Fermi National Accelerator Laboratory

F Fermi National Accelerator Laboratory

FERMILAB-Conf-96/090-A

New Guises for Old Dark-Matter Suspects

Edward W. Kolb Fermi National Accelerator Laboratory P.O. Box 500, Batavia, Illinois 60510

April 1996

Proceedings of the Relativistic Astrophysics: A Conference in Honour of Igor Novikov's 60th Birthday, To be Published by Cambridge University Press

Operated by Universities Research Association Inc. under Contract No. DE-AC02-76CHO3000 with the United States Department of Energy

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New Guises for Old Dark-Matter Susp ects

By ED W AR D W. KOLB

NASA/Fermilab Theoretical Astrophysics Group

Fermi National Accelerator Lab oratory,Box 500, Batavia, Illinois, 60510, USA

The three most p opular susp ects for particle dark matter are massive neutrinos, axions, and

sup ersymmetric relics. There are now well develop ed early-universe plots capable of placing

each of the susp ects at the scene to day to b e the dark matter of the universe. Because Igor

Novikov has contributed in a fundamental way to the idea of a neutrino-dominated universe and

knows that story well, I thoughthewould like to hear something ab out the other two susp ects.

But rather than retelling the standard story, I will p oint out that there may b e new twists to

the dark matter story, and that old, familiar susp ects may hide in unfamiliar guises.

1. Intro duction

One of the simplest questions one might ask ab out the universe is, \What is it made

of ?" It is somewhat of an embarrassment for mo dern cosmologists to b e forced to answer

the question with the shrug of the shoulders. Although to daywe see more of the universe

than ever b efore, everything we do see reinforces the fact that there is more to the universe

than meets the eye. On scales as small as our galaxy to scales as large as the Hubble

radius, most of the mass of the universe seems to b e invisible to us. The nature of the

ubiquitous dark matter is p erhaps the most fundamental question in cosmology to day.

Of particular interest to cosmologists interested in the early universe is the p ossibility

that some fraction of the dark matter is non-baryonic, in the form of some sp ecies of ele-

mentary particle. In most talks on particle dark matter, the rst step in the investigation

is to round up the usual susp ects for identi cation. At the top of the most-wanted list is

the neutrino. Of all the susp ects for dark matter, the neutrino is the only one known to

exist. A mass for the neutrino even as small as a few eV would mean that the neutrino

is dynamically imp ortant in understanding the formation and evolution of structure in

the present universe. Furthermore, without a known principle to require the neutrino to

b e massless, the neutrino cannot use the alibi that it is naturally massless.y

There is still reasonable doubt ab out neutrinos as dark matter b ecause it is very hard

to construct a scenario where light neutrinos, by themselves, are resp onsible for structure

formation. The problem is that by the time neutrinos would come to dominate the mass

of the universe and structure would b egin to form, they would have travelled a rather

large distance since they rst decoupled in the early universe. Because of collisionless

phase mixing or Landau damping, initial p erturbations on scales smaller than the free-

streaming length, 20 Mp c 30 eV =m for neutrinos of mass m ,would b e suppressed.

Such a dark matter sp ecies with a relatively large velo city just b efore matter domination,

likea light neutrino sp ecies, is known as hot-dark matter, or hdm for short.

In order to develop large-scale structure from small initial seeds, most cosmologists

favor a typ e of dark matter known as cold dark matter, or cdm. The matter is called

cold, b ecause at the time when the universe rst b ecame matter dominated the dark

matter was extremely nonrelativistic, and the collisionless damping length was smaller

than any scale of astrophysical interest. The list of dark-matter susp ects is very long,

exceeding in number even the number of characters in a classic Russian novel. But two

y There is no symmetry principle that demands the neutrino must b e massless in the way

that gauge invariance implies that the photon must b e massless. 1

2 EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects

on the cdm list mightbecharacterized as \prime" susp ects: axions and sup ersymmetric

relics. These are the two I will discuss in this talk.

Before starting, it is useful to disp ense with the preliminary de nitions. It is convenient

to express the present mass density of a particle sp ecies in terms of its contribution to

something known as the critical density. If the density of a matter-dominated universe

is greater than the critical density, the universe will eventually recollapse, while if the

density is less than the critical density the universe will expand forever. In terms of

Hubble's constant H and Newton's constant G, the critical densityis 3H =8G =

0 C

2 29 3

1:88h 10 g cm . The annoying factor of h accounts for the imprecision in our

1 1

knowledge of Hubble's constant: H = 100h km s Mp c . The energy densityinsome

sp ecies \i"isgiven as a ratio to the critical density, and denoted as = .

i i C

2. Sup ersymmetric relics

2.1. Supersymmetry and Supersymmetry breaking

There are two fundamental reasons for b elieving that nature is sup ersymmetric. The

rst reason is that sup ersymmetry can rescue the standard electroweak mo del from the

embarrassment of nely tuned coupling constants. The standard electroweak mo del em-

ploys fundamental scalars, usually refereed to as \Higgs" scalars, to break the gauge

symmetry sp ontaneously. But scalar particles havevery bad ultraviolet b ehavior, which

has the e ect of dragging the electroweak Higgs mass up to the mass scale of any encom-

passing theory,such as a grand-uni ed theory. Thus, unless coupling constants are very

nely tuned or some other dynamics enters the picture, light scalar masses of order the

electroweak scale would not b e p ossible. Sup ersymmetry susy is an example of \some

other dynamics." Because of the relative factor of 1 between fermionic and b osonic

lo ops, the addition of fermionic lo ops can mitigate the bad ultraviolet b ehavior of scalar

lo ops. The wayto realize this p ossibilityisifforevery b oson there is a corresp onding

fermion app earing in the calculation of the quantum corrections to the Higgs mass.

This corresp ondence between fermions and b osons implies that both fermions and

b osons app ear in multiplets, and they are transformed into each other by sup ersymmetry

transformations. Thus, susy is intimately related to Poincare symmetry. In fact, the

commutator of susy transformations generates the momentum op erator. susy is the only

knowway to unify spacetime and the internal symmetries of the S -matrix. Thus, susy

seems to b e a fundamental part of any attempt to unify gravity with the fundamental

forces. This aesthetic reason is the second motivation for susy.

The particular realization of susy I will consider is the sup ersymmetric extension

of the standard mo del. Although this mo del, the minimal sup ersymmetric standard

mo del mssm, has many parameters in addition to the plethora of parameters of the

non-symmetric version of the standard mo del, it is suciently restrictivetohavesome

predictivepower.

Of course in nature susy is broken|there is no massless fermionic photon for instance.

I will return to the question of susy breaking in a moment. But the rst relevant

issue for susy dark matter is the existence of something known as R -parity, a discrete

multiplicative symmetry. The R -parity of a particle is given in terms of its spin S ,baryon

3B L+2S

number B and lepton number L,by R =1 . Known particles all haveeven

R -parity, while their susy partners are all R -o dd particles. If R -parity is conserved, then

the lightest R -o dd color singlet lrocsmust b e stable|and hence a candidate for dark

matter. There are inconveniences with any theory if R -parity is broken. Rapid proton

decay, for instance. If one works hard enough these diculties can be overcome, but

EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects 3

the assumption of exact R -parityisvery attractive and naturally leads to dark matter

candidates. So I will assume exact R parity.

Now let's return to the issue of susy breaking. The details of susy breaking will deter-

mine the identity of the lsp,aswell as its mass and interaction strength. Unfortunately,

essentially nothing is known ab out susy breaking. The only reasonable constraintone

might imagine is that any susy-breaking Lagrangian terms must have mass dimension

less than four.y Since wehave no other choice, let's consider all p ossible dimension-two

and dimension-three susy-breaking terms consistent with gauge and Lorentz symmetry:

2 2 2 2 2

LSUSY BREAKING = m jH j m jH j m H H + H H

1 2 1 2

1 2 12 1 2

y y y y y

2 2 2 2 2

e e

e e e e

M e M L e M d L M ue d M Q ue Q

ij Rj ij Lj ij Rj ij Rj ij Lj

Ri Li Ri Ri Li

e e e eu e

L Q

e e e e

H Q h A ue H Q h A d H L h A e

2 Li u u ij Rj 1 Li d d ij Rj 1 Li e e ij Rj

1 1 1

f f

a a a a

g f g f

W G W G : 2.1 M B B M M

1 2 3

2 2 2

The tilde sup erscript denotes the susy partner of familiar particles: e is the selectron, d

a a

f g

is the down squark, G are the gluino elds, W and B are the sup ersymmetric partners

e e

of the familiar SU 2 and U 1 gauge elds, and Q and L are SU 2 doublets containing

the SUSY partners of left-handed quarks and leptons. The elds H and H are the

1 2

two Higgs necessary in susy. The parameters M and A are 3 3 symmetric matrices.

The matrix A has mass dimension one. Note that the op erators app earing in the rst

two lines of 2.1 are op erators of mass dimension two, while the last two lines contain

op erators of mass dimension three.

The usual pro cedure is to cho ose a set of parameters including the constants app earing

in 2.1, requiring that the resulting low-energy theory leads to the usual standard mo del.

The choice of these mass parameters, along with the Higgsino mass parameter , results

in a mass matrix for the neutralinos: the bino B , the zino W , and the Higgsinos H

and H . In terms of the mass of the Z , the weak mixing angle , and tan the

ratio of the vacuum exp ectation values of the two Higgs elds resp onsible for electroweak

f f

0 0

symmetry breaking, the neutralino mass matrix in the basis B; W ; H ; H is given

1 2

1 0

M 0 m cos sin m sin sin

1 Z W Z W

C B

0 M m cos cos m sin cos

2 Z W Z W

C B

A @

m cos sin m cos cos 0

Z W Z W

m sin sin m sin cos 0

Z W Z W

2.2

The susy-breaking masses M and M are commonly assumed to b e of order m or larger,

1 2 Z

and if the susy mo del is emb edded in a grand uni ed theory,then3M =M =5 = .

1 2 1 2

If we assume the relation b etween M and M , then there are three parameters in the

1 2

neutralino mass matrix: ,tan ,andM the zino-bino mass parameter.

1=2

Now the game is set: for a given set of parameters, diagonalize the mass matrix, nd

the mass of the lightest sup ersymmetric particle and its eld content of course in general

it is a linear combination of the four neutralino elds, determine its annihilation cross

section, and put the ab ove information into the freeze-out machinery to determine the

relic abundance.

Di erent groups who study the problem come up with slightly di erent comp osite

y Dimension-four susy-breaking terms su er from the bad ultraviolet b ehavior we are trying to x.

4 EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects

sketches for the dark matter susp ect. Some b elievethe particle content is mostly Hig-

gsino, while some nd mostly bino. However just ab out all groups nd a relatively large

mass for the susp ect, b etween 30 and several hundred GeV.

In this part of the talky Iwould like to prop ose a di erent picture for the wanted p oster:

aparticleoflow mass 500 MeV to 1.6 GeV and \photino-like," with an SU 2 U 1

content almost identical to the photon.

The motivation for this light photino comes from our lackofknowledge ab out susy-

breaking. Referring to the susy-breaking terms in 2.1 we see that there are dimension-

two and dimension-three terms. There are theoretical reasons to b elieve that dimension-

three terms mightbemuch smaller than the dimension-two terms. It app ears dicult

to break susy dynamically in a way that pro duces dimension-three terms while avoiding

problems asso ciated with the addition of gauge-singlet sup er elds. In mo dels where

susy is broken dynamically or sp ontaneously in the hidden sector and there are no

gauge singlets, all dimension-three susy-breaking op erators in the e ective low-energy

theory are suppressed compared to susy-breaking scalar masses by a factor of hi=m ,

where hi is the vacuum exp ectation value of some hidden-sector eld and m is the

Planck mass. Thus, dimension-three terms may not contribute to the low energy e ective

Lagrangian. This would imply that at the tree level the gluino is massless, and the

neutralino mass matrix is given by 2.1 with vanishing 00 and 11 entries. However,

non-zero contributions to the gluino mass and the neutralino mass matrix come from two

sources: radiative corrections such as the top-stop lo ops for the gluino and neutralinos,

and \electroweak" lo ops involving higgsinos and/or winos and binos for the neutralinos

but not for the gluino.

The generation of radiative gaugino masses in the absence of dimension-three susy

breaking was studied byFarrar and Masiero. They found taking 40 GeV that as

the typical susy-breaking scalar mass, M ,varies b etween 100 and 400 GeV, the gluino

mass ranges from ab out 700 to ab out 100 MeV, while the photino mass ranges from

around 400 to 900 MeV. This estimate for the photino mass should be considered as

merely indicative of its p ossible value, since an approximation for the electroweak lo op is

strictly valid only when and M are much larger than m . The other neutralinos are

0 W

muchheavier, and the pro duction rates of the photino and the next-lightest neutralino

in Z decay are consistent with all known b ounds.

The conclusion is that light gluinos and photinos are quite consistent with present

exp eriments, and result in a numb er of striking predictions. One prediction is that the

photino e should b e the relic R -o dd particle, even though it may b e more massivethan

the gluino. This is b ecause b elow the con nement transition the gluino is b ound with a

gluon into a color-singlet hadron, the R , whose mass which is in the 1 to 2 GeV range

when the gluino is very light is greater than that of the photino. In this scenario, lsp

is an ambiguous term: the gluino is lighter than the photino, although the photino is

lighter than the R . As discussed ab ove, a more relevant term would b e lrocs|lightest

R -odd color singlet.

However, mo dels with lightgauginoswere widely thought to b e disallowed b ecause it

had b een b elieved that in such mo dels the relic density of the lightest neutralino would

exceed cosmological b ounds unless R -paritywould b e violated allowing the relic to decay.

In the next subsection I will rehash that argument, and then p oint out how the restriction

can b e evaded if the R mass is close to the e mass here \close" means within a factor

of two.

y Reference to all the material presented here can b e found in Farrar and Kolb 1996 and

Chung, Farrar, and Kolb 1996

EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects 5

2.2. Self-annihilation and freeze out

The reaction rates that determine freeze out will dep end up on the e and R masses, the

0 0

cross sections involving the e and R , and p ossibly the decay width of the R as well. In

turn the cross sections and decay width also dep end on the masses of the e , eg and R ,

as well as the masses of the squarks and sleptons. We will denote the squark/slepton

masses by a common mass scale exp ected to b e of order 100 GeV. Even if the masses

were known and the short-distance physics sp eci ed, calculation of the width and some

of the cross sections would b e no easy task, b ecause one is dealing with light hadrons.

Fortunately, our conclusions are reasonably insensitive to individual masses, lifetimes,

and cross sections, but dep end crucially up on the R -to- e mass ratio, denoted by r .

When we do need an explicit value of the photino mass m, the R mass M , or the

squark/slepton mass M , we will parameterize them by the dimensionless parameters

, r ,and :

8 S

m =800 MeV ; M = rm; M =100 GeV: 2.3

8 S

The standard pro cedure for the calculation of the presentnumb er density of a thermal

relic of the early universe is to assume that the particle sp ecies was once in thermal

equilibrium until at some p oint the rates for self-annihilation and pair-creation pro cesses

b ecame much smaller than the expansion rate, and the particle sp ecies e ectively froze

out of equilibrium. After freeze out, its numb er density decreased only b ecause of the

dilution due to the expansion of the universe.

Since after freeze out the numb er of particles in a comoving volume is constant, it is

convenient to express the numb er density of the particle sp ecies in terms of the entropy

density, which in a comoving volume element is also constant for most of the history

of the universe. This numb er-density-to-entropy ratio is usually denoted by Y . If a

sp ecies of mass m is in equilibrium and nonrelativistic, Y is given simply in terms of

3=2

the mass-to-temp erature ratio x m=T as Y x = 0:145g=g x expx, where

g is the number of spin degrees of freedom, and g is the total number of relativistic

degrees of freedom in the universe at temp erature T = m=x. Well after freeze out Y x

is constant|we will denote this asymptotic value of Y as Y .

If self annihilation determines the nal abundance of a sp ecies, Y can b e found by

2 2

integrating the Boltzmann equation dot denotes d=dt n_ +3Hn = hjv j i n n ,

where n is the numb er density, n is the equilibrium numb er density, H is the expansion

rate of the universe, and hjv j i is the thermal average of the annihilation rate.

There are no general closed-form solutions to the Boltzmann equation, but there are

reliable, well tested approximations for the late-time solution, i.e., Y . Then with knowl-

edge of Y , the contribution to h from the sp ecies can easily b e found. Let us sp ecialize

to the survival of photinos assuming self-annihilation determines freeze out.

Calculation of the relic abundance involves rst calculating the freeze-out value of

x, known as x , where the abundance starts to depart from the equilibrium abun-

dance. Using standard approximate solutions to the Boltzmann equation gives x =

ln0:0481m m 1:5ln[ln 0:0481m m ], where wehave used g =2, g = 10, and

Pl 0 Pl 0

parameterized the nonrelativistic annihilation cross section as hjv j i = x . Us-

A 0

11 2

ing the diagram shown in Figure 1, = 2 10 mb, which leads to x '

0 f

3 4

12:3 + ln = . The value of x determines Y :

f 1

2:4x

4 7

' 7:4 10 : 2.4 Y =

m m

Pl 0

Once Y is known, the present photino energy density can b e easily calculated: =

mn = 0:8 GeV Y 2970 cm . When this result is divided by the critical density,

8 1

6 EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects

2 5 3

=1:054h 10 GeV cm , the fraction of the critical density contributed by the

2 8 2 4

photino is found to b e h =2:25 10 Y . For Y in 2.4, h = 167 .

8 1 1

e e

The age of the universe restricts h to b e less than one, so for = 1, the photino

must b e more massive than ab out 10 GeV or so if its relic abundance is determined by

self-annihilation.

2.3. R -catalyzedfreeze out

Farrar and I pointed out that for mo dels in which both the photino and the gluino

are light, freeze out is not determined by photino self annihilation, but by e ! R

interconversion. The basic p oint is that since the R has strong interactions, it will

stay in equilibrium longer than the photino, even though it is more massive. As long as

e ! R interconversion o ccurs at a rate larger than H , then through its interactions

with the R ; the photino will b e able to maintain its equilibrium abundance even after

self annihilation has frozen out.

Griest and Seckel discussed the p ossibility that the relic abundance of the lightest

sp ecies is determined byitsinteractions with another sp ecies. They concluded that the

mass splitting b etween the relic and the heavier particle must b e less than 10 for the

e ect to b e appreciable. We nd that e ! R interconversion determines the e relic

abundance even though the R maybetwice as massiveasthe e . The di erence arises

b ecause Griest and Seckel assumed that all cross sections were roughly the same order

0 12

of magnitude. But in our case the R annihilation is ab out 10 times larger than other

relevant cross sections.

Iwillnow consider in turn the reactions we found to b e imp ortant in our scenario. The

diagrams for the individual constituent pro cesses can be found in Figure 1. However,

as we shall see, it is not a simple task to go from the constituent diagrams to the cross

sections and decay width.

: For photino self-annihilation at low energies the nal state X is a lepton- e e ! X

antilepton pair, or a quark-antiquark pair which app ears as light mesons. The pro cess

involves the t-channel exchange of a virtual squark or slepton between the photinos,

pro ducing the nal-state fermion-antifermion pair. See the upp er third of Figure 1.

of the squark/slepton is much greater than In the low-energy limit where the mass M

s, the photino-photino-fermion-antifermion op erator app ears in the low-energy theory

=M ,with e the charge of the nal-state fermion. with a co ecient prop ortional to e

Also, as there are two identical fermions in the initial state, the annihilation pro ceeds as

a p-wave, whichintro duces a factor of v in the low-energy cross section. The resultant

low-energy photino self-annihilation cross section is:

2 2

m v

2 4 11 1 2

hjv j i =8 q ' 2:0 10 x mb; 2.5

EM i 8

e e

where we have used for the relative velo city v = 6=x with x m=T , and q is the

magnitude of the charge of a nal-state fermion in units of the electron charge. For the

light photinos we consider, summing over e, , and three colors of u, d, and s quarks

leads to q =8=3.

0 0 0

: In R self-annihilation, at the constituentlevel the relevant reactions are R R ! X

ge + ge ! g + g and eg + ge ! q +q see the middle third of Figure 1, which are unsuppressed

, so the cross section should b e typical of strong interactions. In the byanypowers of M

0 0

nonrelativistic limit we exp ect the R R cross section to b e comparable to the pp cross

section, but with an extra factor of v , accounting for the fact that there are identical

fermions in the initial state so annihilation pro ceeds through a p-wave. Thereissomeen-

EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects 7 ~γ q, l ~ γ q, l ~ q,~ l _ _ _ _ γ~ q, l ~ 2 -2 γ q, l e M~ S

~ g~ g g q g g

~ _ ~g g g q S ~ ~ γ q γ q q~ ~g ~ q q e g M~-2 g

S S

Figure 1. Feynmann diagrams for the constituent pro cesses determining the relic photino

abundance. The top left diagram is for e self annihilation, and on the top right is the e ective

low-energy op erator for that pro cess. The two diagrams in the middle are the diagrams for

0 0

R self annihilation. Finally in the lower left-hand corner is the diagram for the e ! R

interconversion pro cesses all interconversion pro cesses can b e obtained from crossings of this

diagram, and on the lower right-hand side is the e ectivelow-energy op erator.

0 0

ergy dep endence to thepp cross section, but it should b e sucient to consider hjv j i

R R

2 1 1

0 0

to be a constant, approximately given by hjv j i ' 100v mb = 600 x r mb,

R R

where wehave used for the relativevelo city v =6T=M =6=rx, with x m=T .

We should note that the thermal average of the cross section might be even larger

if there are resonances near threshold. In any case, this cross section should be much

larger than any cross section involving the photino, and for the relatively small values of

r we employ it will ensure that the R remains in equilibrium longer than the e , greatly

simplifying our considerations.

: This is an example of a phenomenon known as co-annihilation whereby Re ! X

the particle of interest in our case the photino disapp ears by annihilating with another

particle here, the R . Of course co-annihilation also leads to a net decrease in R -

odd particles. We can estimate the cross section for Re ! X in terms of the e self

annihilation cross section by comparing the lower third of Figure 1 to the upp er third:

2 M 3 4

hjv j i' hjv j i ; 2.6

e R e e

3 8=3 m v

where the ratio of 's arises b ecause the short-distance op erator for co-annihilation is

2 2 4

prop ortional to e g rather than e , the second factor is the color factor coming from

i i

P P

2 4

the gluino coupling, and the third factor comes from the ratio of q = q for the

i i

participating fermions. Wehave replaced m app earing in 2.5 by mM , although the

actual dep endence on m and M may b e more complicated. Finally, the annihilation is

s-wave so there is no v =3 suppression as in photino self-annihilation.

Although the short-distance physics is p erturbative, the initial gluino app ears in a

light hadron, and there are complications in the momentum fraction of the R carried

8 EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects

Table I: Cross sections and the decay width used in the calculation of the relic photino

abundance. The dimensionless parameters and were de ned in 2.3, and F r was

8 S

de ned b elow 2.9. The co ecients A, B , and C re ect uncertainties involving the calculation

of hadronic matrix elements.

Pro cess Cross section or width

0 1 1

R self annihilation: hjv j i 600 x r mb

0 0

R R

11 1 2

2:0 10 x mb e self annihilation: hjv j i

e e

10 2

1:5 10 r A mb co-annihilation: hjv j i

e R

0 14 5

R decay: 2:0 10 F r [ B ]GeV

R !e

0 10 2

e { R conversion: hjv j i 1:5 10 r C mb

by the gluino and other non-p erturbative e ects. For our purp oses it will b e sucientto

account for the uncertaintyby including in the cross section an unknown co ecient A,

leading to a nal expression

10 2

A mb: 2.7 hjv j i'1:5 10 r

e R

0 +

R ! e : In what we call interconversion pro cesses, there is an R -o dd particle in

the initial as well as in the nal state. Although the reactions do not of themselves change

the number of R -o dd particles, they keep the photinos in equilibrium with the R s, which

in turn are kept in equilibrium through their self annihilations. An example is R decay.

It can o ccur via, e.g., the gluino inside the initial R turning into an antiquark and a

virtual squark, followed by squark decayinto a photino and a quark. In the low-energy

limit the quark{antiquark{gluino{photino vertex can b e describ ed by the same typ e of

four-Fermi interaction as in co-annihilation see Figure 1. One exp ects on dimensional

. The lifetime of a free gluino to decay grounds a decay width / M =M

0 EM S

to a photino and massless quark-antiquark pair was computed by Hab er and Kane.

However this do es not provide a very useful estimate when the gluino mass is less than

the photino mass.

In an attempt to account for the e ects of gluino-gluon interactions in the R , which

is necessary for even a crude estimate of the R lifetime, Farrar develop ed a picture

based on the approach of Altarelli et al.: The R is viewed as a state with a massless

gluon carrying momentum fraction x, and a gluino carrying momentum fraction 1 x;y

having therefore an e ectivemassM 1 x . The gluon structure function F x gives the

probabilityinaninterval x to x + dx of nding a gluon, and the corresp onding e ective

mass for the gluino. One then obtains the R decay width neglecting the mass of nal

state hadrons:

5=2

M; r= M; 0 dx 1 x F x f 1=r 1 x ; 2.8

where in this expression the factor M; 0 is the rate for a gluino of mass M to decay

2 2 3 4 5 6

to a massless photino, and f y = [1 y 1 + 2y 7y +20y 7y +2y + y +

3 2

24y 1 y + y logy ] contains the phase space suppression which is imp ortant when

y Of course there should b e no confusion with the fact that in the discussion of the R lifetime

weusex to denote the gluon momentum fraction whereas throughout the rest of the pap er x

denotes m=T .

EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects 9

the photino b ecomes massive in comparison to the gluino. Mo deling K decay in a

similar manner underestimates the lifetime by a factor of 2 to 4. This is in surprisingly

go o d agreement; however caution should b e exercised when extending the mo del to R

decay, b ecause kaon decayismuch less sensitive to the phase-space suppression from the

nal state masses than the present case, since the range of interest will turn out to b e

r 1:2 2:2. For r in this range, taking F x 6x1 x leads to an approximate

behavior

14 5

=2:0 10 F r GeV [ B ] ; 2.9

0 +

R !e

5 1 6

where F r =r 1 r , and the factor B re ects the overall uncertainty. We b elieve

< <

3. Lattice QCD calculation of the relevant B a reasonable range for B is 1=30

hadronic matrix elements would allow a more reliable determination of B .

R ! e : The short-distance subpro cess in this reaction is q + eg ! q + e , again

describ ed by the same low-energy e ective op erator as in co-annihilation and R decay.

0 0

At the hadronic level the matrix elementforR ! Xe is the same as for R e ! X

for any X , evaluated in di erent physical regions. Thus the di erence between the

various cross sections is just due to the di erence in uxes and nal state phase space

integrations, and variations of the matrix elementwith kinematic variables. Given the

crude nature of the analysis here, and the great uncertainty in the overall magnitude of

the cross sections, incorp orating the constraints of crossing symmetry are not justi ed

at present. We will therefore use the same form as for 2.7, letting C parameterize the

10 2

hadronic uncertainty in this case: hjv j i'1:5 10 r C mb.

This completes the discussion of the lifetimes, cross sections, and their uncertainties.

The results are summarized in Table I.

Once the cross sections and decay rate is known, one can develop the Boltzmann

equations for the system and numerically solve them to nd the relic abundance. This is

b eing studied byChung, Farrar, and Kolb. But for the purp ose of illustrating the main

issues, it will suce to compare equilibrium reaction rates to the expansion rate near

freeze out.

To obtain an estimate of when the rates will drop b elow the expansion rate, we will

assume all particles are in lte lo cal thermo dynamic equilibrium. In lte a particle of

mass m in the nonrelativistic limit has a numb er density

g g

3=2 3=2 3

mT expm=T = T=m m expm=T : 2.10 n =

3=2 3=2

2 2

Here g counts the numb er of spin degrees of freedom, and will b e 2 for the R and the

e . Of course all rates are to be compared with the expansion rate. In the radiation-

dominated universe with g 10 degrees of freedom

1=2

2 19 2 2

H =1:66g T =m =2:8 10 x GeV : 2.11

There are two striking features apparent when comparing the magnitudes of the equilib-

0 0

rium reaction rates in Table I I. The rst feature is that the numerical factor in R R ! X

is enormous in comparison to the other numerical factors. This simply re ects the fact

that R annihilation pro ceeds through a strong pro cess, while the other pro cesses are all

suppressed by a factor of M .

The other imp ortant feature is the exp onential factors of the rates. They will largely

determine when the photino will decouple, so it is worthwhile to examine them in detail.

The exp onential factor in e e ! X is simply expm=T , which arises from the equi-

librium abundance of the e . It is simple to understand: the probabilityofone e to nd

another e with which to annihilate is prop ortional to the photino density, whichcontains

a factor of expm=T = expx in the nonrelativistic limit.

10 EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects

Table I I: The ratio of the equilibrium reaction rates to the expansion rate for the indicated

reactions. Shown in [] is the scaling of the rates with unknown parameters characterizing the

cross sections and decay width.

Pro cess =H Scaling

7 1=2 3

e e ! X 1:2 10 x expx [ ]

20 1=2 1=2 0 0

3:5 10 x r exprx [ ] R R ! X

0 7 1=2 5=2 3

e R ! X 8:9 10 x r exprx [ A]

4 2 3=2 3 + 0

7:1 10 x r F r exp[r 1x] [ B ] e ! R

3=2

1 4

0 6 1=2 5=2

e ! R 9:6 10 x r exp[r 1x] exp0:175 x [ C ]

8 8 S

~ ~

Figure 2. Equilibrium reaction rates divided by H for r = 1:25 and r = 2:0, assuming

= = 1, and that the factors A = B = C =1. The rates can b e easily scaled for other

8 S

choices of the parameters.

0 0 0

The similar exp onential factor in R R ! X is also easy to understand. An R must

nd another R to annihilate, and that probability is prop ortional to expM=T =

exprx.

The pro cess e R ! X is also exothermic, so the only exp onential suppression is the

probability of a e lo cating the R for co-annihilation, prop ortional to the equilibrium

numb er densityof R , which is prop ortional to expM=T = exprx

+ 0

In e ! R the exp onential factor is exp[M m=T ] = exp[r 1x], which

is just the \Q"value of the decay pro cess.

For the pro cess e ! R , it is necessary for the collision to have sucient center-

of-mass energy to accountforthe e {R mass di erence, which accounts for a factor of

exp[M m=T ]. The numb er density of target pions is expm =T , so this factor

is also present. Combining the two factors leads to the overall factor app earing in Table

I I: exp[M m + m =T ] = exp[r 1x] exp0:175 x.

Graphs of the reaction rates as a function of r is shown in Figure 2. There are several

things to notice from the graphs: 1 The R self-annihilation rate is always larger than

the other rates. This means that the assumption that the R is in equilibrium during e

EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects 11

~ ~

Figure 3. Assuming e freeze out is determined by e $ R interconversion, this gures shows

3=2

as a function of r the values of [ C ] required to give the indicated values of h . The

uncertainty band is generated byallowing to vary indep endently over the range 0:5 2.

8 8

freeze out is a go o d approximation for the values of r considered here. 2 Even for r as

large as r = 2, the interconversion rates seem to b e slightly more imp ortantthan e self

annihilation in keeping photinos in equilibrium. 3 The pro cess e ! R seems to b e

the most imp ortant pro cess for r<2. Detailed numerical integration of the Boltzmann

equation by Chung, Farrar, and Kolb con rms this. 4 The freeze-out temp erature is

very sensitivetothevalue of r . This traces to the exp onential sensitivity of the reaction

rates up on r . We will make use of this last feature to nd a cosmologically acceptable

range of r .

Assuming that e ! R do es determine the e relic abundance, Figure 3 shows the

sensitivityof h to r . From this gure we can draw some very interesting conclusions.

3=2

C ] < 10 , then r must be less than 1.9. If we demand If we assume that [

2 2

that the relic photinos are dynamically imp ortant say h 10 then r 1:2 if

3=2 3=2

4 4

[ C ] > 0:1. Finally,ifwecho ose our b est guess [ C ] 1 and h 0:25,

8 8

S S

< <

then the allowed range of r is 1:4 r 1:6.

2.4. Testing the scenario

Direct detection of light photinos is not easy. the interaction cross section decreases

with photinos mass, and more imp ortantly, the kick they would givetoamassive target

nucleus also decreases with decreasing photino mass.

The case for light photinos hinges up on lab oratory exp eriments. The scenario dep ends

up on the existence of the R , the gluino|gluon b ound state,y with a mass roughly 1.5

times the photino mass. If the R can be discovered and after all, the discovery of a

1.5 GeV hadron do es not sound imp ossible, from its decay one can learn the e mass,

and hence r ,aswell as the parameters of the short-distance matrix element. While there

is no shortage of candidates for relic dark matter particle sp ecies, this prop osal extends

the idea that photinos may b e the dark matter to a previously excluded mass range by

incorp orating new reactions that determine the photino relic abundance. If this scenario

y In fact, one suggested title for this talk is \The R : sglueball or screwball?"

12 EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects

is correct, direct and indirect detection of dark matter mightbeeven more dicult than

anticipated. However the scenario requires the existence of low-mass hadrons, whichcan

b e pro duced and detected at accelerators of mo derate energy. Thus particle physics ex-

p eriments will either disprove this scenario, or makelight photinos the leading candidate

for dark matter.

So the next step is obvious: nd the R :

3. Relic axions

3.1. The axion

The axion story b egins with PQ symmetry breaking, which o ccurs when a complex

scalar eld with non-zero PQ charge develops a vacuum exp ectation value. This PQ

symmetry breaking can be mo deled by considering a p otential of the standard form

2 2

~ ~

V = jj f =2 . The axion is the Nambu{Goldstone degree of freedom resulting

from sp ontaneous breaking of the global symmetry. However, since the PQ symmetry

is anomalous, it is broken explicitly by QCD instanton e ects, leading to a mass for

the axion. In general the instanton e ects resp ect a residual Z symmetry, and the

2 2

axion develops a p otential due to instanton e ects of the form V a=m f =N [1

cosN a=f ], where m = =f . The axion eld is often represented in terms of an

a a QC D a

angular variable Na=f , and if is taken as the dynamical variable, its potential

for N =1is

2 2

V =m T f 1 cos : 3.12

a a

The ab ove description is only valid at zero temp erature. Because QCD instantons are

, their e ects are strongly suppressed at high temp eratures. large, with a size set by

QCD

Thus, after PQ symmetry breaking at T f , but before QCD e ects are imp ortant

around T , the axion is e ectively massless. For T , the temp erature dep en-

QC D QCD

2 n

dence of the axion mass scales as m T / T=T , where n =7:4 0:2

3.2. The standardscenario

When the eld x is created during the Peccei-Quinn symmetry breaking phase tran-

sition at T f , it should be uncorrelated on scales larger than the Hubble radius at

that time.

The usual assumption in axion cosmology is that PQ symmetry is broken b efore or

during in ation, and the reheat temp erature after in ation is to o low to restore the PQ

symmetry. This would mean that in ation will \smo oth" the axion eld on scales larger

than the Hubble radius to day. While this may indeed b e the case, I will not makethat

assumption here. Since Tkachev and I rst did the work rep orted here, Kofman, Linde,

and Starobinski, as wellasTkachev, have made the interesting observation that symmetry

restoration and breaking in the preheating phase of in ation mayhaveinteresting e ects.

Just one of the interesting e ects may b e to restore PQ symmetry after in ation, even

if the reheat temp erature is to o low to do so. This means that PQ symmetry will be

restored and broken after in ation, pro ducing a spatially dep endent axion eld. While

the dust has yet to settle on the issue, it is clear that it is na ve to think that in ation

automatically results in a smo oth axion eld. So it is worthwhile to explore the p ossibility

that the axion eld is misaligned on scales larger than the Hubble radius at the time of

the QCD transition when the axion mass switches on. In this part of the talky Iwould

y Reference to all the material presented here can be found in Kolb and Tkachev 1993, 1994,1996

EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects 13

like to prop ose another di erent picture for the wanted p oster: axions that are clump ed

to day within our galaxy, rather than uniformally distributed.

Assuming an initially chaotic axion eld after the PQ transition, as the temp erature

decreases and the Hubble radius grows the eld b ecomes smo oth on scales up to the

Hubble radius. This continues until T = T 1 GeV when the axion mass \switches

on," i.e., when m T 3H T , and the axion mass b egins to b ecome imp ortant in

a 1 1

the equations of motion. Coherent axion oscillations then transform uctuations in the

initial amplitude of the axion eld into uctuations in the axion density.

Since the initial amplitude of the coherent axion oscillations on the scale of the Hubble

radius is uncorrelated, one exp ects that typical p ositive density uctuations on this scale

will satisfy 2 , where is mean cosmological density of axions. These overdense

a a a

regions were called axion miniclusters by Hogan and Rees. At the temp erature of equal

matter and radiation energy density, T = 5:5 h eV , non-linear uctuations will

EQ a

14 3

separate out as miniclusters with 10 gcm . The minicluster mass will be

of the order of the axion mass within the Hubble radius at temp erature T , M

1 MC

9 13

10 M . The radius of the cluster is R 10 cm, and the gravitational binding

energy will result in an escap e velo cityofv =c 10 .

It is easy to understand how a minicluster forms. First consider the evolution of the

background axion density. In the limit that the initial misalignment angle is constant

on all scales, the axion energy density scales with temp erature as T =3T s=4 for

a EQ

T , where s / T is the entropy density and T is the temp erature of equal

QCD EQ

matter and radiation energy densities.

Now supp ose that there is a region with axion over-density, = 1 + . Then

a a

the matter density in that region will dominate the radiation density at a temp erature

T =1+T . If is larger than unity, then that region will separate out from the

cosmological expansion, gravitationally collapse, relax, and form a minicluster with the

approximate densityithadatT . A detailed study of this leads to a nal minicluster

densityof

3 14 3 2 3

' 140 1 + T 3 10 1 + h gcm : 3.13

mc a eq a

Even a relatively small increase in is imp ortant b ecause the nal density dep ends up on

for 1.

3.3. Non-linear axion dynamics and production of miniclusters

Igor Tkachev integrated the eld equations near the QCD transition with various choices

for initial conditions where the axion eld is inhomogeneous on scales larger than the

Hubble radius.

In our numerical investigations of the dynamics of the axion eld around the QCD

ep o chwe found that as the oscillations commence, imp ortant, previously neglected, non-

linear e ects can result in the formation of transient soliton-like ob jects we called axitons.

The non-linear e ects result in regions with much larger than unity, p ossibly as large

as several hundred, leading to enormous minicluster densities. We also found that the

minicluster mass scale is set by the total mass in axions within the Hubble radius at

a temp erature around T = T 1 GeV when axion mass is equal to H . Since this

temp erature is somewhat higher than , the e ect is to lower the minicluster mass

QCD

from the estimate of Hogan and Rees to ab out 10 M .

3.4. Detecting axion miniclusters

If a large fraction of axions end up in miniclusters and in our numerical simulations that

is what we found, then the lo cal axion densitymaybevery clumpy. This would mean

14 EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects

Figure 4. This is a two-dimensional slice through a three dimensional b oxofTkachev's sim-

ulation showing the distribution of axion energy densities for T T . The height of the plot

corresp onds to = = = 20, and the width to a length of 4H T this corresp onds to a

a a 1

comoving length of ab out 0.25 p c. The miniclusters are clearly seen.

that direct searches for axions would not b e very ecientunlesswe just happ ened to b e

passing through a minicluster.

However it may b e p ossible to detect miniclusters through femtolensing or picolensing.

The basic idea is that if gamma-ray bursts are at cosmological distances, then miniclusters

could act as gravitational lenses. For source and lens of cosmological distances, the

10 12 1=2

Einstein ring radius is R 5 10 M=10 M cm. This means that if two

gamma-ray burst detectors are separated by distances larger than R , typically one

detector will b e within the Einstein ring and the other will not. Measurement of di erent

uxes in two widely separated detectors would extend the detectable range of lens masses

to M < 10 M . If the angular separation of the images of a point-like source is

comparable to the wavelength of the light, interference b etween the lensed images will

lead to a distinctive fringe pattern both in co ordinate space and in energy sp ectrum.

The interference pattern in the energy sp ectrum will dep end up on the lo cation of the

detector. Since well separated detectors will see di erent patterns, there is a distinctive

signature of the e ect.

4. Conclusions

Although there are solid, well motivated scenarios that lead to one of several susp ects

forming cold dark matter, one must explore vigorously the p ossibility of a non-standard

scenario for standard dark-matter susp ects such as susy and axions. It would be a

crime if we fail to apprehend dark matter just b ecause a prime susp ect app ears in an

unfamiliar guise.

Iwould liketoacknowledge collab oration with Glennys Farrar and Daniel Chung on

the light-photino work rep orted here, and collab orations with Igor Tkachev on the axion-

minicluster work. This work was supp orted by the DepartmentofEnergyandbyNASA

under GrantNAG5{2788.

EDWARDW.KOLB: New Guises for Old Dark-Matter Suspects 15

REFERENCES

Farrar, G. R. & Kolb, E. W. 1996 Light photinos as dark matter. Phys. Rev. D 53, 2990{

3001.

Chung, D. J. H., Farrar,G.R.&Kolb, E. W. 1996 Light photinos as dark matter I I. In

preparation.

Kolb, E. W. & Tkachev, I. I. 1993 Axion miniclusters and b ose stars. Phys. Rev. Lett. 71,

3051{3054.

Kolb, E. W. & Tkachev, I. I. 1994 Non-linear axion dynamics and formation of cosmological

pseudo-solitons. Ap. Phys. Rev. D 49, 5040{5049.

Kolb, E. W. & Tkachev, I. I. 1996 Femtolensing and picolensing by axion miniclusters. Ap.

J. Lett. 400, L25{L29.