A crucial point is that most of the eigenmodes of ALB on a CHM have wavelengths less than Rc, and the number density of these modes is well approximated by the usual Weyl asymptotic formula6 n(k) = (2n)-dn(d_1)Vdkd-1

,

(27)

'There can also be a few lighter supercurvature modes, with wavelengths as large as the longest linear distance in the manifold, and masses thus bounded below by L~1.

239 where fl(d-i) = Area(5 d _ 1 ). Further, bounds on the value of the first nonzero eigenvalue are known. In the best-studied CHM case of d = 2 it can be proven that a large enough volume (and thus genus) d — 2 CHM will have first eigenvalue > 171/(784^). The analogous conjecture in d = 3 is more problematic, but has also been made 14 . In addition, numerical studies of the spectra of even small volume d = 3 CHMs show that they have very few modes with X < Rc 15 . The basic result is that the first KK modes are exponentially more massive than those in the flat case. This drastically raises the threshold for their production, and as a consequence the astrophysical bounds 16,7 completely disappear since the lightest KK mode has a mass (at least 30 GeV), much greater than the temperature of even the hottest astrophysical object. Similarly the large KK masses imply a much higher normalcy temperature T», up to which the evolution of our brane-localized 4D universe can be normal radiation-dominated FRW. Turning to the spin-O(l) excitations, the detailed form of the Laplacian is modified. However, the Mostow-Prasad rigidity theorem for CHMs of dimension d > 3 implies that ALL has no zero modes, and it is conjectured that the gap to the first excited state is of similar size to the Laplace-Beltrami result that is physically reasonable. Finally for the spin-1 fluctuations hi^ recall that any zero modes would correspond to KK gauge-bosons and are directly related to the continuous isometries of the compact space. But, as a result of the quotient by F, CHMs have no such isometries, and thus there are no massless KK gauge bosons. The non-zero KK modes once again have a mass gap that is at least as large as \/L and is more likely close to ~ 1/RC, as in the previous cases. Thus these additional types of fluctuation should not disturb our results. 2.3

Radion Stabilization

In order for the CHM model to work, it is necessary to realize Rc ~ M~l and ea ~ exp((d — \)L/RC) > 1 consistently with the ansatz of a factorizable geometry, a static internal space, and the vanishing of the 4D long-distance (3> L) cosmological constant (CC) A4. To see how this might work, consider a 3-brane embedded in (4 + d) dimensions, with bulk and brane actions Sbuik = J d4+dx^-\g{i+d)\(M^+2n 5brane = | d 4 x v / - | 5 ^ u c e d | ^ / 4 +

+ A-£m) _^

(28) (29)

240

respectively, where Cm is the bulk matter field Lagrangian, and f4 is the brane tension. Note that, since CHMs are just quotients of Hd, there will exist a uniform negative bulk cosmological constant A ~ M4+d, and that to ensure a static internal space, this must be balanced in the field equations by the small curvature radius of the internal space. Dimensionally reducing these actions yields an effective 4D potential for the radion Rc of the form V{RC) = ARdcea - M*ea{MtRc)d-2

+ W(RC) + f4 .

(30)

Here the first two terms arise from the (4+d) bulk CC term, and the curvature of the internal space. Now, in general we may expand W(RC), which comes from Cm, as a Laurent series in Rc M4

with dimensionless coefficients ap. Broadly speaking, there are then two interesting possibilities. If the determination of the minimum is dominated by a competition between any two terms in V, then the condition that the 4D CC vanish (Vm;n = 0) cannot be achieved with a brane tension such that |/ 4 1 < M4. Thus, such a situation is not consistent with the basic assumption that a low-energy effective theory is valid on the brane Fortunately there is an attractive alternative. If three or more i n dependent terms in V{RC) are all important at the minimum (for example the CC and curvature terms, and one of the matter terms from W) then we can tune the coefficients ap such that Vm\n = 0, without needing f4 2> M4. Thus, the basic assumptions remain consistent. Moreover, this tuning is particularly natural in the CHM case precisely because the minimum will naturally occur for a curvature radius close to the fundamental scale Rc ~ M~x, at which the high-scale theory will produce many different terms that contribute roughly in an equal way. (This is exactly the opposite situation from the large flat extra dimension case where the minimum has to occur at a length scale much greater than M~l.) This one fine-tuning is just the usual 4d CC problem. It seems unlikely to me that this one fine tuning will be solved within these models, since in the end one is left with an arbitrary higher-dimensional cosmological constant that one can add to the theory. Nevertheless, at the very least the cosmological constant problem appears in a different guise in these models. It remains to check one important detail. In the usual large extra dimension scenario the radion moduli problem in the early universe provides quite a strong constraint 17 . In the CHM case this problem is much weakened.

241

The radion, which is the light mode corresponding to dilations of the internal space, gets its mass from the stabilizing potential V(RC). Here _ 1 RlV"{Rc)

_, 1

which is close to Ml ~ TeV 2 . Therefore, the radion lifetime is t ~ M^/M^, much shorter than in the case of flat extra dimensions, and only slightly longer than the age of the universe at nucleosynthesis, even if M* ~ TeV.

3

Conclusions and Further Directions

I have briefly reviewed the structure of theories with large extra dimensions, and discussed the main constraints on these models. I have then described an important modification to these theories, in which the internal manifold comprises a CHM. With this modification, the hierarchy problem is solved by a mild tuning of parameters, in an interesting and topologically stable way. While cosmologically and astrophysically much safer, models with internal compact hyperbolic spaces do have testable signatures accessible to collider experiments. Since KK modes abound close to the fundamental scale, Standard Model particle collisions with center-of-mass energies near this scale will result in the production of KK particles, detectable by a distinctive missing energy signature 6 . Although this is qualitatively similar to the scenario of 10 , the spectrum of KK modes one will see is quite distinctive. A full exploration of these experimental signatures will require a more complete investigation of the spectrum of large CHMs, in particular the issues of isospectrality and homophonicity of such manifolds. It is quite likely that such CHMs have other implications for higher-dimensional physics.

Acknowledgements I would first like to acknowledge my collaborators on CHMs: John MarchRussell, Nemanja Kaloper and Glenn Starkman. I would also like to thank all the organizers of COSMO-2000 for an enjoyable and stimulating conference, and for continuing the now traditional focus on particle cosmology. In addition, I am indebted to the Korean Institute for Advanced Study (KIAS) for support during the conference and hospitality in Seoul.

242

References 1. I. Antoniadis, Phys. Lett. B246, 377 (1990); J. Lykken Phys. Rev. D54 3693 (1996). 2. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B429, 263 (1998); I. Antoniadis, et al, Phys. Lett. B436,257 (1998). 3. P. Hofava and E. Witten, Nucl. Phys. B460, 506 (1996); Nucl. Phys. B475, 94 (1996). 4. J. Polchinski, Phys. Rev. Lett. 75,4724 (1995). 5. G. Landsberg, "Minireview on Extra Dimensions", hep-ex/0009038 (2000). 6. I. Antoniadis, K. Benakli, M. Quiros, Phys. Lett. B 3 3 1 , 313 (1994); G. Giudice, R. Rattazzi and J. Wells, Nucl. Phys. B544,3 (1999); E. Mirabelli, M. Perelstein, M. Peskin, Phys. Rev. Lett. 82, 2236 (1999); T. Han, J. Lykken, R. Zhang, Phys. Rev. D59 105006 (1999). 7. S. Cullen, M. Perelstein, Phys. Rev. Lett. 83 (1999) 268. 8. N. Kaloper, J. March-Russell, G. Starkman and M. Trodden, Phys. Rev. Lett. 85, 928 (2000); [hep-ph/0002001]. 9. See for example: W.P. Thurston, Three-Dimensional Geometry and Topology (Princeton UP, Princeton, 1997). 10. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999); ibid 4690 (1999). 11. G. Mostow, Ann. Math. Stud.78 (Princeton UP, Princeton 1973); G. Prasad, Invent. Math.21 255 (1973). 12. h t t p : / / t h a m e s . n o r t h n e t . o r g / w e e k s / i n d e x / S n a p P e a . h t m l 13. http://www.geomview.org/overview/ 14. R. Brooks, private communication. 15. N. Cornish and D.N. Spergel [math.DG/9906017 ] and references therein. 16. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Rev. D59, 086004 (1999). 17. N. Arkani-Hamed, S. Dimopoulos, N. Kaloper and J. March-Russell, Nucl. Phys. B5667, 189 (2000); [hep-ph/9903224].

R A D I O N P H E N O M E N O L O G Y IN T H E RANDALL S U N D R U M SCENARIO

S. B A E , P. K O , H. S. L E E Department

of Physics, E-mail:

KAIST, Taejon, 305-701, [email protected]

Korea

JUNGIL LEE Deutsches

Elektronen-Synchrotron E-mail:

DESY, D-22603 [email protected]

Hamburg,

Germany

Phenomenology of a radion (<j>) in the Randall-Sundrum scenario is discussed. The radion couples to the trace of energy momentum tensor of the standard model with a strength suppressed only by a new scale (A^,) which is an order of the electroweak scale. In particular, the effective coupling of a radion to two gluons is enhanced due to the trace anomaly of QCD. Therefore, its production cross section at hadron colliders could be enhanced, and the dominant decay mode of a relatively light radion is <j> —¥ gg, unlike the SM Higgs boson case. We also present constraints on the mass m^ and the new scale A^, from the Higgs search limit at LEP, perturbative unitarity bound and the stability of the radion/Higgs potential under radiative corrections.

1

Introduction

Despite the tremendous success, the standard model (SM) has several theoretical drawbacks, one of which is related with stabilizing the electroweak scale relative to the Planck scale under q u a n t u m corrections, which is known as the gauge hierarchy problem. Traditionally, there have been basically two avenues to solve this problem : (i) electroweak gauge symmetry is spontaneously broken by some new strong interactions (technicolor or its relatives) or (ii) there is a supersymmetry (SUSY) which is spontaneously broken in a hidden sector, and superpartners of SM particles have masses around the electroweak scale 0 ( 1 0 0 — 1000) GeV. However, new mechanisms based on the developments in superstring and M theories including D-branes have been suggested by Randall and Sundrum J . If our world is confined to a three-dimensional brane and the warp factor in the Randall and Sundrum (RS) theory is much smaller t h a n 1, then loop corrections cannot destroy the mass hierarchy derived from the relation v = e~hroirv0, where v0 is the VEV of Higgs field (~ O(Mp)) in the 5-dimensional RS theory, e^kr°n is the warp factor, and v is the V E V of Higgs field (~ 246 GeV) in the 4-dimensional effective theory of the RS theory by a kind of dimensional reduction. Especially the extra-dimensional subspace need not be a circle S 1 like the Kaluza-Klein theory 1, and in t h a t 243

244 case, it is crucial to have a mechanism to stabilize the modulus. One such a mechanism was recently proposed by Goldberger and Wise 2 ' 3 , and also by Csaki et al. 4 . In such a case, the modulus (or the radion <j> from now on) is likely to be lighter t h a n the lowest Kaluza-Klein excitations of bulk fields. Also its couplings to the SM fields are completely determined by general covariance in the four-dimensional spacetime, as shown in Eq. (1) below. If this scenario is realized in nature, this radion could be the first signature of this scenario, and it would be i m p o r t a n t to determine its phenomenology at the current/future colliders, which is the purpose of this talk 5 . Some related issues were addressed in Refs. 6 | 7 . In this talk, we first recapitulate the interaction Lagrangian for a single radion and the SM fields in the Randall S u n d r u m scenario, and discuss the decay rates and the branching ratios of the radion into SM particles. Then the perturbative unitarity bounds and stability condition for the radion Higgs potential on the radion mass m^ and A^, are considered. Current bounds on the SM Higgs search can be easily translated into the corresponding bounds on the radion. Then the radion production cross sections at next linear colliders (NLC's) and hadron colliders such as the Tevatron and LHC are considered. Then our results will be summarized at the end. 2

Radion in the Randall S u n d r u m Scenario I

In the RS theory I 1 , the hierarchy problem between the Planck scale Api ~ 10 1 8 GeV and the electroweak scale A E W ~ 10 2 GeV can be solved geometrically via v = e~kVaWvo with krc ~ 12, where the warp factor e~ °n is in the classical RS metric, dsls

= e-2kr^riltvdxitdxv

- r2cdy2.

(1)

T h e radius r c is the VEV of the i/t/-component of metric Gab{x,y). When the bulk field Gyy(x,y) is decomposed into Kaluza-Klein (KK) modes, the lowest lying mode is the massless radion ((f)) in the original RS model I, since there is no tree-level stabilization mechanism for the radion. T h e loop corrections can solve the hierarchy problem, but give too light radion to be consistent with experiments 8 . Therefore, a tree-level stabilization mechanism is needed, and the Goldberger-Wise mechanism 2 is a promising one, because they stabilized the modulus without any severe fine-tuning of the parameters in the full theory. In the Goldberger-Wise stabilization mechanism, there is a bulk scalar field $(x,y) which has large quartic self-interactions only on the hidden and visible branes, and an extra-dimension dependent V E V $ ( y ) . After replacing

245 the field <&(x,y) in the original Lagrangian with its V E V $(j/) and integrating the Lagrangian over y, we have the modulus stabilizing potential. Since there is a potential stabilizing the radion, the radion has a mass of order O(10) GeV at the tree level 3 . From a recent analysis ' , it was known t h a t the one-loop allowed radion mass can be ~ O(10) times larger t h a n the tree-level one, but the radion is still lighter t h a n the Kaluza-Klein modes. 3 3.1

S c a l e A n o m a l y a n d t h e I n t e r a c t i o n L a g r a n g i a n for a R a d i o n Scale

Anomaly

If a Lagrangian possesses no dimensionful parameter, there is a scale symmetry at classical level, for which one can construct a conserved Noether current D/j.. One can show t h a t the improved energy m o m e n t u m tensor 0^ satisfies the following relation : dllD
(2)

For example, if one considers QCD as an example, the improved energy mom e n t u m tensor O' 2 " is given by QV

=

_pa\npav

_j_ _„nv papa pa

+ |?(7"£>" + y Z?")g - g^q(irDa

/g\

- mq) q.

It is clear t h a t this theory has traceless energy m o m e n t u m tensor at classical level if we consider the massless quark limit (without dimensionful parameters) . However, this is no longer true in q u a n t u m field theory ( Q F T ) in which the radiative corrections are usually divergent so t h a t renormalization procedure is called for. In the renormalization, one needs t o regularize the theory with a suitable cutoff parameter to make loop integrals finite. Therefore a hidden cutoff scale is necessarily involved with Q F T . If we use the dimensional regularization instead of cutoff regularization, the dimensionless parameter in four-dimensional theory is no longer dimensionless in D-dimensional theory. And the classical scale s y m m e t r y in 4 — D is no longer a good s y m m e t r y in arbitrary D dimensions. To look into the energy m o m e n t u m tensor Q^ more closely at q u a n t u m level, let us consider the scale dependence of q u a n t u m effective action. Since the renormalized coupling gs(fi) has a scale dependence,

^

= /%*),

(4)

246 one finds t h a t

^

dfi

= P{g>)^

' dgs

= %.

(5)

Therefore we end up with

6£(SM)anom =

Y,

PG 9

} °^ ti{F°uFGl"/).

(6)

G = S U ( 3 ) C , •••

Since the classical scale symmetry is broken by q u a n t u m effects, it is called scale anomaly (like the chiral anomaly) 9 . If there were mass parameters in the classical Lagrangian, the scale s y m m e t r y would have been broken already at classical level so t h a t 0 £ is not zero and this should be considered altogether with the scale anomaly term. 3.2

Interaction

Lagrangian for a Radion

The interaction of the radion with the SM fields at an electroweak scale is dictated by the 4-dimensional general covariance, and is described by the following effective Lagrangian via canonical normalizations of the fields 4 ' 3 :

£int = -r-e£(SM) + ...,

(7)

where A^ = (<j>) ~ 0(v). The radion becomes massive after the modulus stabilization, and its mass m^ is a free parameter of electroweak scale 4 . Therefore, two parameters A^ and m^ are required in order t o discuss productions and decays of the radion at various settings. T h e couplings of the radion with the SM fields look like those of the SM Higgs, except for v —>• A^. However, there is one i m p o r t a n t thing to be noticed : the q u a n t u m corrections to the trace of the energy-momentum tensor lead t o trace anomaly, leading to the additional effective radion couplings to gluons or photons in addition to the usual loop contributions. This trace anomaly contributions will lead to distinct signatures of the radion compared to the SM Higgs boson. T h e trace of energy-momentum tensor of the SM fields at tree level is easily derived :

6£(SM)tree = J2mfff -2m^W+W-^ - m | ^ Z " / + {2mlh2-dllhd'1h) + ...,

(8)

where we showed terms with only two SM fields, since we will discuss two body decay rates of the radion into the SM particles, except the gauge bosons

247 of which virtual states are also considered. T h e couplings between the radion 4> and a fermion pair or a weak gauge boson pair are simply related with the SM Higgs couplings with these particles through simple rescaling : grf,_f_f = gfM - v/As, and so on. On the other hand, the — h — h coupling is more complicated t h a n the SM h — h — h coupling. There is a m o m e n t u m dependent part from the derivatives acting on the Higgs field, and this t e r m can grow up as the radion mass gets larger or the CM energy gets larger in hadroproductions of the radion. It m a y lead to the violation of perturbative unitarity, which will be addressed after we discuss the decay rates of the radion. Finally, there is no h — <j> — <j> coupling, since the radion couples to the trace of the energy m o m e n t u m tensor and there should be no h — <j> mixing after field redefinitions in t e r m s of physical fields. In addition to the tree-level T ^ ( S M ) t r e e , there is also the trace anomaly term for gauge fields 9 , Eq. (6). This trace anomaly couples with the p a r a m eter of conformal transformations in our 3-brane. And the radion plays t h e same role as the parameter of conformal transformation, since it belongs t o the warp factor in the 5 dimensional RS metric 4 . Therefore, the p a r a m e ter associated with the conformal transformation is identified with the radion field <j>. As a result, the radion <j> has a coupling to the trace anomaly term. For QCD sector as an example, one has }CD 2gs

( 2 \ as _ as U n b -^ f)^Z QCD, \ 3 V 8TT = -JZ 8TT

(9)

where nj = 6 is t h e number of active quark flavors. There are also counterparts in the SU{2) x U(l) sector. This trace anomaly has an i m p o r t a n t phenomenological consequence. For a relatively light radion, the dominant decay mode will not be <j> —y bb as in t h e SM Higgs, b u t —>• gg. 4 4-1

Radion Phenomenology Decay Properties

of a Radion

Using the interaction Lagrangian Eq. (7), it is straightforward to calculate the decay rates and branching ratios of the radion into / / , W+W~, Z°Z°, gg and hh.

T{^ff) = Nc^±(l-Xff\

r(* _• w+w~) = ^ - ^ y r ^ ; (i - Xw +

6 2

-x w),

248 m" = ^32TTA T 2^

ZZ

)

3

^

(1 - * z +

-z;>

™3 £/i ^2 r

( ^ ^

=

r(«^^^)

3 2 ^ ^ ^ ( 32^3Ai

1

+

&QC£> +

Y ) ' ^2lq{xq

(10)

where ar /)t y ]Z)/l = 4 m j | B , | Z ] i / m J , and / ( z ) = z[l + (1 - z ) / ( z ) ] with rfj/

/(*)

ln[l - - y ( l - j/)] o 2/ arcsin ( l / ^ / z ) ,

-(^i§)-H

:

z > 1 z < 1.

(11)

Note t h a t as m ( - } oo, the loop function approaches I(xt) —> 2 / 3 so t h a t the top-quark effect decouples and one is left with bqcD with rif = 5. For <j> —• WW, ZZ, we have ignored SU(2)i x C/(l)y anomaly, since these couplings are allowed at the tree-level already, unlike the (j>—g — g or )2T(Alp = v). T h e decay rate scales as (v/A^,) 2 , but the branching ratios are independent of A^. In Fig. 1, we also show the decay rate of the SM Higgs boson with the same mass as <j>. We note t h a t the light radion with A^ = v could be a much broader resonance compared to the SM Higgs even if m^ < 2m\\r. This is because the dominant, decay mode is —>• gg (see Fig. 2), unlike the SM Higgs for which the 66 final state is a dominant decay mode. This phenomenon is purely a q u a n t u m field theoretical effect : enhanced <j> — g — g coupling due to the trace anomaly. For a heavier radion, it turns out t h a t —> VV with V = W or Z dominates other decay modes once it is kinematically allowed. Also the branching ratio for <j> —> hh can be also appreciable if it is kinematically allowed. This is one of the places where the difference between the SM and the radion comes in. If A^ 3> v, the radion would be a narrow resonance and should be easily observed as a peak in the two jets or WW(ZZ)

249

10

! • • • ' ! ' • • • I • • • • I • • . • I • • • • 1 1 • 1 1 I •

.. • I . . .

• I •

100 200 300 400 500 600 700 800 9001000

m,

(GeV)

Figure 1. The total decay rate (in GeV) for the radion <j> f°r m/, = 150 GeV with a scale factor ( A ^ / ^ ) 2 . The decay rate of the SM Higgs boson is shown in the dashed curve for comparison.

100 200 300 400 500 600 700 800 9001000 m , (GeV)

Figure 2. The branching ratios for the radion into the SM particles.

250 final states. Especially —> ZZ -> (II)(I I') will be a gold-plated mode for detecting the radion as in the case of the SM Higgs. Even in this channel, one can easily distinguish the radion from the SM Higgs by difference in their decay width. 4-2

Perturbative

Unitarity

The perturbative unitarity can be violated (as in the SM) in the ViVi —> VLVL or hh —> VLVL, etc. Here we consider hh —> hh, since the <j> — h — h coupling scales like s/A^ for large s = (pht + Ph2)2 • The tree-level amplitude for this process is M(hh-^hh) =

~

3 6 A V

[—^2

+ ^~ZJ

(12)

+ TT^J)

where A is the Higgs quartic coupling, and s + t + u = ^m2h. Projecting out the J = 0 partial wave component (ag) and imposing the partial wave unitarity condition |cto|2 < Im(cto) (i.e. |Re(ao)| < 1/2), we get the following relation among rnh^v^rn^, and A^,, for s » m j , m\ :

l + ml

3A

87rAi

8TT

2m

<\-

(13)

This bound is shown in the lower three curves of Fig. 3. We note t h a t the perturbative unitarity is broken for relatively small A^ < 130 (300) GeV for m<£ ~ 200 GeV (1 TeV). Therefore, the tree level results should be taken with care for this range of A^> for a given radion mass. 4-3

Radion Productions

at hadron and linear

colliders

The production cross sections of the radion at hadron colliders are given by the gluon fusion into the radion through quark-loop diagrams, as in the case of Higgs boson production, and also through the trace anomaly term, Eq. (4), which is not present in the case of the SM Higgs boson : a(pp (or pp) -> <j>X) = K o-ho(gg

-» <j>) I JT

- g(x, Q2)g(r/x,

Q2) dx,

(14)

X

where r = m2,/s and yfs is the CM energy of the hadron colliders (y/s = 2 TeV and 14 TeV for the Tevatron and LHC, respectively). T h e K factor

251

.1200

> O


800

600

200

100 200 300 400 500 600 700 800 9001000

m, (GeV)

Figure 3. The excluded region in the m$ and A^, space obtained from the recent L3 result on the SM Higgs search (the left three curves) and perturbative unitarity bound (the lower three curves).

I

|

I

I

I

|

,

,

10-


10'

r

10

r

1

r

1

1 ' ' ' 1' ' Radior, • Higgs

\\

T

NT"**-.

\

>v

LHC

"I

"""--...

~s

-1

10

T

n

Xjevatron^

^

10~'

mrrn—pmn i

J IV,

*]

l

w~ fcf CTEQ5L 10~

.

i

i

l

,

200

] ,

,

!

,

400

,

600

800

1000

m, (GeV)

Figure 4. The radion production cross section via gluon fusions at the Tevatron (s/s = 2 TeV) and LHC (y/s = 14 TeV) with a scale factor (A^/v)2. The Higgs production cross sections are shown in dashed curves for comparison.

252 includes the QCD corrections, and we set K = 1.5. T h e parton-level cross section for gg —>• <j> is given by 2

Vho(gg

«2.(Q) 256TTA2

1

bqcD +J2 ?

(15)

where I(z) is given in the Eq. (11). For the gluon distribution function, we use the C T E Q 5 L parton distribution functions 1 2 . In Fig. 4, we show the radion production cross sections at the Tevatron and LHC as functions of m^ for A^ = v. We set the renormalization scale Q = m^ as shown in the figure. When we vary the scale Q between m^/2 and 2m^, the production cross section changes about + 3 0 % to —20%. The production cross section will scale as (v/A^)2 as before. Compared to the SM Higgs boson productions, one can clearly observe t h a t the trace anomaly can enhance the hadroproductions of a radion enormously. As in the SM Higgs boson, there is a great possibility to observe the radion up to mass ~ 1 TeV if A^ ~ v. For a smaller A^,, the cross section becomes larger but the radion becomes very broader and it becomes more difficult to find such a scalar. For a larger A^, the situation becomes reversed : the smaller production cross section, but a narrower width resonance, which is easier to detect. In any case, however, one has to keep in mind t h a t the perturbative unitarity may be violated in the low A^, region. At the e+e~ colliders, the main production mechanism for the radion <j) is the same as the SM Higgs boson : the radion-strahlung from Z° and the WW fusion, the latter of which becomes dominant for a larger CM energy 13 . Again we neglect the anomaly contributions here. Since both of these processes are given by the rescaling of the SM Higgs production rates, we can use the current search limits on Higgs boson to get the bounds on the radion. W i t h the d a t a from L3 collaboration 1 4 , we show the constraints of A^ and •Hi,], in the left three curves of Fig. 3. Since L3 d a t a is for y/s = 189 GeV and the mass of Z boson is about 91 GeV, the allowed energy for a scalar particle is about 98 GeV. If the mass of the scalar particle is larger t h a n 98 GeV, then the cross section vanishes. Therefore, if m^ is larger t h a n 98 GeV, there is no constraint on A^. And the forbidden region in m^ — A^, plane is not changed by m/j > 98 GeV, because there is no Higgs contribution to the constraint for rrih > 98 GeV. T h e radion production cross sections at NLC's and the corresponding constant production cross section curves in the (A^, m^) plane are shown in Fig. 5 and Fig. 6, respectively. We have chosen three different CM energies for NLC's : V ^ = 500 GeV, 700 GeV and 1 TeV. We observe t h a t the relatively light radion (m^ < 500 GeV) with A^ ~ v (up to ~ 1 TeV) could be probed at NLC's if one can achieve high enough luminosity, since

253 .10

I I | I I I I I I I I I I I I I I | I I I I I I I I I I I I I I | I I I I | I I I I I I I I I.

CL

>/s=1000GeV Vs=700 GeV Vs=500GeV .

100 200 300 400 500 600 700 800 900 1000 m , (GeV)

Figure 5. The production cross section for the radion at NLC's at y/s — 500 , 700 and 1000 GeV, respectively.

.1200

-T—i—I—i—i—r-

O
800

600

400

200 -

0

200

400

600

800

1000

m , (GeV)

Figure 6. The constant production cross section curves at next linear colliders (NLC's) for y j = 500 , 700 and 1000 GeV

254 the production cross section in this region is less t h a n a picobarn. There are also studies of the radion effects on low energy phenomenology such as muon (g — 2) 6 and the weak mixing angle 1 5 . T h e effects are generally small in the region where perturbative unitarity is not violated. 5

Conclusions

In summary, we demonstrated t h a t the radion t h a t stabilizes the fifth dimensional modulus in the RS I scenario has characteristic signatures at colliders due to the scale anomaly. Were it not for the scale anomaly in Q F T , the radion would have behaved exactly the same as the SM Higgs except t h a t the dimensionful parameter v relevant for the SM Higgs is replaced by the radion decay constant A^. T h e radion would have decayed preferentially into 66 final state for rritf, < 2mw, and into WW/ZZ for a heavier radion (m^ > 2mv=w,z), like the SM Higgs boson. Because of the scale anomaly, however, the situation drastically changes and (f> —>• gg is greatly enhanced over (f> —)• 66, and could be a dominant decay channel for a light radion. Also this enhanced (j> — g — g ((f> — 7 — 7) coupling makes the gluon (photon) fusion into a radion the dominant radion production mechanism at hadron (photon) colliders.

Acknowledgments This work was supported in part by BK21 program (HSL), D F G - K O S E F exchange program (PK) and by K O S E F SRC program through C H E P at Kyungpook National University ( P K ) . References 1. 2. 3. 4.

L. Randall and R. Sundrum, Phys. Rev. Lett. 8 3 , 3370 (1999). W . D . Goldberger and M.B. Wise, Phys. Rev. Lett. 8 3 , 4922 (1999). W . D . Goldberger and M.B. Wise, Phys. Lett. B 4 7 5 , 275 (2000). C. Csaki, M. Graesser, L. Randall and J. Terning, Phys. Rev. D 6 2 , 045015 (2000). 5. S. Bae, P. Ko, Hong Seok Lee, Jungil Lee, Phys. Lett. B 4 8 7 , 299 (2000). 6. U. M a h a n t a and S. Rakshit, Phys. Lett. B 4 8 0 , 176 (2000). 7. G. F. Giudice, R. Rattazzi and J. D. Wells, Nucl. Phys. B 5 9 5 , 250 (2001) ; U. M a h a n t a and A. D a t t a , Phys. Lett. B 4 8 3 , 196 (2000) ; Saebyok Bae and Hong Seok Lee, KAIST-TH-2000-13, hep-ph/0011275, to appear in Phys. Lett. B ; K. Cheung, Phys. Rev. D 6 3 , 056007 (2001).

255 8. J. Garriga, O. Pujolas and T . Tanaka, hep-th/0004109 ; W . D . Goldberger and I.Z. Rothstein, Phys. Lett. B 4 9 1 , 339 (2000). 9. R. Crewther, Phys. Rev. Lett. 28, 1421 (1972) ; M. Chanowitz and J. Ellis, Phys. Lett. B 4 0 , 397 (1972) ; Phys. Rev. D 7, 2490 (1973) ; J. Collins, L. Duncan and S. Joglekar, Phys. Rev. D 16, 438 (1977). 10. W. D. Goldberger and M. B. Wise, Phys. Rev. D 60, 107505 (1999). 11. T . Gherghetta and A. Pomarol, Nucl. Phys. B 5 8 6 , 141 (2000). 12. H.L. Lai et al. ( C T E Q Collaboration), Eur.Phys.J. C 12 375 (2000). 13. E. Accomando et al, Phys. Rep. 2 9 9 , 1 (1998). 14. L3 Collaboration, hep-ex/9909004. 15. Jihn E. Kim, B. Kyae and J. D. Park, SNUTP-00-017, hep-ph/0007008.

COSMOLOGY OF R A N D A L L - S U N D R U M

MODELS

HANG BAE KIM Department

of Physics,

Lancaster

University,

Lancaster

LAI

l^YB,

Great

Britain

There are many interesting issues in the brane world with a large/warped extra dimension. We focus on the cosmological aspects. We review the cosmological solutions of the brane world and how the conventional four-dimensional cosmology is recovered by including the effect of stabilization. Its implications on the mass hierarchy and the cosmological constant are discussed.

1

Introduction

For past two years, many particle physicists and cosmologists were excited by the development of two ideas, the brane world and the warped extra dimensions, both of which are based on the existence of extra dimensions. The basic idea of the brane world is that standard model particles are localized on a (3+l)-dimensional brane (or a set of branes) embedded in a higher dimensional spacetime, while gravity propagates in the whole bulk 1'2. The warped extra dimension assumes that the background metric is curved along the extra dimensions, mainly due to the negative bulk cosmological constant 3 ' 4 . Why are these two ideas so exciting? They have brought us fresh views and perspectives in gravity, cosmology, particle physics and string theory. We have seen many interesting issues discussed so far, such as the localization of gravity, the gauge hierarchy problem, the cosmological constant problem and self-tuning models 5 , the construction of supersymmetric RS models and the role of supersymmetry 6 , the connection to string theory or Horava-Witten model and warped compactification, the interpretation in light of AdS/CFT holographic duality 7 , the collider signatures of the KK modes and the radion, etc 8 . In this talk, we focus on the cosmological aspect of the two ideas, mostly in five dimensional models with one extra dimension. There has been much interest in this because the five dimensional nature of gravity and the brane setup might lead to the non-conventional cosmology even at low temperature as well as at high temperature above TeV scale. It was found that the Friedmann equation of the brane shows a H ex p behavior and has an additional dark radiation term 9 . There was also a difficulty concerning the negative tension brane, and it was not very clear what happens at temperatures above TeV, which can alter the early universe cosmology including inflation. However, in the early models, a few important ingredients are not ad256

257

dressed, such as the mechanism for the localization of fields on the brane, the stabilization of the brane systems 10 and the way to achieve necessary fine tunings of parameters. They inevitably involve bulk matter and dynamics, and can change the whole picture, for example, changing the exponential warp factor to power law. For the cosmological consequences of brane world models, taking the stabilization into account is found to be crucial n>12>13. The inclusion of the effect of stabilization recovers the conventional FRW cosmology at temperatures below TeV. The talk is organized as follows. First, we derive the effective fourdimensional brane equations to see the generalities of brane dynamics, though its usefulness is limited by the lack of the knowledge of bulk effects. Then, we try to solve the five dimensional equations with an appropriate ansatz. The solution can be obtained in very restricted cases, but a framework can be found where we can study brane cosmology with the effect of stabilization taken into account. This is done through the T | component which is adjusted to stabilize the extra dimension in the presence of brane matter. Based on this, we analyze the background spacetime where we also discuss the mass scales and the hierarchy problem, and one and two brane models in turn. 2 2.1

Effective four-dimensional equations on the brane Framework

We consider the five-dimensional spacetime with coordinates ( r , x l , y ) , and 3-branes embedded in it. The action describing our framework is "M 3

= / 'd x^/ —g -— R — A& + CtM JM

+Y2 f rf4*y^w [-A,+c iM \

(i)

i •*

where M is the fundamental gravitational scale of the model, A& and A$ represent the bulk cosmological constant and brane tensions, respectively. CbM and CiM are Lagrangians for the bulk fields and for the fields localized in the branes. To investigate the role of bulk and bulk fields in brane dynamics 14 we introduce a bulk scalar <£, with CbM = j;(d<&)'2. We also allow that A&, A, and Ci can be functions of $. The bulk Einstein equations obtained from the action (1) are GMN

= TMN

= OM^QN^

-

9MN

2(5$)

2

+ A,($)

(2)

258

and the bulk scalar equation is

D$-^0=O.

(3)

The existence of branes imposes the junction conditions on the above bulk equation, giving discontinuities in the first derivatives across the branes of metric and scalar field. First let us proceed the general setup. A brane can be described by the normal vector UM- Then the induced metric on the brane is given by QMN = 9MN — n-MnN, while the extrinsic curvature by KMN — 9MTIN- The junction conditions are [K»A = - ^ ( r ^ - \ g » v T ^

(4)

where the bracket in the left hand side means the difference across the brane and X"M„ is the energy momentum tensor of brane matter, i.e., A,- and £JMSuppose that we are localized on a certain brane and want to study the brane dynamics as observed by us. We may take two different approaches. The first approach is to derive the effective 3-brane equations localized on our brane. The second is to directly solve the whole bulk equations. In this section, we follow the first approach. The second will be dealt with in the next section. 2.2

The effective 3-brane Einstein equation

To derive the effective 3-brane equations, we need to know the extrinsic curvature and the intrinsic curvature of our brane in terms of the bulk metric QMN and the normal vector UM, which are provided by the Codacci equation QMK™ - 8,K = g™GMNnN,

(6)

and the Gauss equation G

2 GMN9™9? ""-3

+ (GMNnMnN

+KKIIV - K™KMv where E^v = CMNOpnMn°g^g^

- ±G)

9lil

- - {K2 - KMNKMN) and CMNOP

9VLV

- £„„,

is the bulk Weyl tensor.

(7)

259

Now we take y as the Gaussian normal coordinates and impose Z^ symmetry, y ~ —y. We expand $ around the brane, $(x,y)

= 4>{x)

+

*I(I)|J/|

+ 2*2(a:)j/2 + C>(2/3)-

(8)

Then from the bulk equations (2) and (3), and the junction conditions (4) and (5), we obtain the equation for

n

dAh

_

T

d(j>

12M

3

(dA .d

_$,

and the four-dimensional effective Einstein equation for the brane G^ = Aeff()srM„ +

(9) 14

T + 77^-n M„ - E^ + nv{4>), 3M 3 M6

-^JJT^ 3

6M "

(10)

where Aeff(<£)

2M 3

A6 +

A2 6M 3

1 UA 8 \dcp

(11) (12)

?liV{) = dll4>dv-\g»u(d<j>)2.

(13)

The first three terms in the right hand side of (10) deal with the sources on the brane. The first two terms are same as four-dimensional Einstein gravity, if we identify the four-dimensional Planck mass A{4>) (14) 6Me' The third term gives a correction quadratic in energy-momentum tensor. The last two terms in Eq. (10) are bulk effect terms, which reflect the existence of bulk in brane dynamics. They are inputs from bulk dynamics and not determined in the brane equations. The brane equations (9) and (10) are not closed equations. In this sense, they would not be very useful without the knowledge of the bulk effect terms. However, they reveal a most general structure of the brane equations. In this regard, we note that the Planck mass (14) seems at first to be determined solely by the brane tension. This seems strange because the graviton (and its zero mode) comes from the bulk fields. Therefore there must be some correlation between the brane tension and bulk dynamics, which are incorporated in the brane equations through these bulk effect terms. We will see this in the following sections where the bulk solutions are treated. Mp2

260

3 3.1

Five-dimensional cosmological solutions Framework

In this section, we investigate the cosmological bulk solutions of fivedimensional models. For the two brane models, the fifth dimension y is assumed to be an orbifold S1 /Z2 with y ~ y + 1 and y ~ —y identified. Two 3-branes reside at two fixed points (boundaries) y = 0, \. Since we are interested in the cosmological solution, we consider the metric where the 3-dimensional spatial section is homogeneous and isotropic. ds2 = -n'2(T,y)d,T2 + a2{r,y)'yijdxidxj

+ b2(r,y)dy2,

(15)

where jij is the 3-dimensional homogeneous and isotropic metric, and we will use K = —1,0,-1-1 to represent its spatial curvature. Einstein equations are given by GMN = (1/M3)TMN where (16)

a2 b2

fa" [a

(17)

n a \ a

nJ )

(18) (19)

f$

= diag[-p,p,_p,j3,p 5 ]+

^ —^-dia.g[-ph

pi, pi, Pi, 0]

(20)

i=0i

In addition to Einstein equations, we use the energy-momentum conservation equation, d^iT^f = 0 dp

„. „

„. a

.„

„ .b

-f + 3(p + p)- + {p + p5)dr

a

P'5+P5 \1 — +3-)+—p-3-p n aj n

b

a

0,

(21)

= 0.

(22)

Brane sources in Eq. (20) can be converted to boundary conditions (Junction conditions): n, a and b must be continuous and n', a1 are discontinuous

261 due to boundary sources by y+

'

b n 3.2

2Pi+3Pi 3M 3

la' y'+ _ ba

'

Pi

3M3'

Five-dimensional spacetime with the bulk cosmological constant and brane tensions

First, we consider the five-dimensional spacetime supported by the negative bulk cosmological constant p = — p = Aj < 0 and brane tensions pi = —pi — A*. We define the parameters

fc

A

1/2

\

=W

'

ki =A

(24)

^

In general cases with ko, —ki/2 > k, the metric is given by

15

+ 8 dx dx + (fcr6 )o;dy ds2 =-dr[krz smh(kb T v e\y\r . + ^c v) :'" 7 + g 2

i

j

2

ij

2

0

0

0

0

(25)

where bo and CQ are determined by /ccosh(co) = ko,

kcosh(-kbo

+ Co) =—k^-

(26)

The metric (25) describes a slice of AdSs space inflating both in extra dimension and in spatial dimensions. We note two special cases. If we have a fine tuning (—k'2 — k)/(ki — k) = ekbo(—hi + k)/{k\ + k), we obtain a inflationary solution with static extra dimension 16 d s

,

=

i -dr2 +• ~e2kT5^iidx dx' j - , —

+, b^ 20^dy2

( 2 ? )

2

sinh (A;6o|j/| + c0) With two fine tunings k = ko — —k±, we can get a static solution, the Randall-Sundrum model 4 ds2 = e-'^oMri^dx^dx"

+ b20dy2

(28)

This model attracted much attention because it convert the gauge hierarchy problem into a geometric problem. The four-dimensional Planck scale in this model is given by Mp = (Mz/k)[l — e~kb°]. Any mass parameter mo on the visible brane corresponds to a physical mass m = moe~^kb°, which we identify as the weak scale. Hence the large hierarchy between the Planck scale and the weak scale Mp/Mw ~ 1016 can be explained by a warp factor e~^kb° with 7;kbo ~ 37. The property is not spoiled by i? 2 corrections, if R2 corrections are given by Gauss-Bonnet interactions 17 .

262

Note that the space described by the metric (25) is locally AdSs, because it shares the same bulk equation. The metric (25) can be transformed to (28) by a coordinate transformation e-WRs«/RS =

fcrsinh^&oy + c0) + g0,

m s = T cosh(kb0y + c 0 ).

(29)

Therefore, we can say that above metrics are the slices of AdSs with different boundary geometries. 3.3

Cosmological solutions with static extra dimension

Let us turn to the more interesting situation where the matter is added on the brane(s) and possibly in the bulk. But we require the extra dimension to be static, that is b = 0. This requires a fine tuning between matter densities. For the bulk energy-momentum, we assume p = Ab,

j3=-A&,

p5 = -Ab+p5(r,y).

(30)

where A& < 0. The form of Ps(r, y) is constrained by Eq. (22) to be PS{T)

(31)

n(T,y)a6{T,y)

With the gauge fixing n(r,y = 0) = 1, we obtain the solution a2(r,y)

18 13

'

= a20

1+

K

F

C +

~A

1/2

sinh(2/c6y)

(32)

and n{r,y) = a(T,y)/a0(T). Here a 0 (r) = a(r,y = 0) and C(T) is determined by i% (T) up to a constant through the equation 2a 0 (r) _ (33) PZ(T). 3M 3 CIO(T) is fixed by the boundary condition. Suppose that a brane with the energy density po is placed at y — 0, and assume Z-2 symmetry y ~ —y. The junction condition (23) at the brane gives the evolution equation for ao(r) C =

Ka0j

ag

V6M 3 /

C_ -A a\

y

!

If we further assume that we are living on the brane, this equation is nothing but the Friedmann equation of our universe. However, in the simple case

263

where A.b — 0 and p 5 = 0, this equation differs from the four-dimensional Friedmann equation in two aspects. First, the Hubble parameter H = a0/a0 is proportional to po instead of y/po. Second, there is a Cja\ term which looks like a radiation term. This means that we can have dynamic solution without any matter on the brane or in the bulk. This term seems to have an interesting interpretation in view of AdS/CFT 7 . Both of these alter the big bang nucleosynthesis result: H ex po is not compatible and C is required to be small enough. If we consider the negative bulk cosmological constant together with the positive brane tension, p 0 = A0 + POM, the above equation is written as

(£)' + 5 = ( « ' *> + T S > M + SJP*« - 1 -

<35>

The four-dimensional Friedmann equation is obtained for k% — k2 = 0, C = 0, POM ^ Ao. We can obtain a viable cosmology for the positive tension brane attached to the infinite size extra dimension. The effective cosmological constant is given by Aeff <x (kfi — fc2) At high energy/temperature, that is, in the very early universe p\M term dominates and results in very interesting consequences 19 . 4 4-1

The stabilized RS models Stabilization by balanced bulk matter

In the previous section, we saw that the brane at y = 0 fixes the whole bulk solution. If we have the second brane at y = -|, the junction condition at y = \ gives _ sinh(/c6) + \{p$ - C - 1) sinh(fc6) - p0 cosh(kb) ^ ~~ cosh(kb) + i(pg - C - l)(cosh(fc6) - 1) - p0 sinh(kb)'

P

(36

^

where

A

= 6fe^ = °'^ ° = lk-

(3?)

This is a constraint between energy densities pi on two branes. The reason why this constraint is necessary is obvious. We used the ansatz with the static extra dimension, which is not the general case for the two brane model. But the (almost) static extra dimension is required from the phenomenological view point. We need a stabilization mechanism to make the extra dimension static.

264

Here we consider a simple modeling of how the stabilization mechanism works 1 3 ' n . It must involve some bulk dynamics. We suppose that it is through the role of C. The basic idea is that the back reaction by the stabilization mechanism to the brane energy densities which, if left alone, would destabilize the extra dimension, induces the bulk energy momentum T55 which forces C fitted to the constraint (36) through (33) and keep 6 = 0. The constraint (36) can be solved to give the necessary C _2 C = (Po-l)-2

po + pi - (1 + Popi) tanh(fcfe) 2 t a n h ( ^ ) { 1 _ pi_ t a n h ( f c 6 / 2 ) } '

(38)

and the bulk energy-momentum component ps through 6M3 ... ,C. (39) 4a {T,y)a{T,y) Inserting (38) into (34), we obtain the 3-brane Friedmann equation for the stabilized two brane system P5(r,y)=

oo\ aoj

2

K__ a.Q

2

3, 3

Po + Pi ~ (1 + popi) tanh(fc6) tanh(kb){1 — px tanh(kb/2)}

We have two comments here. First, in general it is expected that the stabilization mechanism induces T\ as well as T^, and b may not be strictly static but be shifted somewhat as pi changes in time. Considering T55 only and requiring strictly static b seems to give a limit of infinitely steep stabilization potential. It is of course unrealistic, but gives the leading behaviors of RS models with a certain, unknown stabilization mechanism. Second, we can see from (39) that the induced ps does not fix the additive constant of C. This cannot be controlled by T§, but required to vanish to satisfy the constraint. This may require another mechanism behind. Actually non-vanishing C implies the breakdown of conformal symmetry in the bulk, and there might be a connection between the stabilization mechanism which requires non-trivial C and the conformal symmetry breaking. Now we rephrase the metric for the stabilized RS model ds2 = -n2(T,y)dT2 where 6(T, y) = b — constant,

+ a 2 (r, y)6ijdxidxj

+ 6 2 (r, y)dy2

(41)

265

a{r,y) = a 0 (r) cosh(2kby) - po sinh(2kby)

H

{po + Px) cosh(kb) - ( l + p 0 p i ) . ,,.,

—

T-TTTT

,.„,,,

,,

1/2

7^r-{cosh{2kby) - 1}

sinh(/t6) — p±{cosh(kb) — 1}

(42)

and ao(r) satisfies (40). ^.2

The static five-dimensional spacetime: The mass hierarchy and the cosmological constant

With the metric in (41) and (42), we first consider the case where there is no matter and only the bulk cosmological constant and the brane tensions are involved. The metric is now given by a(r,y) = ao(r)n(y) and n(y) = cosh(2kby) — /c0 sinh(2kby) (ko + k±) cosh(kb) — (1 + fco^i) +- sinh(/c6) — ki{cosh(kb) — 1} (cosh(2A%) - 1}

1/2

(43)

From (38) and (39), the balanced ps is found to be P5(y) =

6Mj3fc1.2 n(yy

(k0 + kx) - (1 + k0ki) tanh(A;6) (ko - 1) - 2

tanh(fc6){l - kx tanh(A;6/2)}

(44)

The scale factor a(r, y) undergoes inflation, and the Hubble parameter can be defined independently of y because H = o(r,y)/a(r,y) — do(r)/ao(r). The static background spacetime is obtained when we make a fine tuning to satisfy the condition for the vanishing cosmological constant (kQ + kx) - (1 + k0ki) tanh(A;6) = 0.

(45)

With this condition, (43) and (44) are simplified to n

(y)

=

[cosh(2kby) - ko sinh(2kby)] 6M3(feg -k2) Pn(y) = n(y)4

.

(46) (47)

We can identify the four-dimensional Planck scale in two ways. Firstly, we can get it from the 4-dimensional effective theory which is obtained by integrating

266

the action over the extra dimension, Mf3

2

/ ^ bdyn{yf = — [sinh(fc&) - fc0{cosh(A;6) - 1}] . (48) Secondly, we can deduce it from the 3-brane Friedmann equation (40), which leads to M3 tanh(fcfc){l - ki tanh(kb/2)\ Ml

=

2

L-LJ1

(49)

P

{ k l-kLtanh(kb) ' The two derived Planck scales (48) and (49) coincide under the condition (45), that is, when the cosmological constant vanishes. Let us consider the gauge hierarchy problem in this background spacetime. Physical mass scale at two branes are given by n

Mw

= MII{T, 0) = M,

MH = Mn(r, h=M

(50)

[cosh(A;6) - k0 sinh(fc6)]1/2 .

(51)

Note that we placed the visible brane at y = 0 and identified the physical mass scale of the visible brane as the weak scale. The physical mass scale on the hidden brane, called the hidden scale here, is in general different from the Planck scale. Now the ratio of the electroweak scale and the Planck scale is M% (M\ , . , . , „ r . , / 1 ) N i n , „, (Y J [sinh(A:6) - k0 {cosh{kb) - 1}] w 10 32 . (52)

Mh

The condition for the vanishing cosmological constant and the solution to the gauge hierarchy problem impose two conditions among four parameters k, ko, ki_ and 6. Hence, in this model, there are continuous set of static background spacetimes which solve the cosmological constant problem and the gauge hierarchy problem together, specified by, for example, two parameters (k/M, kb) and ko and ki can be expressed in terms of them from (45) and (52) sinh(kb) - 10 32 (k/M) k ° ~ cosh(Jfei) - 1 ' ~kl = ~~ko + tanh(fefc) = 1 - fco tanh(kb)'

(53)

V

;

The original RS model with k = —ko = fci is a special case where p$ in (47) vanishes. Note that while the hidden scale is larger than the weak scale by the ratio of warp factors at two branes, n(r, | ) / n ( r , 0), it is in general different

267

from the four-dimensional Planck scale by a factor M/k and a different warp factor combination. In the original RS model where k = —ko = fci, ekb S> 1 and M/k ~ 1, the difference disappears. If we introduce another hierarchy M/k ^> 1 (without any feasible reason at the moment), this will show up in the hierarchy of the hidden scale and the Planck scale. 4-3

One brane model

Let us turn to the cases where matters are added on the brane. First, we look at the case pi. = 0. This corresponds to compactifying the extra dimension without the second brane. This is made possible by the tuned ps distribution along the extra dimension 1 6 M d r - -kcoth(kb){p0

Ps

- 3p0) -

PoiPo + 3p0) 12M 3

(55)

The 3-brane Friedmann equation (40) becomes —^ a0J

+^=2k2[-l+ ai

p0coth(kb)}.

(56)

If we split out the brane tension from the brane energy density, the above equation takes the form of four-dimensional Friedmann equation

a0J

kn —coth(fcfc) — 1 ) + coih(kb)poM

aLQ

(57)

with the identifications M3

tanh(fc&),

(58)

Aeff = A0 - (6M 3 A 6 )2 tanh(fcb).

(59)

Mi

An interesting characteristic of this model is its implication for the cosmological constant problem. Suppose that we have the relation k — ko in some way, that is, assume a solution to the 'big' cosmological constant problem. Then the size of the cosmological constant has an exponential dependence on the size of extra dimension. This is just the RS-type solution to the 'small' cosmological constant problem. The currently observed cosmological constant ft A ~ 1 can be fitted with kb « 140.

268

4-4

Two brane model

Now we turn to the two brane case. We split the brane energy densities into the brane tensions and the brane matter energy densities as we did in the one brane model, and impose the vanishing cosmological constant condition (45). The the 3-brane Friedmann equation (40) can be written as POMP^M

tanh(A;6)

+ P\M doV , _, * + ' 6M3fc l-ft 0 tanh(fc6) + = P\M t&nh(kb) tanh(&6/2) aoj % 3MJ, 2 ~ 6MJ.A; l-k0t<mh(kb) K

POM

(60)

where p±M = Pi-Mn{k)4 = P-M [cosh(A;6) — ko sinh(A;6)] , which is the physically observed energy density on the hidden brane. Up to small correction, this equation is nothing but the four-dimensional Friedmann equation. The energy densities on both branes contribute equally. So the matter on the hidden brane acts as dark matter for our brane. The inclusion of the effect of stabilization mechanism, through the balanced T55 component in this simplified model, gives a ordinary FRW cosmology at least up to TeV scale, resolving the peculiarities caused by the existence of the negative tension brane. Above the TeV scale, we meet a complicated situation where in addition that the quadratic terms become important, we need to consider the excitation of other dynamical variables which may spoil the stabilization. Further study is required to clarify it. 5

Conclusion

There are many interesting issues in the brane world models and large/warped extra dimensions, such as the mass hierarchy, the cosmological constant, localization of gravity, confinement of fields on the brane, fine tunings of the bulk cosmological constant and brane tensions, stabilization of the extra dimension, the role of supersymmetry, the role of AdS/CFT duality, connection to string theory, and collider signals, etc. In this talk, we focused on the cosmological implications, together with the cosmological constant and the gauge hierarchy. In cosmological side, the model with one positive tension brane and the infinite warped extra dimension (RS2) has a viable cosmology without the need of stabilization. But for models with two or more branes or with the compact extra dimension, taking the effects of stabilization into account is very crucial in studying the cosmology of the models. We have shown that, through the simple method using the balaced T | component, the inclusion of the effects of stabilization recovers the

269 conventional four-dimensional FRW cosmology. Therefore, the stabilized RS models have viable cosmology below TeV scale, with interesting new perspective on the mass hierarchy and the cosmological constant. Further study is required to clarify the cosmology of these models at temperatures above TeV scale. Acknowledgments I would like to express my gratitute to the organizers of the "International Workshop on Particle Physics and the Early Universe", COSMO2000 and acknowledge the hospitality of KIAS. I have benefited from discussions with K. Choi and H. D. Kim. References 1. V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 1 2 5 , 136 (1983) Phys. Lett. B 1 2 5 , 139 (1983) K. Akama, Lect. Notes Phys. 1 7 6 , 267 (1982). 2. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 4 2 9 , 263 (1998); I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos a n d G. Dvali, ibid. 4 3 6 , 257 (1998). 3. M. Gogberashvili, h e p - p h / 9 8 1 2 2 9 6 ; Europhys. Lett. 4 9 , 396 (2000); Mod. Phys. Lett. A 1 4 , 2025 (1999). 4. L. Randall and R. Sundrum, Phys. Rev. Lett. 8 3 , 3370 (1999); Phys. Rev. Lett. 8 3 , 4690 (1999) 5. N. Arkani-Hamed, S. Dimopoulos, N. Kaloper and R. S u n d r u m , Phys. Lett. B 4 8 0 , 193 (2000); S. Kachru, M. Schulz and E. Silverstein, Phys. Rev. D 6 2 , 045021 (2000); S. Forste, Z. Lalak, S. Lavignac a n d H. P. Nilles, Phys. Lett. B 4 8 1 , 360 (2000); J H E P 0 0 0 9 , 034 (2000); J. E. Kim, B. Kyae, H. M. Lee, h e p - t h / 0 0 1 1 1 1 8 ; h e p - t h / 0 1 0 1 0 2 7 . 6. T. Gherghetta and A. Pomarol, Nucl. Phys. B 5 8 6 , 141 (2000); A. Falkowski, Z. Lalak and S. Pokorski, Phys. Lett. B 4 9 1 , 172 (2000); E. Bergshoeff, R. Kallosh and A. Van Proeyen, J H E P 0 0 1 0 , 033 (2000); M. Zucker, h e p - t h / 0 0 0 9 0 8 3 . 7. S. S. Gubser, h e p - t h / 9 9 1 2 0 0 1 ; N. Arkani-Hamed, M. Porrati, and L. Randall, hep—th/0012148; R. Rattazzi and A. Zaffaroni, hep-th/0012248. 8. C. Csaki, M. L. Graesser a n d G. D. Kribs, Phys. Rev. D 6 3 , 065002 (2001); S. C. Park, H. S. Song and J. Song, h e p - p h / 0 0 0 9 2 4 5 ; 9. P. Binetruy, C. Deffayet, and D. Langlois, Nucl. Phys. B 5 6 5 , 269 (2000)

270

10. W.D. Goldberger and M.B. Wise, Phys. Rev. Lett. 83, 4922 (1999) 0 . DeWolfe, D. Z. Freedman, S.S. Gubser, and A. Karch, Phys. Rev. D62, 046008 (2000) 11. P. Kanti, K. A. Olive and M. Pospelov, Phys. Rev. D62, 126004 (2000); P. Kanti, 1.1. Kogan, K. A. Olive and M. Pospelov, Phys. Lett. B468, 31 (1999); Phys. Rev. D61, 106044 (2000); Phys. Lett. B481, 386 (2000). 12. C. Csaki, M. Graesser, L. Randall and J. Terning, Phys. Rev. D62, 045015 (2000) J. M. Cline and H. Firouzjahi, Phys. Lett. B495, 271 (2000) 13. H. B. Kim, Phys. Lett. B478, 285 (2000) 14. K. Maeda and D. Wands, Phys. Rev. D62, 124009 (2000); T. Shiromizu, K. Maeda and M. Sasaki, Phys. Rev. D62, 024012 (2000). 15. H. B. Kim and H. D. Kim, Phys. Rev. D61, 064003 (2000). 16. N. Kaloper, Phys. Rev. D60, 123506 (1999); T. Nihei, Phys. Lett. B465, 81 (1999). 17. J. E. Kim, B. Kyae and H. M. Lee, Phys. Rev. D62, 045013 (2000); Nucl. Phys. B582, 296 (2000) 18. P. Binetruy, C. Deffayet, U. Ellwanger, and D. Langlois, Phys. Lett. B477, 285 (2000) 19. R. Maartens, D. Wands, B. A. Bassett, I. Heard, Phys. Rev. D62, 041301 (2000)

Precise calculation of neutralino relic density in the minimal supergravity model" Takeshi Nihei Department

of Physics,

Lancaster

University,

Lancaster

LAI

^YB,

UK

We study precise calculation of neutralino relic density in the minimal supergravity model. We compare the exact formula for the thermal average of the neutralino annihilation cross section times relative velocity with the expansion formula in terms of the temperature, including all the contributions to neutralino annihilation cross section analytically. We confirm that the expansion formula fails badly near s-channel poles. We show that the expansion method causes 5-10 % error even far away from the poles.

1

Introduction

In recent years, observations of cosmic microwave background radiation have been providing experimental values for cosmological parameters with some accuracy. Current experimental bound on the fractional energy density of cold dark matter is [2,3] ficDM/i2 = 0.1 - 0.3,

(1)

where h m 0.7 is the parameter in the Hubble constant Ho = 100 h km/sec/Mpc [4]. This bound (1) is expected to be improved in the near future, so precise calculation of the relic density becomes important. Particle physics should provide a particle-theoretical explanation for dark matter. The minimal supergravity model [5] is one of the most promising candidate for new physics beyond the standard model. In this model, a discrete symmetry, so-called R-parity, is introduced to avoid rapid proton decay. One of the consequences of introducing the R-parity is that the lightest superparticle (LSP) is stable, and the LSP becomes a good candidate for cold dark matter [6]. The LSP in this model is mostly the lightest neutralino x which is a mass eigenstate given by a linear combination of neutral gauginos and Higgsinos X

= NnB+N12W3+N13H°

+ NliH°2.

(2)

In general, supersymmetric models have a huge number of unknown parameters. However, the minimal supergravity model has only a small number of parameters, so it has reasonable predictive power. There are five unknown "This talk is based on the work [1].

271

272

parameters in this model m 0 , ml/2,

A, tan/3, sgn(/x),

(3)

where mo, m i / 2 and A represent a common scalar mass, a common gaugino mass, and a common trilinear scalar coupling, respectively. They parametrize soft supersymmetry breaking terms at the GUT scale « 2 x 10 16 GeV. tan/3 is the ratio of vacuum expectation values of the two neutral Higgs fields, y, denotes the Higgs mixing mass parameter. In the case of the minimal supergravity model, the absolute value of n is determined from the condition for consistent radiative electroweak symmetry breaking, and only the sign is a free parameter. In this work, we assume that there are no CP violating phases in these parameters (3) for simplicity. 2

Calculation of the relic density

In this section, we first briefly review the standard calculation of the neutralino relic density [7] in the minimal supergravity model. After that, we explain what is necessary to calculate it precisely. We evaluate the relic density at present, starting from thermal equilibrium in the early universe. The time evolution of the neutralino number density in the expanding universe is described by the Boltzmann equation

+ 3Hnx = -<™M0i> K - nT) >

^

(4)

where H is the Hubble expansion rate. nx describes the actual number density of the neutralino, while nxq is the number density which the neutralino would have in thermal equilibrium, a denotes the cross section of the neutralino pair annihilation into ordinary particles. WM01 is so-called M0ller velocity which can be identified as the relative velocity between the two neutralinos. (eri>M0i) represents the thermal average ofCTWM01• In the early universe, the neutralino is in thermal equilibrium nx = nxq. As the universe expands, the neutralino annihilation process freezes out, and after that the number of the neutralinos in a comoving volume remains constant. In the annihilation cross section a in eq. (4), there are a lot of final states which contribute: \X r~* ff> hh> WW> ZZ' Zh' e t c - A m o n g t h e s e final states, fermion pairs / / usually give dominant contributions. Using an approximate solution to the Boltzmann equation (4), the relic density px = rnxnx at present is given by

= lM

*

w, (10 ^r
(5)

273

where x — T /mx is a temperature of the neutralino normalized by its mass. Tx and T 7 are the present temperatures of the neutralino and the photon, respectively. The suppression factor (Tx/T^)3 « 1/20 follows from the entropy conservation in a comoving volume [8]. Mpi denotes the Planck mass, XF ~ 1/20 is the value of x at freeze-out, and is obtained by solving the following equation iteratively: X F = ln

~

V ^ V 2^{aVM«l)*FXF

(6)

)'

where G;v is the Newton's constant, and g* represents the effective number of degrees of freedom at freeze-out. The relativistic formula [9] for thermal average in eq. (4) can be written as an integration over one variable [10] <<™M0I>

= - 4T„2, 7^7 / dsa(s)(s -4m2)^K1 x 8m'xTKi{mx/T) J4m^

(^ ) , \T J

(7)

where Ki (i = 1,2) are the modified Bessel functions. The cross section a(s) is a complicated function of s in general, so we have to evaluate the above integration numerically to calculate the thermal average. If the normalized temperature x is small enough, we may be able to use an expansion formula in terms of x for the thermal average (7) {crvM0i) = a + bx.

(8)

This expansion, neglecting the higher order of x, is widely used in literatures. In the case that the neutralino is the LSP, the a coefficient for a fermion pair production is proportional to the fermion mass due to the Majorana nature of the neutralino. On the other hand the b coefficient for the same final state includes a contribution which is not proportional to the fermion mass, so the b coefficient is much larger than the corresponding a coefficient for each fermionic final state: a
274

In this work, however, we concentrate on (i) and (ii) to compare the expansion formula with the exact one, and we don't take into account (hi) and (iv) for simplicity. Namely, we use the approximate solution (5) to the Boltzmann equation, and we neglect coannihilation effects. We have derived analytic expressions for the exact annihilation cross section and have obtained a and b coefficients analytically for every contribution including interference terms. These interference terms are neglected in most literatures, but we found that they can give significant contributions in some cases. Some of the analytic expressions can be found in literatures for both the exact cross section [16,17] and the expansion coefficients [18,19]. 3

Thermal average — exact formula vs. expansion

Let's see the rough behavior of the integrand in eq. (7). In the case of interest, the argument of the function K\ is much larger than unity, since T ^ mx/20 and T/J > 2mx. Therefore the thermal average can be written as a convolution of the cross section with a function which decays exponentially: /•OO

(CVM01) ~ /

dsa(s)F(s),

(9)

where the function F(s) has an exponential suppression factor as F(s) <x e-^s'T.

(10)

Naively, it seems that the expansion should converge quickly, since the function F(s) in eq. (10) decays exponentially as s increases [9]. However this is not always true. It is known that the expansion method fails badly when the annihilation cross section changes rapidly with s [10,12,16]. This happens, e.g, near s-channel poles and thresholds of new channels. In the following we compare the expansion result with the exact one, including all the contributions to the annihilation cross section. 4

Numerical results

The spectrum and the couplings in the minimal supergravity model at the weak scale are calculated by solving renormalization group equations with boundary conditions at the GUT scale and implementing radiative electroweak symmetry breaking. In this analysis, we use an existing code "SUSPECT" [20] to calculate the spectrum and the couplings. In the following, we present the results of our calculation. Fig. 1 shows the integration of the thermal average / 0 F dx (avMai) as a function of m ^ 2

500

0 m1/8

Mil! pill

•in Lini

103 10 2 10' r 10° r 10-' r 10-* r 10"»

500

250 500 (GeV)

n 1

r^~~"—^

1:

T

[ I l l

— io* A, 10' > b 10°

iox •O

250

(GeV)

500

10

-;

i "

i I

zz

r r r

1

n

r

__J

r • i l l

il:l i i i

250

V

-» 10"! '1/2

Figure 1: Integration of the thermal average P = 2, m 0 = 200 GeV, A = 0, n > 0.

n 500

(GeV)

F

500 L

i/t

(GeV)

dx (crvM0l) as a function of m 1 / 2 fort

276 tan/J = 2, m 0 1

1

'

1

1

1

r\

1

.

1.5

1


/S

1 \

= 200 GeV, A = 0, p > 0

10

-

01

-/"^

c:

\G

S 0.5

0 w

g

CO a

[ex

Q,

•

,

c:

A l l , , , 250 i1/2 (GeV)

0.5

500

500 '1/2

(a)

(b) 2

Figure 2: The relic density Clh (a) and the ratio n e x p a n s j o n /f2 e xact (b) for tan/3 = 2, mo = 200 GeV, A = 0, /i > 0.

tan/? = 50, 1

i

i

|

i

,

i

i

|

i

i

i

i

m0 |

,

i

1

= 600 GeV, A = 0, /i > 0 1

« c

^

1

'

1

1

1

I

1

1

1

1

1

1

1

1

1

I

1

1

1

1

o CO

" -

•

X

01

£ 0.5 —

" • 3

—

qT

CO

0.

(ex

I

"

"

1a 1

\

i

i

1

l

i

i

i

1

,

i

l

r

1

i

250 500 750 m1/2 (GeV)

(a)

i

i

i

1000

•

•

_

0.5 C)

i

i

i

i

1

i

•

i

i

1

i

i

i

i

1

•

250 500 750 m1/2 (GeV)

i

i

t

1000

(b)

Figure 3: Similar results to Fig. 2 for tan/3 = 50, m0 = 600 GeV, A — 0, /t > 0.

277

for tan/3 = 2, m0 = 200GeV, A = 0, fi > 0. Note that the relic density in eq. (5) is proportional to the inverse of this integration. The solid line and the dashed line represent the exact result and the expansion result, respectively. We have ploted the total contributions and the individual contributions for relevant final states. The exact results show similar behaviors with those in a previous analysis [21]. There are two peaks in the total contributions. One is from the Z boson contribution, and it takes place when 2m x « mz is satisfied. The other is from the lightest Higgs (h) contribution, and it happens when 2mx « m/j. Around the poles, we see huge difference between the exact result and the expansion. Away from the poles, the difference is not so huge. We plot the relic density vs. mi/2 in Fig. 2 (a) for the same parameters as those in Fig. 1. In the region of the poles, we see a dip of the relic density. Away from the poles, the relic density is too large to satisfy the experimental constraint (1). In F i g . 2 ( b ) , w e p l o t t h e r a t i o

fiexpansion/Oexact,

Where ftexact (^expansion)

denotes the relic density calculated with the exact (expansion) formula for the thermal average. In the pole regions, the expansion is quite different from the exact one by a large factor [16]. Also we find a difference of about 5 % in this ratio even far away from the pole. In a very large tan/3 case, the result has a different feature. In Fig. 3 (a) and (b), we show similar results for tan/3 = 50, mo = 600 GeV, A = 0, /i > 0. The relic density is small due to enhancements of the pseudoscalar Higgs exchange contribution and the heavier neutral Higgs exchange contribution in the bb final state for a large tan/3. The are two reasons for these enhancements. First, couplings of the pseudoscalar Higgs and the heavier neutral Higgs to the bottom quark become large for a large tan /3. Secondly, masses of heavy Higgses become smaller in a large tan/3 region. Because of these enhancements in the annihilation cross section, the relic density in Fig. 3 (a) becomes small enough to satisfy the experimental constraint (1) for a wide region of 350 GeV < mi/-2 < 800 GeV. As for the ratio of the expansion result to the exact one in Fig. 3 (b), we see a difference more than 10 % for 600 GeV < m 1 / 2 < 800 GeV. Note that we find this relatively large difference in the interesting region where the relic density satisfies the experimental constraint (1). We show a contour plot for the quantity 2mx — TUA for tan/3 = 50, A = 0 and n > 0 in Fig. 4, where TJIA denotes the pseudoscalar Higgs mass. Along the thin solid line, this quantity vanishes so that there are enhancements in the annihilation cross section from the pseudoscalar Higgs pole contribution and the heavier neutral Higgs pole contribution through the bb final state. Fig. 4 shows that the region 600 GeV < rrii/2 < 800 GeV in Fig. 3 (b), where the

278

tanp = 50,

A = 0, fi > 0 /

"\l00GeV

BOO _ Charged LSP f \ =

/

600

400

/ ^ T

200

OGeV

\

):

i {

20p CeV.

/ y ^ No EWSB

V \ -

-

400

S00

m0

(GeV)

Figure 4: A contour plot of 1mx — THA in the m o - m i / 2 plane for tan 0 = 50, A = 0 and fi > 0. The region 'Charged LSP' is excluded because the lighter stau is the LSP in this region. In the region 'No EWSB', the electroweak symmetry breaking does not occur.

279 deviation of fiexpailsion from fiexact is about 10 %, differs from the pole region (i.e. the zero of the quantity 2mx - TTIA) in m 0 by about 150 GeV. Despite of this difference, we still have a sizable error of 10 % in the expansion result, as we have seen in Fig. 3 (b). In Fig. 2 and Fig. 3, we find 5-10 % difference even away from the pole. Thus the expansion never succeeds to approximate the exact result within the accuracy of 5 %. This implies t h a t the coefficient of the next order ex 2 in the expansion (8) has the same order with the b coefficient c/b « 0(1). 5

Conclusions

We have presented results of precise calculation of neutralino relic density in the minimal supergravity model. We have calculated all the contributions to neutralino annihilation cross section analytically. We have compared the exact formula for the thermal average of the annihilation cross section times relative velocity with the expansion formula in terms of the t e m p e r a t u r e . We have confirmed t h a t the expansion formula fails badly near s-channel poles. We have shown t h a t the expansion method causes 5-10 % error even far away from the poles. Acknowledgments The author was supported in p a r t by P P A R C grant P P A / G / S / 1 9 9 8 / 0 0 6 4 6 . References 1. 2. 3. 4. 5.

T. Nihei, L. Roszkowski and R. Ruiz de Austri, in preparation. P. de Bernardis et.al., Nature 4 0 4 , 955 (2000). A. Balbi et.al, astro-ph/0005124. W. Freedman, Phys. Rep. 3 3 3 , 13 (2000). For reviews on the minimal supergravity model, see for instance, H.P. Nilles, Phys. Rep. 1 1 0 , 1 (1984); P. Nath, R. Arnowitt and A.H. Chamseddine, Applied N = 1 Supergravity, World Scientific, Singapore (1984). 6. J. Ellis, J.S. Hagelin, D.V. Nanopoulos, K A . Olive and M. Srednicki, Nucl. Phys. B 2 3 8 , 453 (1984). 7. For reviews on calculations of the relic density, see for instance, E.W. Kolb and M.S. Turner, The Early Universe, Addison-Wesley (1990); G. J u n g m a n , M. Kamionkowski and K. Griest, Phys. Rep. 2 6 7 , 195 (1996). 8. K.A. Olive, D. Schramm a n d G. Steigman, Nucl. Phys. B 1 8 0 , 497 (1981).

280

9. M. Srednicki, R. Watkins and K.A. Olive, Nucl. Phys. B 310, 693 (1988). 10. P. Gondolo and G. Gelmini, Nucl. Phys. B 360, 145 (1991). 11. H. Goldberg, Phys. Rev. Lett. 50, 1419 (1983). 12. K. Griest and D. Seckel, Phys. Rev. D 43, 3191 (1991). 13. S. Mizuta and M. Yamaguchi, Phys. Lett. B 298, 120 (1993). 14. J. Edsjo and P. Gondolo, Phys. Rev. D 56, 1879 (1997). 15. J. Ellis, T. Falk, K.A. Olive and M. Srednicki, Astropart. Phys. 13, 181 (2000). 16. J.L. Lopez, D.V. Nanopoulos and K. Yuan, Phys. Rev. D 48, 2766 (1993). 17. K. Griest, M. Kamionkowski, and M.S. Turner, Phys. Rev. D 41, 3565 (1990). 18. J. Ellis, L. Roszkowski and Z. Lalak, Phys. Lett. B 245, 545 (1990); K.A. Olive and M. Srednicki, Nucl. Phys. B 355, 208 (1991). 19. M. Drees and M. Nojiri, Phys. Rev. D 47, 376 (1993). 20. A. Djouadi, J.-L. Kneur and G. Moultaka, the code available on the web at http://www.lpm.univ-montp2.fr:7082/~kneur/suspect.html. 21. H. Baer and M. Brhlik, Phys. Rev. D 53, 597 (1996).

MICROLENSING B Y N O N - C O M P A C T ( N O N - B A R Y O N I C ) OBJECTS (NEUTRALINO STARS): THEORY A N D POSSIBLE I N T E R P R E T A T I O N OF OBSERVATIONAL DATA ALEXANDER F. ZAKHAROV Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya, Moscow, 117259, Russia E-mail: [email protected]

25,

Microlensing distant stars by non-compact objects such as neutralino stars is considered. Recently Gurevich and Zybin considered the objects as microlenses. Using the non-singular density distribution we analyse microlensing by non-compact objects. We obtain the analytical solutions of the gravitational lens equation and the analytical expression for the amplification factor of the gravitational lens. We show that using the model of microlensing by non-compact objects it is possible to interpret microlensing event candidates having two typical maximums of light curves which are usually interpreted as binary microlenses.

1

Introduction

The first results of observations of microlensing which were presented in the papers 1>2'3 have discovered a phenomenon, predicted in the papers. 4 ' 5 The basics of microlensing theory and observational data are given in the reviews 6,7,8,9 a n c j m t n e book. io ^ m a t t e r of the gravitational microlens is unknown till now, although the most widespread hypothesis assumes that they are compact dark objects as brown, red or white dwarfs. Nevertheless, they could be presented by another objects, in particular, an existence of the dark objects consisting of the supersymmetrical weakly interacting particles (neutralino) has been recently discussed in the papers. u ' 1 2 The authors showed that the stars could be formed on the early stages of the Universe evolution and to be stable during cosmological timescale. Using the singular model distribution microlensing by non-compact lenses were analysed in the papers. 13'14>15 The geometric optics is used in our model and effects connected with diffraction and mutual interference of images and analysed in the papers 16>17>18 are neglected.

281

282 2

N o n - s i n g u l a r m o d e l for c o m p a c t m i c r o l e n s e s

We approximate t h e density of distribution mass of a neutralino star in t h e following form PNes(r)=2p0-^-^,

r

2

(1)

where r is t h e current value of a distant from stellar center, p0 is a mass density of a neutralino star for a b o u n d a r y of a core (or for a distance rc from a center), rc is a radius of a core. So we use the nonsingular isothermal sphere model (or the model of an isothermal sphere with a core). T h e dependence is approximation of the dependence which has been considered in t h e paper, 8 where the authors considered the model of non-compact object with a core. It is clear t h a t the singular (degenerate) dependence is the limiting dependence of (1) for rc -> 0. So, it is not difficult t o obtain surface density mass, according t o expression (1)

£ ( 0 = Wo0ir?2c /

— ,.„

Jo

— , „d/i -an = = 4p 4po—, , L9 0

c

atan—. . atan-

In t h e case, if R0 3> £, then E ( f ) — • 27rpo _ 7^ fi ^=- In t h a t case the lens equation has the following form

ff=%rZ-Dd,aNeS&,

(2)

i->d

where Ds is a distance from the source t o the observer, Dd is a distance from the gravitational lens t o the observer, Dds ia a distance from the source t o the gravitational lens, vectors (ff, £) define a deflection on the plane of the source and the lens, respectively

«^) = JR2dS--^WZ1--

(3)

We calculate the microlens mass 2

Mx = 8irp0r

c

rRx r2dr / — - = 8irp0r2c(Rx

R - r c a t a n — ) « 8np0r2Rx.

(4)

We use characteristic value of a radius r c , corresponding the microlens "mass" Mx = 8np0r2Rx, thus we obtain lens equation in the dimensionless form. We

283

introduce t h e dimensionless variables by t h e following way x=~,

y=—, Vo

rc Ecr =

,

k(x) =

"

Vo = rc—-, Dd

,

(5)

d2x'k(x')

a(x) = -

—.

As we supposed that surface density is an axial symmetric function then the equation of the gravitational lens may be written in the scalar form 19,10 y = x — a(x) = x

,

mix) — 2 /

x

x'dx'k(x').

Jo

We remind, that we have the following expression for the function k(x) k{x) =

k0

2-jrporo Scr

G DdDds c2 Ds

2TTMX

rcRx

k0

n AGMX DdDds 4rcRx c2 Ds

Hence, the lens equation has the following form y = x-D

n R\ 4rcRx

10

,

(6)

where D — 2fco3

Analysis of the gravitational lens equation

We will show that gravitational lens equation has only one solution if D < 2 and have three solutions if D > 2 and y > yCT (we consider gravitational lens equation for y > 0), where ycr is a local maximal value of right hand of Eq. (6). In order to illustrate the analytical relationships obtained in the previous section, we show in Fig.l the right hand side of the gravitational lens equation for different values of the parameter D. The values xcr and yCT for D > 2 characterize the position of the local maximum of this function. It is possible to show that we determine the value xCT which corresponds to y„ using the following expression 2D - 1 - V4D + 1

2

<

=

7T

.

(7)

284

Figure 1. The right hand side of the gravitational lens equation for various values of the parameter D.

It is easy to see that according to (7) x2r > 0 if and only if D > 2 and „ -v/l + x%r — 1 ya = xa-DX£ ,

(8)

If we choose xCT < 0 then ycr > 0 So, if D < 2 then gravitational lens equation has only one solution for (y > 0), if D > 2 then gravitational lens equation has one solution (if y > ycr), three distinct solutions (if y < yCT), one single solution and one double solution (if y = yCI). It is possible to show that gravitational lens equation is equivalent to the following equation x3 - 2yx2 - (D2 - y2 - 2D)x - 2yD = 0,

(9)

jointly with the inequality x2 - yx + D > 0.

(10)

Thus it is possible to obtain the analytical solutions of gravitational lens equation by the well-known way. We perform z — x — 2j//3 and obtain incomplete equation of third degree z3 + pz + q = 0,

(11)

285 2 y2 and „i q _ _ y fy^2- - D(D + 1) , so we have the where p = 2D - Dl 22 - ?= -?-

O \ y

o

y

following expression for the discriminant

Q=

(f ) 3 + (I)' = %j [-y4 + y 2 ( 2 j ° 2 + 10Z? " 1} + D(2 ~ DW •

(12)

If Q > 0 then Eq. (11) has unique real solution (therefore the gravitational lens equation (6) has unique real solution). We use Cardan expression for the solution -q/2+y/Q+y-q/2-y/Q

+ 2y/3.

(13)

We suppose the case D > 2. If y > ycr then the gravitational lens equation has unique solution. If Q > 0 then we use the expression (13) for the solution. If Q < 0 then we have the following expression x = 2^cos

^±^L+2y/2,,

(fc = 0,l,2)

(14)

where cosa =

q

2x/rW3F

,

(15)

and we select only one solution which corresponds to the inequality (10) which corresponds to k = 0 in (14) because if the gravitational lens equation has only one solution then we have a positive solution x for a positive value of impact parameter y therefore there is the inequality x > y which is easy to see from (8). It is possible to check that maximal solution of (9) corresponds to k — 0 therefore the solution is the solution of (8). If V < 2/cr then the gravitational lens equation has three distinct solutions and we use the Eqs. (14-15) to obtain the solutions. We consider now the case D < 2. We know that the gravitational lens equation has unique solution for the case. If Q > 0 then we use the expression (13) for the solution. If Q < 0 then we have the following expressions (14-15) and we select only one solution which corresponds to the inequality (10) which also corresponds to k = 0 as in the previous case. It is known that magnification for gravitational lens solution Xk is defined by the following expression (J-k

1

£>(vTT^-i)\ L | ^ x / T T ^ - i X

j \

X

1

D

i_ VTT:

, (16)

286 5.0

3.0

1.0 -3.0

-0.0

3.0

Figure 2. The light curve corresponds microlensing by noncompact object for D = 1.9.

so the total magnification is equal

Aot(y) = Y, Vk,

(17)

where the summation is taken over all solutions of gravitational lens equation for a fixed value y. 4

Two types of light curves

Let us consider examples of light curves for various parameter values. Fig. 2 shows a light curve corresponding to microlensing by non-compact body for D = 1.9. In this case the light curve qualitatively resembles the light curve for a compact (Schwarzschild) microlens, as well as noncompact microlens whose density distribution is described by a singular isothermal sphere model when rc = 0. 13 Fig. 3 shows a light curve corresponding to microlensing by a non-compact body when D = 4. When the source has a finite size characterized by the radii Rs = 0.01,0.03 and 0.1, the light curve undergoes a change: its maxima decrease with increasing Rs (fig. 4). In this case, the light curve has two maxima and is qualitatively similar to light curves observed in the MACHO 20 and OGLE 21 experiments (see fig. 5). Such light curves have usually been interpreted as the result of microlensing by binary lens. 9 , e However, they could be associated with microlensing by a single compact gravitational microlens if the external gravitational field is taken into account. 15

287 80.0

60.0

40.0

20.0

0.0 -0.6

-0.2

0.2

0.6

Figure 3. The light curve corresponds microlensing by noncompact object for D = 4. The light curve resembles the light curve for OGLE # 7 event. 60.0

40.0

20.0

0.0 -0.6

-0.2

0.2

0.6

Figure 4. Light curve corresponding to microlensing by a non-compact object for D = 4 and different source radii, namely Ra = 0.01, 0.03 and 0.1.

5

Conclusions

The light curve corresponding to non-compact microlens is shown in Fig.3 (for D = 4). The finite maximal value of the amplification in Fig.3 is connected with a calculation of amplification for a finite set of times and if we consider the

288

750

800

850

1100

1150

J.D. hel. - 2448000

Figure 5. Microlensing event candidate: the OGLE # 7 event, which is often interpreted using a model with binary compact microlens. 2 1 The regions a and b of the caustic intersections are shown on an enlarged scale in the two insets. The MACHO collaboration obtained several dozen additional data points to the OGLE observations) in the two wavelenght bands: these data demonstrated the achromatic character of the light curve.

amplification on the all interval then the maximal value of the amplification must be infinite. It is easy to see that the light curves resemble the light curve for OGLE # 7 event candidate that is usually interpreted by binary lens model. 21 We recall that the appearance of two types of light curves for a toy density distribution model for noncompact object was discussed by Ossipov and Kurian. 23 More detailed analysis of the nonsingular model and its consequences are presented in the paper. 24 Detailed discussion of these degenerate properties of the singular model is given in the paper. 25 Acknowledgements I would like to thank M.V. Sazhin for valuable discussions of various aspects of microlensing. This work was partially supported by the Russian Foundation for Basic Research (project # 00-02-16108). References 1. C. Alcock et al., Nature 365, 621 (1993). 2. E. Aubourg et al., Nature 365, 623 (1993).

289 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

A. Udalski et al., ApJ 426, L69 (1994). A.V. Byalko, AZh 46, 998 (1969). B. Paczinsky, ApJ 304, 1 (1986). B. Paczinsky, preprint astro-ph/9604011 (1996). E.Roulet, S. Mollerach, preprint astro-ph/9603119 (1996). A.V. Gurevich, K.P. Zybin, V.A. Sirota, Uspekhi Phys. Nauk 167, 913 (1997). A.F. Zakharov, M.V. Sazhin, Uspekhi Phys. Nauk 168, 1041 (1998). A.F. Zakharov, Gravitational Lenses and Microlenses, (Janus, Moscow, 1997, in Russian). A.V. Gurevich, K.P. Zybin, Phys. Lett. A 208, 276 (1995). A.V. Gurevich, K.P. Zybin, V.A. Sirota, Phys. Lett. A 214, 232 (1996). A.F. Zakharov, M.V. Sazhin, Journ. Exper. Theor. Phys. 110, 1921 (1996). A.F. Zakharov, M.V. Sazhin, Journ. Exper. Theor. Phys. Lett. 63, 894 (1996). A.F. Zakharov, M.V. Sazhin, Pis'ma v Astron. Zhurn. 23, 403 (1997). A.F. Zakharov, Astron. Astrophys. Trans. 5, 85 (1994). A.F. Zakharov, Astron. Lett. 20, 359 (1994). A.F. Zakharov, A.V. Mandzhos, Journ. Exper. Theor. Phys. 104, 3249 (1993). P. Schneider, J. Ehlers, E.E. Falco, Gravitational Lenses, (Springer, Berlin - Heidelberg - New York, 1992). C. Alcock et al., preprint astro-ph/9606165 (1996). A. Udalski et al., Ap J 436, L103 (1994). V.A. Belokurov, M.V. Sazhin, Phys. Lett. A 239, 215 (1998). D.L. Ossipov, V.E. Kurian, Phys. Lett. A 223, 157 (1996). A.F. Zakharov, Astron. Reports 43, 325 (1999). A.F. Zakharov, Phys. Lett. A 250 67 (1998).

A FEASIBILITY STUDY FOR DARK MATTER SEARCH U S I N G CsI(Tl) CRYSTAL

H.J. AHN, S.K. KIM, T.Y. KIM, I.H. PARK, E.I. WON Physics Department, Seoul National University, Seoul 151-742, Korea M.J. HWANG, H.J. KIM, S.Y. KIM, T.H. KIM, Y.J. KWON Physics Department, Yonsei University, Seoul 120-749, Korea W.K. KANG, Y.D. KIM, E.Y. OH Physics Department, Sejong University, Seoul 143-747, Korea E-mail: [email protected]

Physics Department,

Physics Department,

S.Y. CHOI Chonbuk National University, Chonju, 561-756 Korea M.H. LEE University of Maryland, College Park, MD 20742, USA

T W . KIM Disposal R&D Group, Korea Electric Power Corporation, Daejon 305-600, Korea We report a feasibility study of CsI(Tl) scintillator to use for an experiment of dark matter search. The intrinsic backgrounds from the internal radioisotopes of CsI(Tl) are studied. The photomultiplier tubes matching better for CsI(Tl) has been studied. We also investigated the pulse shape discrimination power of this scintillator using the low energy alpha source. The overall sensitivities expected with this scintillator for future dark matter experiment is estimated.

1

Motivation

Scintillators have been widely used in the experiments for the direct dark matter search 1 . Among them Nal(Tl) , CaF(Eu), Liquid Xe have been used for the previous experiments. Recently DAM A experiment 2 reported a positive signal for a WIMP (Weakly Interacting Massive Particle) interaction with nuclei in Nal(Tl) crystal detector. CsI(Tl) crystal is another interesting scintillator because it has a high scintillation efficiency and a strong pulse shape discrimination power. In this report, we describe the studies on the feasibility of utilizing CsI(Tl) crystal for a WIMP search experiment. The main disadvantages of Csl crystal are the high intrinsic backgrounds from Cesium radioisotopes and smaller number of photoelectrons due to the mismatch between emission 290

291

spectra and the photocathode sensitivities. We studied CsI(Tl) crystals specially on these two points to improve the detector characteristics, so as to have a similar or better sensitivity than the present WIMP search experiments. 2

Background measurements of Csl crystals

To use Csl crystals in ultra low background experiment such as dark matter search 7 ' 8 or neutrino experiment 9 , it is essential to reduce the intrinsic backgrounds of the crystal to the required levels of each experiments. The main intrinsic backgrounds in Csl crystals are conjectured to be due to the Cesium radioisotopes of 137Cs and 13ACs . l37Cs (t1/2 = 30.07 y, Q=1175.6 keV) is the beta-emitter decaying with 95% branching ratio to the meta-stable state(i 1 / 2 = 2.55 minutes) of 137Ba at 661.6 keV state. Therefore the beta electron and the subsequent gamma is not correlated and the backgrounds below 20 keV is significant. l3ACs (i 1 / 2 = 2.065 y, Q=2058.7 keV) is also a beta emitter, but the subsequent gammas are correlated with the beta electron. Therefore the backgrounds due to the 13iCs is not significant at low energies. Another significant background could be 87Rb since Rubidium is very close to Cesium in chemical properties. 87 Rb (ti/2 = 4.75 x 1010y , Q=282.3 keV, 27.8% relative abundance of Rubidium) beta decays to the ground state with 100 % branching ratio, and will generate significant backgrounds at low energies even with small contamination. The expected background spectra in a CsI(Tl) crystal with a size of 10cm x 10cm x 30cm due to 137Cs , 13iCs , and 87Rb are simulated with a Geant3 Monte-Carlo program. 25 identical detectors are assumed to be in 5X5 configuration, and only inner 9 detectors are counted, and assumed the outer layer detectors work as Compton veto counters specially for 13iCs background rejection. Figure 1 shows the expected count rates in the unit of cpd (counts/kg day keV). The background rate below 20 keV energy region is set to be about 0.3 cpd for each radioisotopes totalling to be around lcpd. The contamination levels for such a low background rates should be below O.OOlBq/kg, O.lBq/kg, and 0.2ppb for 137Cs , 134Cs , and 87Rb respectively. There are very few measurements on the contamination level of these isotopes in Csl crystals until now. Texono9 group cited 137Cs contamination in CsI(Tl) crystal less than 0.0128 Bq/kg from the high purity germinium detection, which is much less than one of previous data. Also 87Rb contamination level is set to be less than 9ppb with mass spectrometry by the same group. We measured the background spectra of large CsI(Tl) crystal detectors to obtain 137Cs and 13ACs contamination level at low background environment, and also with HPGe(High Purity Germanium) detector. The crystals were in

292

0.8 0.9 Energy (MeV)

Figure 1. Background count rates expected from 10cm x 30cm Csl crystals in 5X5 configurations

137

Cs

A

Cs , and

87

Rb

for 10cm x

a size of about 5 — 6cm x 5 — 6cm x 30cm. Two 9265B 3" photomultiplier tubes(PMT) were attached at both sides, and the discrimination threshold was set at single photoelectron level. At least 2 photoelectrons are required for each PMT, so the total number of photoelectrons for the hardware trigger was 4. The two PMT signal forms have been digitized with LeCroy digital oscilloscope, and transferred to the PC through GPIB interface. The timing range was 5 usee, and the timing bin was 10 ns. The 1st 1 fisec was used to reject after pulse events in off-line analysis. Our previous study 5 demonstrated that the 6 keV x-rays from 57Co source are cleanly measured with this trigger scheme. The measurements were performed at a site located inside an empty tunnel in a storage water power plant, which is about 400 meter underground, where the muon flux is 1/10000 of that at the ground. The detector was shielded in a simple shielding structure of 2cm copper and 10cm lead layers. We have tested 25 crystals supplied by 3 companies. The normalized background spectra of two typical crystals are shown in Figure 2. The main shapes of the spectra can be explained with the decay of 137Cs , 134Cs , and 87Rb isotopes though there are distinguishing features. The upper spectrum shows a large count rate from the beta decay of 87Rb at low energy, and the bottom spectrum shows the characteristic shapes of the decay from 137 Cs and 13iCs isotopes. The curves in the spectra are the results of fitting

293

with the simulations which include the response of the CsI(Tl) detector to the decays of 137Cs and 134Cs in the energy region of 500-2500 keV. The environmental (extrinsic) background level is not considered here assumed to be small. The contamination level of 87 Kb is also analyzed with ICP-MASS spectrometer for each crystal of two types, and resulted in 20 ppm(a) and < 20 ppb(b) consistent with the measured CsI(Tl) spectra and also with the previous chemical analysis 6 . The Rubidium contents in the crystals depends on the chemical processes in producing the Csl powder. In case of the crystal with high Rubidium contents, the supplier didn't make specail efforts to reduce it 15 ' 6 .

250

500

750

1000

1250

1500

1750

2000

2250 2500 Energy (keV)

250

500

750

1000

1250

1500

1750

2000

2250 2500 Energy (keV)

Figure 2. The background spectra of a 5kg CsI(Tl) crystal in a simple shielding structure.

The 137Cs and 134Cs contamination levels of all detectors tested are shown in the Figure 3. Although there are variations in crystal by crystal by an order of magnitude, the background rates at low energies expected from the contamination level of the three isotopes are too high to perform direct WIMP search experiments. Our preliminary measurements on the commercially available Csl powder shows similar contamination level though it is less. Cesium is obtained mainly from the pollucite ore as Cs2Al2SiiOi2{H20) at

294

T

\14/\ / \l/\ mn

fl Csl37(Bq/kg)

Figure 3. The contamination levels of different suppliers.

137

Cs

and

134

C s . Open and close symbols are for

the mines of Canada, Zimbabwe and Ukraina. It should be interesting to see the contamination level of 137Cs and 134Cs if any in the pollucite ore. At present we are studying the contamination levels in the Csl powder and raw materials in detail. 3

Light yields

The number of photoelectrons from a low energy deposition in Csl crystals is another important issue for the WIMP search experiment. In this respect CsI(Tl) crystal has a disadvantage that the emission spectra don't match well with the normal bialkali photocathodes. We measured the number of photoelectrons with a CsI(Tl) crystal(3cm x 3cm x 3cm) attached to a 3" photomultiplier tubes with different photocathodes. Figure 4 shows the single photoelectron spectra obtained with the events of 6.4 keV energy deposition(left-up corner) and the energy spectra(number of photoelectrons) from 57Co source with a RbCs PMT. For these low energy deposition the photoelectrons are well separated from each other in their arriving time in the signal. The single photoelectron spectrum shows a well separated peak. Table 1 shows the number of photoelectrons for the photons of various energies with a normal bialkali photocathode(9265B PMT) and RbCs photocathode. The number of photoelectrons per keV was about 4.5 and 2.5 with RbCs and normal bialkali

295 Table 1. Number of photoelectrons from each peaks from Numbers in () are the energy resolutions in %. Energy(keV) 6.4 14.4 122.1

57

Co

source for different P M T s .

No. of photoelectrons 9265B RbCs 15.2(35.9) 27.3(29.6) 67.9(21.7) 40.1(26.4) 578(8.12) 375(10.0)

PMT respectively. RbCs photocathode has an enhanced quantum efficiency at around 550 nm region than the normal bialkali photocathode by about factor 3. If we fold the quantum efficiencies14 with the emission spectra of CsI(Tl) crystals, the increase in the number of photoelectrons in case of RbCs photocathodes with respect to normal bialkali tube is found to be 37%, which is consistent with the measurements qualitatively. The quantitative discrepancy between the expectation and measurements at low energies are under investigation.

7000 6000

RbCs Photocathode

5000 4000 3000

VTP*L

2000

X.

1O0O

, , . . , , . . 0.004

O.O06

t^^s^te

0.008

0.O1

0.012

100

200

Cluster size(Arbitrary Unit)

Figure 4. The energy spectra of RbCs P M T with single photoelectron spectrum.

300

400

500

600 n

57

Co

700

(Vc^spe)

source. The inserted graph is the

296 4

Pulse shape discrimination

The pulse shape discrimination(PSD) power of CsI(Tl) crystal has been used at rather high energy even to separate hydrogen isotopes in nuclear and high energy physics experiments. The difference in pulse shapes of various particles is due to the difference of stopping power 13 . There are a few studies on the discrimination power of CsI(Tl) crystal detector at very low energies 7 ' 8 with neutron beam. We have measured the discrimination power between alpha and gamma particles with a CsI(Tl) crystal of size 5cm x 5cm x 5cm. We used a 2WPo alpha source with a continuous energy distribution and various gamma sources. The pulse shape of each signal is parameterized with a single value of mean time, r = 4<^ Ai ' ' . Figure 5 shows the mean time distributions from

E

alpha and gamma sources at different energy ranges. The solid line is a fit 2 for each spectrum with a function of / ( r , w ) = Ti0'1we <'°9,") , where r is the mean time and w is the width of the r distribution. The PSD power dependent on the doping type and concentration is under study now.

Figure 5. Mean time distribution for a, and 7 particles at various energy gates measured with a CsI(Tl) crystal

297

5

Expected sensitivity

We have estimated the sensitivity of CsI(Tl) crystal detector for a direct WIMP search experiment with the scheme of Gaitskell et al. 12 assuming the same quality and quenching factors measured by previous works 7 ' 8 . The WIMP halo parameters were p — 0.3GeV/cm3, v0 = 230km/sec, vesc = 650fcm/sec, and the Helm's form factor was used for Cesium and Iodine nuclei. Figure 6 shows the limits expected with the different background rates( counts/(kev kg day)) and detector masses for spin-independent WIMPnucleon interactions. This simple estimation doesn't include the systematic uncertainties in PSD methods yet. In addition it doesn't include the increase in the quality factor for a large crystal compared with the measurements of small size crystal. Therefore it is a rather optimistic estimation. It should be worth while to improve the qualities of the present Csl crystals for WIMP search experiments to be compared with the recent DAM A experiment.

Figure 6. The limits expected with (a) 100 kg, lOcpd, (b) 100kg, 2cpd, (c)1000kg, 2cpd of CsI(Tl) crystal detector. The threshold is 2 keV in all cases.

6

Conclusion

We have studied the feasibility of CsI(Tl) crystals for a direct WIMP search experiment specially for intrinsic backgrounds and light yields. The 1Z7Cs contamination levels in the present CsI(Tl) crystals are about 0.01 - 0.1 Bq/kg,

298 and the level of 13iCs are 0 - 0.2 Bq/kg. The light yields of 3cm cubic CsI(Tl) crystal with green extended RbCs photocathode was about 4.5 photoelectrons/keV. For a comparable or better experiment than the current Nal(Tl) experiment, the intrinsic background should be reduced more than an order of magnitude and be controlled. Measurements and understanding of the pulse shape discrimination power with large size of crystal is under study. Acknowledgments The authors appreciate the staffs in Korea Electric Power Corporation(KEPCO) for us to use their HPGe detector. The authors acknowledge the financial support by Ministry of Science and Technology through National Creative Research Initiatives(NCRI). This work was partially supported by Korea Research Foundation Grant (1999-015-DP0078). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

G. Jungman et al, Physics Reports 267 195 1996 R. Bernabei et al, Phys. Lett. B 480, 23 (2000) R. Bernabei et al, IL Nuovo Cimento 112 545 1999 CDMS collaboration, Phys. Rev. Lett. 89, 5699 (2000) H.J. Kim et al, NIM A457 471 2001 K. Kazui et al, NIM A394 46 1997 S. Pecourt et al, Astroparticle physics 11 457 1999 V.A. Kudryavtsev et al, NIM A456 272 2001 H.B. Li et al, hep-ex/0001001 2000 R. B. Firestone et al., "Table of Isotopes", 8th Edition, 1997 J.D. Rewin and P.F. Smith, Astroparticle physics 6 87 1996 Gaitskell , Nucl. Phys. B 51, 279 (1996) J.B. Birks, Theory and Practice of Scintillation counter, Pergamon Press , 1964 14. data from Electron Tubes Ltd. 15. private communication with Chemetall company.

INFLATION WITH S U P E R S Y M M E T R Y

BREAKING

L. C O V I Deutsches

Elektronen-Synchrotron E-mail:

DESY, Notkestrasse Germany [email protected]

85, D-22603

Hamburg,

The grand unification scale MQ ~ 10 16 GeV can generate the inflation scale Mj ~ JQ13 —14 Q e v for small inflaton coupling. We show that in this case the scale of supersymmetry breaking, Ms ~ 10 1 0 GeV may dominate the dynamics of the inflationary phase. We study this effect in a hybrid inflation model whose ground state breaks supersymmetry.

1

Introduction

T h e C O B E measurement x of the cosmic microwave background (CMB) anisotropy shows t h a t the scale of inflation Mj has to be lower t h a n the Planck scale, but larger t h a n the electroweak scale. It is therefore clear t h a t low energy supersymmetry may play an important role during inflation. On the other hand we know t h a t supersymmetry is not exact in nature, and t h a t the supersymmetry breaking (SSB), caused by the inflaton displacement from the minimum, can strongly affect the inflationary phase 2 . Models of inflation in a general supergravity framework have been constructed 3 , but it is still very often thought t h a t a globally supersymmetric model with the spontaneous SSB due to the inflaton field is sufficiently accurate to describe the inflationary phase as well as the reheating process, since the low energy SSB terms, which are important in the true vacuum, are suppressed by m3f2/Hj ~ M j / M j during inflation. Here m 3 / 2 — M j / M p is the gravitino mass in the true vacuum and Hi — M] /(\/?>Mp) the Hubble parameter in the inflationary era. As we shall see in the following, this is not always the case. We will present an explicit model where supersymmetry is broken at all stages, studied in detail in 4 . On the one hand, in such a case the SSB sector is influenced by the inflaton dynamics, and the true vacuum, with the present SSB particle spectrum, is reached only at the end of inflation. On t h e other hand, also the inflationary potential is modified by the presence of the "low energy" SSB terms. This modification turns out to be very i m p o r t a n t in the case of a very weakly coupled inflaton field.

299

300 2

The model

We will consider the hybrid inflation scenario based on the superpotential W = WG + Ws = AT (Ml

- Y,2) + M2S (/3 + S) ,

(1)

where T, E and S are chiral superfields a . The first term of (1) has traditionally been used 5 to break a global or local symmetry in supersymmetric theories. Apart from being the simplest choice giving hybrid inflation 6 , it has also the advantage of avoiding large supergravity corrections in the case of a canonical Kahler potential, thanks to the linearity in the inflaton field T. In (1) higher powers of T can be forbidden by R-invariance. In order to break supersymmetry, we add also the second t e r m Ws, the Polonyi potential 7 , where Ms is the SSB scale and is related to the gravitino mass m 3 / 2 — M j (we will use units with Planck mass Mp — 1 if not explicitly s t a t e d ) . For a gravitino mass of order the E W scale, one needs Ms ~ 1 0 1 0 - 1 1 GeV. T h e Polonyi superpotential is also linear in S and in the limit of global supersymmetry the potential is perfectly flat. Hence, early a t t e m p t s have been made to identify the Polonyi field with the inflaton 8 . However, the supergravity corrections turn out to be too large and to spoil the flatness of the potential. T h e constant /? in the Polonyi potential is completely irrelevant in the global supersymmetric limit, but it plays a key role in the supergravity framework. There it breaks R-invariance, and is adjusted t o have a global m i n i m u m with vanishing cosmological constant. As we shall see it also may play an imp o r t a n t role during inflation. Our conclusions will generally apply to any SSB effective potential where the constant and the linear t e r m d o m i n a t e in an expansion in powers of S. Note, t h a t the superpotential (1) is just a variant of the O'Raifeartaigh model 9 . This becomes apparent after a change of variables. Defining

$=

_i^_

+

_^_

*= _^

1ZL_

(2)

a We will consider a singlet S, but all our conclusions can be easily generalized to the case when T interacts with charged fields by the substitution T,2 —> XX, with X(X) charged under a local U(l), or even S 2 —>• Tr | X2 |, with X belonging to the adjoint representation of a non abelian gauge group. In such case an additional D-term would be present, but it can be made to vanish during inflation with additional constraints on the X(X) multiplet(s).

301 with f = M | / ( A M Q ) ~ m3/2/Hi, W =

A

yrr^

one obtains

Ae =*E2 + MJ/3

$(^(i+C2)-^2

yrre

(3)

In this basis, the inflation T and the Polonyi field S are on the same footing, so t h a t it would be n a t u r a l to identify the inflaton with some field in the SSB sector. 2.1

The scalar

potential

Let us consider now the potential coming from (1). T h e supergravity scalar potential is given by 10 V = eK

dW

3\W\7

+ z*W

(4)

where a sum over all fields z,- is implied and we have chosen for all fields canonical Kahler potential, K = J ^ |z;| 2 . In the limit of global supersymmetry Mp —> oo we have V

Mi + \2 Ml

+ A 2 |E| 2 |T| 2 ]

(5)

and the vacuum corresponds to (T) = 0, (|E|) = MQ, while (5) is undetermined. We have computed in 4 the modifications due to the supergravity terms and they correspond just to small shifts of the v.e.v. of the T and E fields and of the Polonyi field, compared to the global supersymmetric limit. Also the constant /? has to be slightly readjusted to keep a vanishing cosmological constant. We could then be t e m p t e d to think t h a t also for the inflationary phase, the supergravity corrections are practically negligible. Let us see if this really happens. Considering t h a t during inflation E is driven to zero by the large v.e.v. of T, the potential is very simply computed in the basis $ , "if. We have at tree level, neglecting t e r m s of order £ 2 , V

~K l\2

\ M" M2

l-2V2Hp
+ \x\2

V2£f3 9 6)

where £ = jj^fr, X denotes the scalar component, of the chiral multiplet ^ and ip is the real part of the scalar part of $ ~ T. Note t h a t we are assuming t h a t ^ C l since otherwise the potential would not be flat enough to inflate. This happens as long as Ms
302 and we will see later t h a t this is not only consistent with but also required by the C O B E normalization. A one loop correction to the potential is also present, coming from the SSB due to the inflaton field and S. Again up to order £ we have AV

A4M4

2,„2 1 - yfefo3) log 2AV

-oW.f

(7)

where /i is the renormalization scale. We see clearly the "spontaneous SSB" as the first term, while all the rest depends on £ oc rriziz/HiIn both expressions (6) and (7), it is straightforward to take the limit of global supersymmetry, and the potential reduces then to the inflationary potential proposed in n , Vsusy

3

2L

\2M%

8TT

2

2,-2 2A-V

log

0{MG/^)

(8)

T h e inflationary phase

In general inflation is supposed to start at large values of T, where both the masses of S and S are large and we can expect t h a t this fields are stabilized at the origin. Inflation then ends at
\2M%

1 - 2^i

/3 f-