e..t. chun and a. lukas- axino mass in supergravity models

8/3/2019 E..T. Chun and A. Lukas- Axino Mass in Supergravity Models

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IC/95/29TUM-HEP 215/95

hep-ph/9503233

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

AXINO MASS IN SUPERGRAVITY MODELS

INTERNATIONAL

ATOMIC ENERGY

AGENCY

UNITED NATIONS

EDUCATIONAL,

SCIENTIFIC

AND CULTURAL

ORGANIZATION

E..T. Chun

and

A. Lukas

MIRAMARE-TRIESTE



rf



IC/95/29

TUM-HEP 215/95hep-ph/9503233

International Atomic Energy Agency

an d

United Nations Educational Scientific- and Cultural Organization

IN T E R N A T IO N A L C E N T R E FO R T H E O R E T IC A L P H Y S IC S

AXINO MASS IN SUPERGRAVITY MODELS

E. .1. Chun1

International Centre for Theoretical Phvsics, Trieste, Italy

am i

A. Lukas J

Physik Institut, Teehniseho Universitiit Miinrhen,

D-857'17 Garching, Germany

an d

Max P lanck Inst i tu t i'iir Physik, Werner Heisenberg lust itut,

P. O. Box 41) 12 12, Munich, Germany.

A B S T R A C T

We analyze Ihe mass of the axino, the fennioiiic superpartuer of the axion, in general

supe rg rav i ty mode ls incorpora t ing a Pccce i Quiim symmetry mid de te rmine the cosrno-

logieal cons train ts on this mass . In particular, we derive a simple criterion to identify

models with an LSP axino which has a mass of O(nifu-,( fpq) = O(keV) atid can serve

as a candidate for (warm) dark matter. We point out that such models have very special

proper ties and in addition, th e small axino mass has to l>e protected against radiati ve

correcti ons by demanding small couplings in tin- Pecrei Quimi sector. Geuerieally. we

find an axino mass of order J»3/a. Such masses air con strained by the requirement of

an axin o decay which occurs before the decoup ling of the. ordina ry LSP. Especially, for a

large Peceei Quinu scale fPQ > ]( )" GeV this constr aint might be difficult to fulfill.

M I R A M A R E T R I E S T E

March WJ5

'E mail:chuin<*iet|>.triestc.itaE mailialukasidpliysik.tu-mucncSicn.iie

The implications of axions have been examined oxtensively since their existence was

suggested by an attrac tive mech anism for resolving the stron g CP problem [I. 2, 3].

Even though axions are very weakly interacting, their astrophysical and eosmological

effects are stron g enough to narrow down the window of the Peccei- Quin n scale //.g to

10in

GeV < jpq < ldu GeV [5 ] , On the same foo t ing the ax ino a s the supe rsymmrtric

partner of the axion can play an important role in astrophysics and cosmology [U]. An

interesting feature is that axinos may receive a mass of order keV which would render

them a good candidate for warm dark matter. If axinos are heavier than a few keV they

have to decay fast enough not to upset any standard prediction of big bang cosmology-

Given the weakness of their interactions, a constraint on their lifetime put a rather severe

limit on the lower bound of their mass. Therefore it is very impo rtan t to know the axin o

mass in discussing the cosinologicai implications of supersynnnetric axion models.

In g loba l supc rsymmetrv (SUSY) the < alcula tiou of the axino m<is.s was perfo rmed

in refs. [7. 8]. In this paper , we will provide the com putat ions in model?, with local

super sym metr y (s upcrgr avity). Some partial results have been obtained in refs. [!), 10]. In

the case of spontan eously broken global SUSY t he axino mass is of the order m'*,.,/ fpq ~

keV where m : ^( S ITeV) is taken to be the global SUSY break ing scale [7, H]. On t he

contrary, in the context of supergravitv. the axino mass can be of order niA/-2 as lirst

noticed in ref. [9]. Soon after th is if vvas realized that the axino m ass is truly model

dependent and the global SUSY value ni\ rljfnj may be obtained in supergrnvity models

as well [10]. We will extend those results in a generic treatment of supergravity models

also including radiative corrections.

The prime motivat ion for supergravity is well known. Realistic supersym inetrie gen-

eralizations of the standard model are based on local SUSY spontaneously broken in a

so called hidden sector at a mass scale of order W s ~ K) "Ge V [12]. The induced SUSY

breaking scale in the observable sector is determined by a value of the order of the grav-

i t ino mass in^'-i -- M^jMp where ,!/;• is the Planck scale. Axionic extensions of the

minimal siipersytiunctric standard model (MSSM) inevitably incorporate an extra sector

which provides spontaneous breaking of the Percei Quinn U(\) symmetry at the scale

fi'Q, This sector (PQ sector) is considered as a part of the observable sector. In the

framework of effective supergravity theories with a Lagrangian composed out of a global

SUSY part and soft terms the hidden sector dependences are encoded in the soft terms.

We will rely mostly on this effective approach as it makes the calculations tractable.

A color anomal y in the PQ sector can be introduced in two ways. T he fields S in this

sector can be coupled to the standard Higgs doublets /f,, H? of the MSSM like gSH^Hi

(DFSZ axion) [2, 4] or to new heavy quarks Q,, Q-2 like gSQ,Q2 (KSVZ axion) [1]. Our

analysis of axino mass will lie concerned with the tree level result in the effective theory

which is obtained after breaking the PQ sym metry and SUSY and should therefore not



depe nd on which implem entation of the axion is chosen. We will find that this mass

crucially depends on the structure of the 1'Q-sector as well as on the hidden sector. It

should, however, he mentioned that in addition the above couplings and the corresponding

tril inear soft terms lead to a one-loop radiative mass of order m^mp — 'Ag 2Ani v,/l(inz

(A is the triline ar soft, couplin g) [11. 9]. For the DFSZ axion this con tribu tion is clearly

negligible since the coupling g has to be quite small (y ~ 10' 7 ) in order lo generate

reasonab le Higgs masses . In the KSVZ-implem entation there is no such restriction on

g and the above one loop contribution can modify the result for t in- axino mass to be

obtained below.

An interesting observation is that the axino mass depends on whether the PQ sector

adm its additiona l accidental zero modes on the global SUSY level. As a general statement

we find thni the axiuo muss cannot lie bigger thiin Otiir lyj) if the axion niodc is the onlv

global zero mo de in Ilie PQ secto r. In order to see this, it is useiul to analyz e the

full supe rgra vitv Lag raugia n. Late r, we will confirm I his result by using the eflective

Lagraugian approach.

The P Q secmr consists of an arbitrary number ul singlets S" wit h charge r/,, under the

PQ symmetry and provides i ts spontaneous breaking at the scale / , .y = (5Z,, i / , ' | | i '" | ' !)1 'a.

where ?'" = {.*"). The jixioii multiplct <I is given bv 'I> — T l: i}at'"S" i frq- Its component

field co ntent lead s <> ~ {a + in,a) with the axion <: . the sax ion ,s- ami the ax ino a.Supersynimclrv is broken mainly by the hidden sector with siuglel l iclds Z' To simplify

the argu men t we assum e minimal kinetic UTin for the PQ lields as well as tor the liiddcli

sector fields. The scalar potential reads

where G = I\ + M'f, In [ U ' / A / ^ p an d IV - If ($") + \V[Z'). Spontaneous super sv inmet ry

breaking implies {6",) ~ A /, , for some i a n d genera t ion of a gravi t ino m a s s . iii :ir2 ~

Mp expjfr"/2.1 If.]. W e also have to admi t th e possibil i ty thai (G,,) '*• Mr for some <i. W e

no w took ul I he i iumtui/ .at ion condit ion (V,) - tl In es l i i na t e th e axino m a s s . Vanishing

of t he cosiuoiogicnl constant (I ' ) = I t is assumed- Then we have

(V,,) = (G,J;I + G,I,G; +G:)- W

If we take a massive mode 5" in the PQ sector. (G,,i,) is dominated by

since (W ) ~ Mf.n>M2- I" afidition, we can estimate the maximal order of magnitude of

(4)

since {W t) ~ Mp m 3/2 maximally arid (H;,) S fj, Q. Therefore the condition (V a) - 0

gives (G a) ~ JPQ- In the axion direction G* = EaW^Ga/fPQ, we also have (G*) =

( E n <laVaK,,[f,>q) = Y.a 'hV'"^I fi'Q - fi'Q- Since the absence of accidental zero modes is

requi red in the PQ sec tor one finds that (6',,) ~ JFQ for all the fields S".

It is now straightforward to estimate the maximal order of the axino mass by using the

fermiou mass matrix in supergravity models. M tJ= Mpf:\\i{G/2M'p)\G,li, + G aGi,/MP].

Along the axino direction,

Gn Gb(5)

Hence theVom the ('(1) invariance of (I, we gel Yi,,<lit''<'(G ni,) = fl!,{i'b — (Gj,)) S

axiuo mass is niaximally of order ra ;i,-;.

The above order of magnitude estimation is indeed insensitive to the specific forms

of the kinetic term or the superpotential as long as the hidden sector fields have only

non renonmilizable couplings ID the oliservable se(t"i". This hapjieus because higher

power terms in A' or W are naturally suppressed by powers of Mp which renders their

contribution negligible. T herefore we conclude that the a.xino mass in general siipergravity

models is at most of the order m :i/2 if no other zero mode than the axion is present in the

PQ sector.

On the other h and if there arc extra zero modes we are not able to constrain further

the order of (G,,) for a zero m ode d irect ion <i so that the above argumentation breaks

down, hi fact, ax ino mass es ;§> m; .L> ale possible in those models ;is we will sec below.

We will now calculate the actual value of the axino mass relying on the effective super-

gravity Lagraiigian with arbitrary soft terms. This allows to consider axino masses down

to O{m\r2lfrq) where the next to leading order in the I/A/; .-expansion of supergravity

becomes impor tant .

The superpotenl ial \Y (if the PQ sector is expanded as

i r = /,,.s-" i- -rL,.S"s'' i-]-jnhr s"s bsr

a n d a departure from standard soft terms is encoded in

N = dj asa+ ^U f

Then the scalar potential reads

V = ifWOfiW + m.f.v<ir

+ »';)/2

(6 )

W + (A - 3)IV + N + h.cJ (8)

T o minimize this potential we apply th e following strategy : T h e V E V s ? / ' a r e spli t into

a global SUSY value 7<" wi th 0aW(u) = 0 a n d cor rec t ions iua

d u e t o t h e soft t e r m s .



= u" + w". Expand ing daV around the global minimum («") results in

+(1

with the global mass matrix

A-l)M*t + <UMu> b(9)

(10)

and the definitions

/„. = m,n.ln + m fil,, , ./„ = AUu'1 + ((>,,.V}(//) . (ID

Can- should he taken on I he choice o! (it"). With any global minimum also a rotation

«a

—» cxp(r /,,;)n " w ith : = .r h ill undri the couiplexitied fVy(l) leads to such a mini-

mum. Desp ite the y dependen t part, of this symm etry which is clearly present in the full

theory the r dependent part (present, because of the holoniorphy of the superpotential)

is broken by the soft terms. Therefore an appropriate tixinjf for the .r dependent, part of

the symmetry should be applied such that I lie j;lnbal minimum comes close to t.hc local

values (» '") resulting in small expansion cocfiicients (»•"). Such a fixing is provided by

the cond it ion wn = Y.u'lalt''

lll"/frQ = " implying that the correction in the global axion

direction (denoted by an index <v) vanishes.

As can be expected the axino mass is expressible in terms of the corrections ("'") -

(Ma)a =

1 ,,(12)

The se corr ecti ons have to be dete rmi ned fnn [, (!)). Lei n.s work in a basi s with

d i a g o . n l g l o b a l m a s r , m a t r i x M,,i, = . A / , / \ , j , . W e d e n o t e m a s s i v e m o d e s w i t h i n d ic e s I.J. • • •

and possible additional zero modes with indices ^ . 1 . • ' ' Then an importan t obse rva t ion

is tha t the co l lec t ions m, are basically detemiiued by the linear term \Mj\2u< 1ami /, in

eq. (9). In zero mode directions the situation might be more complicated since \M li\2ii'

!3

can be small compared to other terms in eq. (!)). Taking this into account we conclude

that the expression

(M(l}n ~JV Q

gives the correct, order of the axino mass. For the moment we leave the values of w tl

unspecified. Instead we concent rate on the hist term in eq. (13) and split the expression ./„

into its Planck (or GUT ) scale value J™ and corrections .7^' arising from ren ormali zation

down to fyq. The generic value of the corrections can be roughly estimated as

M,

(14)

(15)

with appropria te combina t ions Xa. X' u of the superpotenti al couplings. Accordin g to

eq. (11) Jf is naively of the order /;. y . Therefore a necessary condition for the ax-

ino mass to be much smaller than vi:i/2 I* that Jf = 0.

In general this condition implies a relation between the structure of the superpotential

and the soft ter ms. It is instructive to analyze this relation for a certain subclass of

models, namely those with independent soft coupling A.B.C for the triluieai, bilinear

and linear terms in the superpoten tial. respectively. A comp utatio n leads lo

.if = (B - C)M,,hi,u + -y(A - 2B f C) fnll,n'V . ( l ( i )

Depending on the properties of the superpotenlial (and assuming that at least one coupling

/„ is nonzero to force the symmetry breaking) several cases can be distinguished :

• fai.'t''- /„(„•"'' ' ' ' '• Mni,ii h/ °

:Tl "' " •/,',"' = (I 'I' •"«! only if A = B -• C. No special

property of the superpoteutial is required.

• .f,,i"'' = I'. Mu 4- ":

Then ./,(,'" = i) if and only if A = C . A simple supe rpo lcn t ia l

which fulfills th is require ment is e. g. \Y — \(SS' - /i2)Y since all /„(, - 0.

. fnlll.n''ii'~ = 0, Mn. ? 0 : Then .If = (I if and only if B = C.

• Mn = (I, f,,i,u'' 4 0 : Then ,/,i'" = 0 if and only if A - IB + C = (I. As the

only ones these models allow for the full standard pn1t.cn! B = A - 1, ('• = A - 2.

They possess, however, at least, one additional zero mode Yl,, n"S". An example is

provided by the superpoteutial It" = \(SS' - Z2)Y ••- X'(Z - u) '

1[III].

Observe that in [jarlicnlar for A = B — (' I he expression Jf vanishes in any model.

If no additional zero mode is ])resent the axino mass is already com pletely de term ined

by the first term in eq. (L3). On tree level this means

. / <"> ^ 0 ™ = O( m 3 /2 )(17)

We have therefi>re found a simple criterion to decide about the magnitude of the axino

mass which for soft terms specified by the couplings A,B.C singles out. the particularly

simple patte rns listed above. W e remark tha t the small axino mass in no scale models



observed in ref. [9] can also be understood In term s of our analysis since in those m odels

A = B = C = 0. A full supergravi ty computat ion of the axino mass in no scale models

shows that their tree-level mass even vanishes. Therefore the first line of eq. (17) has to

be underst ood as a generic result which in cert ain special cases might be too large. In

the first case of a light axino mass radiative corrections to the potential parameters can

become impo rtant. Using eq. (14) this leads to a contr ibuti ons of

rn8 = ktn,,n (18)

To keep the order w,-, = ()(mj/.2/f,>c}) an upper bound on the couplings has to be required.

For m-t/2 ̂ 1()J

GeV and /> Q ~ 10" GeV this implies A -- ID"'1. A syste mati c way to

avoid such small couplings is to consider no scale models. Since gangino masses are the

only source of SUSY breaking in those models the standard model singlet fields in the

PQ secto r will not receive any radiative soft, terms.

We see that without additio nal /.ero modes a complete answer can be given. In par-

ticul ar we recover t he result m,-, £ O(ni.iri)-

If addit ional zero modes are present the situati on becomes more complic ated since

the corrections «•< in the zero mode directio ns can become large. In additi on we have

to consider that the /.ero entry M,, of the imiss matrix receives a radiative contribution

Mft = Xi'iifrQ with l':i in analogy t.o eq. (15) unless it is prote cted by an additi onal

cont inuous or discrete symme try. Let us discuss some relevant cases. First we discuss a

model with .!}")= 0, e. g. a model of the last type in the above list for ,4, D.C type soft

torms. Then a tier level axinn mass O^ur^.Jf,,Q) is not guaranteed as opposed to the

ease without additional zero modes : If the terms in e<[. (!)) linear in «•,., vanish (which

e. g. occurs for /.t> = (I. / ^ = 0 or .4 = /J) n value tvj = O(lJ' ) results which causes

an nxino mass given by rii^ — (H(in-M-i/frij'\'"'"'.(/-J)- '"mLV

case a small axino mass

<£ m-.\p has to be .stabilized against radiati ve corrections. For superpo teii tial co uplings

A = (){l) the axi no mass is shifted lo ;»,-, = (J(/;;il;L.)

Now we assume th at ./^"' ^ 0. Then for values A = ()([) the linear term | A/,j | ^ H',J in

eq. (9) will dominate and the axino mass is given by

•m,; = O(m;y,.,/l;i) . (19)

It, on the ot her hand, th e couplings A are very small we can have «;; = 0(1,^ ) leading

to an axino maws in,-, = O((TH^,i//-y)1/

''t). We we that axino masses 3> ;n;1/2 are indeed

possible. An example featuring all these aspects is the superpotential in the fourth entry

of the above list for A,D,C-type. soft terms,

Now we turn to a discussion of the cosmological constraints on the masses of the

axino and the saxion. We begin by noticing the fact tha t self couplings among t he axion

supermultiplet arise after integrating out the heavy fields in the PQ-sector :

(20)(I + -z—s) -<) aO,,a + -0^0.1 + ta-j dILa - — - „ u

JfQ V2 2 / fPQ

where x = Y.,<livf/frQ • I'

1some cases, 111 partic ular in a model with supe rpot euti al

IV -. \(SS' - ;tJ)V and universal scalar soft masses, x is zero at the Planck scale and

receives a contributi on .r, ~ A

2

ln(jUp/7ry)/G4?r

2

when the RG improved potential atthe PQ scale is considered [13]. Genetically, however, x is of order 1. In this case the

self coupling becomes impor tant since a saxion can decay into two axions faster than

e.g., into two ghions. Decay produced axions do not heat the universe. Therefore the

cosmological effect ot saxion decay is different from what lias been investigated assumi ng

vanishing j [14. 15]. If .r is of order 1, a stronger bound on t he saxion mass can be

expec ted since the decay p roduced axions are not ihcnnal im. l but. red shifted away. Th e

standard nucleosynthesis constrains the energy density of the universe due to this red

shifted axions to be less than what is contri buted by one species of neutri nos at the time of

nucleosynt hesis. This gives the constraint »O.«.'/./) < T/> where 7j> = 0.55jv,/> \/TA?r is

the decay temperature of the saxion and I' = .'"''"^/^^/ry ifs

decay rate. The relativistic

degrees of freedom at Tp are counted by i/.j]. We get

\100J '2.1 TeV (21)

This bound on the saxiou mass which receives ;\ contribution O(m:,/-j) from scalar soft

masses might be difficult to fulfill for large values of fi>Q,

Cosmological i mplic ations of axinos were first discussed in ref. [6] assuming unbroke n

R parity. The axino mass can be constrai ned in two ways. F irst , the axino can be the

lightest supersymnictnc particle (LSP). Then it should be lighter than a few keV in order

not to overdose the universe . Othe rwise, the axino decays into at least one I.SP compo sed

out of the neutr aliuos in the MSSM. In this case, the decay pro duced neutralinoK t end

to overdomi nate the energy density of the universe. To avoid this the axino should be

heavy enough to decay before the neutraliuo s decouple. Consider ing the axino decay in to

photino plus photon, it was obtained that the axino mass should be bigger than a few

Te.V [6]. Then, the axino decay into top quark and scalar to p can be allowed. From t his

decay channel one finds a less restrictive bound

m,i > 90 GeV Ifnq/X,

(22)

v40GeVy V1 { ) U G c V

/v m

'

Here Xt is the PQ c harge of the top quark. We see that a wide range of axino masses

between O(keV) and O(102

GeV) is excluded.



Let us now analyze how the above constraints modify if an inflationary expansion is

taken into account, The decoupling temperature of axino or saxion is around thp range

of the reheating temperature Tn ~ 1010

GeV which is the maximally allowed value to

cure the gravitino problem in supergravity models [IB]. If decoupling of the axino occurs

before1

inflation the primordial axino relics are diluted away. The above consideration,

then, has to be applied to the regenerated population of axinos. We recall that the axino

decoupling is determined by its interactions with gluinos, quarks and anti quarks via

gluoti exchange [(>] The axino decouples at the t emperature

10" GeVHI11

GeV

0.1

The regene ra ted number di usily per en tropy is given bv [Hi]

r ~ 2 xH i" GeV Tlt

(23)

(24)

Depend ing on the lunge of the axino mass we can distinguish three cases as follows.

F irst , for a stable axino. the constraint from overcfosurc gives the following loose

b o u n d on the ax ino mass in t e rm s off,, :

m a < 1G0 keVfrQ

10 " GeV

K)"1

GeV

Tit(25)

Second , an axino with a mass satisfying the lower bound in eq. (22) is still allowed by

cosmology. Finally, for an unstable axiuo with mass between the est ima tions in eq. (25)

am i it) eq. (22). one1

gels a hound on the rehea t ing tempera tu re by replacing the ax ino

m a ss in eq. (25) by the mass of the usual LSP since (lie decay produe'ts of the nx iuo

con ta in at least one LSP:

I GeV(20)

This rep resen ts a qu ite st r ingen t bound mi T/i.

In I his letter we have analyzed the ax ino mass in genera l supe rg rav i ty mode ls and the

eosmeilogieal constraints on such models. We have distinguished models with and w i th o u t

additional zero modes in the PQ sector. For I he latter we found the ax ino mass to be

a t most of Oiwyi). hi the con tex t of an effective approae-h encoding supersymmetry

break ing in so t t te rms we were able to derive a simple necessary criterion for a small tree

level nxiKO mass -C m:l/2 given in te rms of supe rpo ten t ia l and soft term properties. For

uniform t.rilinear, bilinear and linear .soft couplings A. D. C the criterion is always fulfilled

for A = D = C whereas for A = D ^ C or ,4 ^ B = C add it iona l p rope rt ie s of the

supe rpo ten t ia l had to be requ ired . If the global vacuum (») rep resen ts an add it iona l ze ro

mode of the g loba l ly supe rsymmetrie theory , i. e. Mu = 0 with the g loba l mass ma tr ix

Af, the re la t ion A - 2B + C = 0 which admits the s t a n d a rd p a t t e rn B = A-l,C = A-2

is sufficient for the criterion to hold.

We showed that in models without aelditional zero modes our criterion is sufficient,

i. e. it. guaran tees a tree level axino mass of nt most mf - 0{m\j2lfp Q}. In the presence

of other zero modes it serves as a good indication for such a sma ll mass but aelditional

conditions (like e. %.A j= B for mexiels with an A,B,C pa tte rn ) are required to have

'"s" = 0(mlr,/f lv).

From the cosinological point of v iew ax iuo masses O(; ; i^ /2 / / / . y ) = O(keV) are very

a t trac t ive . In this ca.se the ax ino is the LSP and can contribute, a relevant purl of the mass

in the universe as (warm) da rk ma tte r . Though mode ls wil h such an ax ino mass can be

construc ted as we have seen they correspond lo very special points in the space spanncel

by the supc rpo ten t ia l and solt term parameters . Moreover, such small masses are not

stable under raeliative corrections arising from rcliormalization effects between A/,, and

th e PQ scale frQ- Taking these elfects into account the axino mass will be generie'ally

given by ml" - <)(hti:tri) with k = A'-'hi( 1V,' / //.g ) / ; i2 ,Tiand a typ ica l supe rpo len t ia l

coupling A- The one loop contribution horn the cliarneleristic coupling of the ax iuo to

the Higgs h 'e 'ds or heavy quarks will be given by /",-,.i,«,p — St/'1 Am-^^f Hiir' which is only

relevant m the KVSZ implementa t ion of the axion. As the LSP the axiuo has to be

lighter than a few keV which in tu rn pu ts a severe limit on the couplings A (and IJ in the

KVSZ case), typically A i H)"4. If the decoup ling tempera tu re of the ax iuo is larger than

the rehea t ing tempera tu re of inflation the overelosure bound on the regene ra ted ax ino

popu la t ion is weakened resulting in a sejineiwhaf weaker bound on A, typically A i 10"3.

In any case we conclude that a e-osmolivgical relevant LSP axino though possible in

principle is not very likely to occur : Special models are needed anel in add it ion sma ll

couplings have to be chosen in orde r lo avoid a conflict with the overelosure bound.

At this poinl it should be mentioned thai no scale supergravitv models can prov ide a

naturally light axino [!)]. Those1 models are1

ehtiriictcrbicd by a special pattern of the ,soft

te rms ; The only IKHI vanishing soil t.emis ;it the Planck scale are gaugino musses and

therefore A = B = C = I) at tree level. A pplying the above sta tements a small axino

mass re su l ts in this ca.se. In fact, a full supergravity calculation shows that the m a ss

van ishes on tree level. Other soft terms for the gauge non singlets can be genera ted dueto renormalizatioii group effects below the Planck scale in those models. Since the PQ-

sector consists of singlets their soft terms are not affected by renormalizatioii effee ' ts and

the axino (.saxion) remains massless [9j. Non vanishing masses can be however generated

via the one loop contribution w.s,ir»p- In the DFSZ-implementa t ion they are so sma ll tha t

cosmological effects of the axino ami the saxino are negligible. This is clearly different

in the KVSZ-case . However, the limit on g necessary to keep the axino mass below the



overclosure bound will be somewhat weakened with respect to the ordinary case since the

tr i l inea r coup ling A for g$QiQi originates from radiative corrections.

As a generic situation we consider models which do not fulfill our criterion for a small

tree level axiiio mass and possess couplings A — O{1). The axino mass in such models

is given by nia = O(m 3 /j) (no additional zero modes) or rnA = O(m :(/i./A-) (additional

zero modes) . First of all this mass has to be linger than the m ass of the ordinary LSP

in the MSSM to allow for a decay of axinos. Secoml, these decays have to occur before

the L SP decouples which trans lates into a liound on /»,-, of typically ma i. 1IKI GeV for

JPQ = 10" GeV if the decay channel into top and stop is possible. O therwis e an even

stronge r bo und /»,-, ~ TeV is required. We see tliat these generic models are significantly

constrained liv cosmological considerations, however, a linal decision depends on details

of the model like t lie exact iixino mass, the sfeniiioii masses, t lie PQ scale etc. T hereliiri:

it might, he iiiiiTesi ing to hliidy models which put 111 it her i oust rail its mi these param eters

like e. g. siip crsv ijiiu etnc unified mod els uicorporot inf; an axion . F<n- a 1'Q s cale in the

upper hull'of the allowed range fpQ > It!" GeV the lower bounds on niA and the saxion

mass m s (which both increase qvmdratieally with //>y) become very stringent and it might,

be difficult to construct, viable models, hi this context, having axmo masses larger than

TH.ty2 in model s with ad ditiona l zero modes might be an interest ing o ption.

Ac kn ow le dg me nt s Th is work was Mipporied liv the EC under con trac t No . SC]

CT91-072!). We would like to thank K. Choi and D. Mainlliolakis lor stimulating discus-

sions in the earlier development of this work, Iv.l.C, would like to thank Professor Abdus

Salam, the Intel-national Atomic Energy Agency, 1'NESCO mid the In te rna t iona l Cen tre

for Theoretical Physics, Trieste, for support.

References

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e..t. chun and a. lukas- axino mass in supergravity models

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