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Is axino dark matter possible in supergravity?
Physics Letters B 276 (1992) 103-107 North-Holland
PHYSICS LETTERS B
Is axino dark matter possible in supergravity? Toru G o t o ~ and Masahiro Yamaguchi 1,2 Department of Physics, Tohoku University, Sendal 980, Japan
Received 20 August 1991
The mass spectrum of an axion supermultiplet is examined in the context of supergravity. It is shown that in a no-scale supergravity a light axino can be realized, which serves as an interesting candidate of dark matter. Cosmological implications of an associated light saxion are also discussed.
Peccei-Quinn ( P Q ) symmetry solves the strong C P p r o b l e m [ 1 ]. The original model p r o p o s e d by Peccei and Quinn, however, causes a serious problem, because it predicts a light pseudoscalar, an axion, which should be detectable in accelerator experiments [ 2 ]. In order to a v o i d this trouble, the axion must be invisible, i.e. it must very weakly interact with o r d i n a r y particles [ 3-5 ]. In a s u p e r s y m m e t r i c ( S U S Y ) theory, the axion is inevitably associated with its fermionic partner (an axino) and a real scalar (in this paper, we call it a saxion), and they form a chiral supermultiplet. Since the axino very weakly couples to o r d i n a r y particles, it will not lead to any significant consequences on accelerator experiments just like the axion. H o w e v e r such a weakly interacting particle m a y play an i m p o r tant role in astrophysics a n d cosmology. In particular, when the axino is light ( < O ( 1 ) k e Y ) a n d becomes the lightest superparticle ( L S P ) , it will be an interesting c a n d i d a t e o f d a r k m a t t e r [ 6 ]. This possibility has been pursued by several authors. In spontaneously broken global SUSY, the axino gets a mass o f the o r d e r of m ~ / M r , Q ~ keV at the tree-level [7], where ms is the SUSY-breaking scale (ms < O ( 10 3 ) G e V ) and MpQ is the breaking scale o f the PQ s y m m e t r y which is typically taken as ~ O ( 1012) GeV. I f the fermion mass does not receive Fellow of the Japan Society for the Promotion of Science. Address after September 1, 1991; Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3255, U.S.A.
a significant radiative correction, the axino remains light. It is sometimes argued that the soft SUSYbreaking terms do not affect the result, because the mass term o f a chiral fermion is not a soft term. Since it is natural to assume supergravity [ 8 ] when considering SUSY models, one is willing to ask what happens in the supergravity. In this letter, we will exa m i n e the mass spectrum o f the invisible axion chiral supermultiplet in the context o f supergravity. As far as we know, the mass spectrum has not been discussed precisely in the literature, although it is primarily i m p o r t a n t in discussing cosmological and astrophysical implications o f the axion multiplet. In m a n y phenomenologically consistent supergravity models, SUSY is assumed to be b r o k e n in a hidden sector which interacts with our observable sector only gravitationally. It will turn out that the mass spect r u m crucially depends on how the observable sector couples to the hidden sector. First, we will see in the m i n i m a l coupling to the hidden sector, the masses o f the axino and a saxion, the scalar c o m p o n e n t o f the axion supermultiplet, are always o f the o r d e r o f a gravitino mass m3/2, in contrast to the global SUSY case. If we take m3/: ~ O (100) GeV, the axino is not astrophysically interesting. Next we will explore the possibility o f a light axino. We will find that a massless axino can be realized in a no-scale supergravity m o d e l [9] at the tree-level up to an a n o m a l y effect. I f the axion multiplet couples to some heavy gauge non-singlet fields with an a p p r o p r i a t e strength, the axino and the saxion receive radiative corrections so
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
1 03
Volume 276, number 1,2
PHYSICS LETTERS B
that their masses lie in the cosmologically interesting region. Before calculating the masses of the axino and the saxion, we will summarize our notations and conventions and give some useful formulae [ 10]. Let 0 i be a scalar component of a chiral supermultiplet and 0" be its complex conjugate. It is convenient to introduce a total K~ihler potential G(0', 0*) = K ( 0 ' , 07) + l n l W ( 0 ' ) 12
( 1)
6 February 1992
Here 2 is a coupling constant o f the order of unity and the mass p a r a m e t e r f i s of the order of the PQ breaking scale MpQ. The PQ symmetry is spontaneously broken with the expectation values of the fields A and A':
(A)=(A'),.~f.
(6)
The direction of the Goldstone boson is then identified with
Here a K~ihler potential K is a real function of {0 i} and {0*} and a superpotential W is a holomorphic function of {0i}. We have used the units of Mel/ x ~ = 1. A scalar potential is given as
A,-(A-A')/x/2,
V=exp(G) [Gi(G-I)~GJ-3],
The invariance of the total K~ihler potential (4) under the U( 1 )pQ transformation implies
(2)
where G~- OG/OO~and G ~- OG/00* etc. From eq. (2), we can evaluate scalar mass terms. A gravitino mass is m 3 / 2 = ( e x p ( G / 2 ) ) where ( A ) represents the vacuum expectation value of a quantity A. A mass term of a chiral fermion is calculated as
(3) In eq. (3), we have taken an effect of the super-Higgs mechanism into account. First we consider the minimal coupling to the hidden sector. We are concerned with a model which has a (spontaneously broken) global symmetry, that is, the total K~ihler potential is invariant under a certain global transformation. In this case, it is known that mass terms for the fermionic partners of the Goldstone bosons are induced when SUSY is broken in the hidden sector and their magnitudes are of the order of the gravitino mass [ 11 ]. Instead of repeating general arguments, we will give an explicit example of a linear realization of the U ( 1 ) pQ symmetry in the minimal supergravity [12]. Let A and A' be fields with the PQ charge -t-2 and - 2 , respectively. Let Y be a PQ singlet field. Consider the total K~hler potential of the following form G = Iz[2+ IAI2+ IA' 12-1- I YI 2 (4)
where z is a field in the hidden sector. The superpotential of the observable sector g is assumed to be
g=2(AA'-f2)Y. 104
and the field orthogonal to this direction is given by
A2=_(A+A' ) / x / 2 .
GAA--GA,A' --GAA* + GA'A'* =O .
(8)
(9)
Differentiating this identity with respect to A and taking the expectation value, we find
(GAA) ( A ) + ( G A ) -- (GAA,) ( A ' ) -- ( G A ) ( A * )
M o = ( e x p ( G / 2 ) [ G o + ~ G , @ - G k ( G - ~ ) lkG i jt] ) .
+lnlh(z)+g(A,A', Y) 12,
(7)
(5)
--0.
(10)
The conditions for the vacuum configuration which are written as ( V i ) = 0 and ( V ) = 0 imply that ( G A ) < < ( A ) in our case. Using eq. (6) and ( GA A ) = 1, we find
( G A A ) - - ( GAA, )>~I .
(11)
Similarly, we get
(GA'A')-- (GAA') ~ 1.
(12)
Then the axino mass can be calculated as
MA,A, = ½(M~A --2M~,, +MA'A' ) ½m3/2((GAA) --2(GAA, ) + ( GA,A, ) ) ,~ m 3 / 2 .
(13)
Therefore we see that the axino has a mass of the order of m3/2 in the case at hand. We can also see that the saxion has a mass of the order of the gravitino mass. The fact that the axino and the saxion have masses ~ m3/2 is a general feature and the same spectrum is obtained even when we consider a non-linear realization of the global symmetry instead of a linear realization described above.
Volume 276, number 1,2
PHYSICS LETTERS B
We should note that (GAA,) ~ 1 implies ( Y ) ~
m3/2. In the global SUSY theory with the superpotential (5), we get ( Y ) = 0, which results in a vanishing axino mass. The shift of the vacuum expectation value is inevitable in the supergravity model described above and we see that the masses of the fermionic partner of a Goldstone boson are different in the global SUSY and in the local case. Let us now explore the possibility of a light axino. In fact, the axino mass depends on how the axion (axino) sector is coupled to the hidden sector. Specifically, when we take a no-scale supergravity model [ 9 ] as the hidden sector, we get an axino with a vanishing mass at the tree-level. Consider a model with a total K~ihler potential G:
G= - 3 ln(z+ z* +aK) +lnl WI 2 ,
(14)
where K is a real function of scalar fields in the observable sector and W is the superpotential which is independent of a hidden sector field z. a is a negative constant which is set so that the kinetic terms for the fields in the observable sector are properly normalized: ( G~ ) = ~ . Here, the suffices indicate the fields in the observable sector only. In this case, the gravitino mass is obtained as m 3 / 2 = ( I Wl(z+z*+ otK) -3/2) and the scalar potential becomes 1
V=-- 3a(z+z, +otK) 2 W*i(K-I)~Wj.
(15)
Since a is negative, the scalar potential is positive semi-definite, therefore the vacuum configuration satisfies ( W i ) =0. Hence the fermion mass term in the observable sector becomes
M..__/_ ( W~/ W) 1/2 'J - - \ (Z+ Z* + OlK)3/2
× ( Wo- ~ W~Wj-K~(K-~ )~ Wt) ) = ( ( z(W~ / W )K/)~/ z Wij) .
(16)
We can see from eqs. ( 15 ) and ( 16 ) that the scalar potential and the fermion masses in the no-scale model have the same form as those in the unbroken global SUSY case despite that SUSY is spontaneously broken in the hidden sector. Thus, the axino and the saxion remain massless at the tree-level.
6 February 1992
As is well-known, the PQ symmetry is not an exact symmetry but is explicitly broken by an anomaly effect. Due to this effect, the axion gets a mass ma
f,~m,~ MpQ
10-
5(1012 GeV) ~ eV.
(17)
As we have seen above, the masses of the axino and the saxion are the same as the axion mass at the treelevel. Such a light scalar saxion causes a fifth force whose interaction range is ~ (MI,o/1012 GeV) cm with much stronger coupling than the gravitational interaction and therefore it is experimentally ruled out. This difficulty can be overcome by radiative corrections. The axino and the saxion can receive non-negligible radiative corrections when relevant diagrams pick up soft SUSY-breaking terms. In the no-scale model, the only soft terms at the tree-level are gauge fermion mass terms. The soft breaking masses for non-singlet scalars emerge at the one-loop level. One may expect the soft breaking mass parameters to be roughly of the order of 100 GeV, although the precise values depend on the detail of the model one considers. On the other hand, a gauge singlet scalar such as the Y field in the above example can have a soft breaking mass only through higher loop corrections and it is negligible. Therefore the radiative corrections to the masses of the axino and the saxion crucially depend on how the axion supermultiplet couples to gauge nonsinglet fields. When the axion very weakly interacts with the gauge non-singlets as in the DFSZ axion [4 ], the radiative corrections are negligible. Then the saxion remains too light and hence this case is excluded experimentally as explained above. Next consider the case where the axion supermultiplet not-so-weakly couples to gauge non-singlets. To be specific, we will consider a SUSY version [ 12 ] of the KSVZ axion model [ 3 ]. Generalizations to other models including a left-right symmetric theory are straightforward. In this model, two chiral supermultiplets Q and Q are introduced. Q and ~ transform under the S U ( 3 ) c × S U ( 2 ) L × U ( 1 ) y × U ( I ) p Q as (3, 1, 0, - 1 ) and (3, 1, 0, - 1 ), respectively. They couple to the A field in the superpotential g' = kA(~(~,
(18)
where k is a coupling constant whose magnitude will 105
Volume 276, number 1,2
PHYSICS LETTERS B
6 February 1992
be discussed below. After the U(1)pQ s y m m e t r y breaking, the quarks Q, O have a mass
mQ ~ kMpQ .
Q
(19)
The diagram depicted in fig. 1 gives a radiative correction to the axino mass [ 13 ]
"k.
+ s......>....@.....>.._.s.
"
"" S s....__>..':::...... .::_....>_ ...... k2
Q
k2 m ~ ~ (4n)2AQ ,
(20)
where Aom Q denotes an off-diagonal mass term of the scalar c o m p o n e n t s o f the multiplets:
AQmQQQ+h.c. ,
(21)
Fig. 2. Feynman diagrams for the radiative corrections which generate a saxion mass.
m s2 ~ ~
where 0 ( 0 ) is the scalar c o m p o n e n t o f 0 ( ~ ) . AQ can be estimated by using renormalization-group equations and it is roughly AQ=O(102) GeV.
(22)
Using eqs. (21) and (22), we find that the axino mass m~ ~ 1 keV, which is relevant for the axino dark matter [6], can be acquired by choosing the coupling k as
k ~ 1 0 -3 .
(23)
In this case, the heavy quark mass becomes 9 MpQ mQ ~ kMpQ ~ l O (-l O-~-~ev) GeV .
(24)
Let us now turn to the saxion mass. The diagrams which contribute to the mass at one-loop level are given in fig. 2. The radiative correction is logarithmically divergent and we introduce a cutoff A, which we should take as the Planck scale or the G U T scale. Using this cutoff, it is easy to see that the saxion mass ms becomes
k2 (4n) 2 (m~+mLft)ln
>
ik
k:
<
mQ Fig. 1. Feynman diagram for the radiative correction which generates an axino mass. 106
(
2 + m soft."~
re°A2
k2 - 2 c ( 4 n ) 2 msoft ,
]
(25)
where resort is the soft SUSY-breaking mass o f the scalar and c is some constant o f the order o f 10. Substituting k ~ 10- 3 and resort ~ 102 GeV, we find rns ~ 101-2 M e V .
(26)
The interaction length due to this scalar exchange is now 1~ms < l0 -12 cm. Before closing this paper, we would like to discuss cosmological implications o f the light scalar al. First consider the case where the saxion mass ms > 10 MeV. Such a saxion emits high energy photons when it decays, and thus it m a y destroy the light nuclei i f the lifetime is longer than 10 2 s. However, when the saxion mass is heavy enough to decay into a m u o n pair, the lifetime of the saxion becomes 1
AQ mQ
mQ2 in
3//
MpQ
2
2
"~ [ 10 MeV'~
tlO' Oev) t
.)s.
Therefore if m s > O ( 10 2) MeV and M v Q < O ( 1 0 1 ° ) GeV, then rs becomes much below 1 s and such a scalar does not give any contradiction with the nucleosynthesis scenario. On the other hand, the scalar with m s < 10 MeV is a dangerous source o f the distortion o f the cosmic background radiations [14]. The unobserved distortion o f the cosmic background ra~ We would like to thank T. Yanagida and M. Kawasaki for discussions on this issue.
Volume 276, number 1,2 diations requires that 10~2GeV.
PHYSICS LETTERS B
MpQ is
m u c h greater t h a n
A m o d e l - d e p e n d e n t analysis is necessary to evaluate the masses o f the a x i o n s u p e r m u l t i p l e t m o r e precisely. Such an analysis enables us to discuss t h e i r c o s m o l o g i c a l i m p l i c a t i o n s m o r e closely. We w o u l d like to t h a n k T. Y a n a g i d a for suggesting the subject to us a n d carefully r e a d i n g the m a n u script. W e are also grateful to K. I n o u e , M. K a w a s a k i , Y. O k a d a a n d T. Y a n a g i d a for helpful discussions. T h e w o r k o f T . G . is s u p p o r t e d in part by the G r a n t i n - A i d for Scientific R e s e a r c h f r o m the M i n i s t r y o f E d u c a t i o n , Science a n d C u l t u r e o f J a p a n N o . 0446 a n d the w o r k o f M . Y . by the G r a n t - i n - A i d for Scientific R e s e a r c h f r o m the M i n i s t r y o f E d u c a t i o n , Science a n d C u l t u r e o f J a p a n No. 029 52019.
References [1]R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. 38 (1977) 1440; Phys. Rev. D 16 (1977) 1791. [2] S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilczek, Phys. Rev. Lett. 40 (1978) 279.
6 February 1992
[ 3 ] J.E. Kim, Phys. Rev. Lett. 43 ( 1979 ) 103; M.A. Shifman, V.I. Vainstein and V.I. Zakharov, Nucl. Phys. B 166 (1980) 493. [4] M. Dine, W. Fischler and M. Srednicki, Phys. Lett. B 104 (1981) 199; A.P. Zhitnitskii, Sov. J. Nucl. Phys. 31 (1980) 260. [5] For a review, see, J.E. Kim, Phys. Rep. 150 (1987) 1. [6] K. Rajagopal, M.S. Turner and F. Wilczek, Nucl. Phys. B 358 (1991) 447. [7] K. Tamvakis and D. Wyler, Phys. Lett. B 112 (1982) 451; J.F. Nieves, Phys. Rev. D 33 (1986) 1762. [ 8 ] H.P. Nilles, Phys. Rep. 110 ( 1984 ) 1. [ 9 ] A.B. Lahanas and D.V. Nanopoulos, Phys. Rep. 145 (1987) 1; J. Ellis, C. Kounnas and D.V. Nanopoulos, Nucl. Phys. B 247 (1984) 373. [ 10] E. Cremmer, S. Ferrara, L. Girardello and A. van Proyen, Phys. Lett. B 116 (1982) 231; Nucl. Phys. B 212 (1983) 413. [ I 1] K. Inoue, A. Kakuto and H. Takano, Prog. Theor. Phys. 75 (1986) 664; A.A. Anselm and A.A. Johansen, Phys. Lett. B 200 ( 1988 ) 331. [12] J.E. Kim, Phys. Lett. B 136 (1984) 378. [ 13 ] P. Moxhay and K. Yamamoto, Phys. Len. B 151 (1985) 363. [14]M. Fukugita, M. Kawasaki and T. Yanagida, Tohoku University preprint TU-347 (1989), unpublished.
107
PHYSICS LETTERS B
Is axino dark matter possible in supergravity? Toru G o t o ~ and Masahiro Yamaguchi 1,2 Department of Physics, Tohoku University, Sendal 980, Japan
Received 20 August 1991
The mass spectrum of an axion supermultiplet is examined in the context of supergravity. It is shown that in a no-scale supergravity a light axino can be realized, which serves as an interesting candidate of dark matter. Cosmological implications of an associated light saxion are also discussed.
Peccei-Quinn ( P Q ) symmetry solves the strong C P p r o b l e m [ 1 ]. The original model p r o p o s e d by Peccei and Quinn, however, causes a serious problem, because it predicts a light pseudoscalar, an axion, which should be detectable in accelerator experiments [ 2 ]. In order to a v o i d this trouble, the axion must be invisible, i.e. it must very weakly interact with o r d i n a r y particles [ 3-5 ]. In a s u p e r s y m m e t r i c ( S U S Y ) theory, the axion is inevitably associated with its fermionic partner (an axino) and a real scalar (in this paper, we call it a saxion), and they form a chiral supermultiplet. Since the axino very weakly couples to o r d i n a r y particles, it will not lead to any significant consequences on accelerator experiments just like the axion. H o w e v e r such a weakly interacting particle m a y play an i m p o r tant role in astrophysics a n d cosmology. In particular, when the axino is light ( < O ( 1 ) k e Y ) a n d becomes the lightest superparticle ( L S P ) , it will be an interesting c a n d i d a t e o f d a r k m a t t e r [ 6 ]. This possibility has been pursued by several authors. In spontaneously broken global SUSY, the axino gets a mass o f the o r d e r of m ~ / M r , Q ~ keV at the tree-level [7], where ms is the SUSY-breaking scale (ms < O ( 10 3 ) G e V ) and MpQ is the breaking scale o f the PQ s y m m e t r y which is typically taken as ~ O ( 1012) GeV. I f the fermion mass does not receive Fellow of the Japan Society for the Promotion of Science. Address after September 1, 1991; Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3255, U.S.A.
a significant radiative correction, the axino remains light. It is sometimes argued that the soft SUSYbreaking terms do not affect the result, because the mass term o f a chiral fermion is not a soft term. Since it is natural to assume supergravity [ 8 ] when considering SUSY models, one is willing to ask what happens in the supergravity. In this letter, we will exa m i n e the mass spectrum o f the invisible axion chiral supermultiplet in the context o f supergravity. As far as we know, the mass spectrum has not been discussed precisely in the literature, although it is primarily i m p o r t a n t in discussing cosmological and astrophysical implications o f the axion multiplet. In m a n y phenomenologically consistent supergravity models, SUSY is assumed to be b r o k e n in a hidden sector which interacts with our observable sector only gravitationally. It will turn out that the mass spect r u m crucially depends on how the observable sector couples to the hidden sector. First, we will see in the m i n i m a l coupling to the hidden sector, the masses o f the axino and a saxion, the scalar c o m p o n e n t o f the axion supermultiplet, are always o f the o r d e r o f a gravitino mass m3/2, in contrast to the global SUSY case. If we take m3/: ~ O (100) GeV, the axino is not astrophysically interesting. Next we will explore the possibility o f a light axino. We will find that a massless axino can be realized in a no-scale supergravity m o d e l [9] at the tree-level up to an a n o m a l y effect. I f the axion multiplet couples to some heavy gauge non-singlet fields with an a p p r o p r i a t e strength, the axino and the saxion receive radiative corrections so
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
1 03
Volume 276, number 1,2
PHYSICS LETTERS B
that their masses lie in the cosmologically interesting region. Before calculating the masses of the axino and the saxion, we will summarize our notations and conventions and give some useful formulae [ 10]. Let 0 i be a scalar component of a chiral supermultiplet and 0" be its complex conjugate. It is convenient to introduce a total K~ihler potential G(0', 0*) = K ( 0 ' , 07) + l n l W ( 0 ' ) 12
( 1)
6 February 1992
Here 2 is a coupling constant o f the order of unity and the mass p a r a m e t e r f i s of the order of the PQ breaking scale MpQ. The PQ symmetry is spontaneously broken with the expectation values of the fields A and A':
(A)=(A'),.~f.
(6)
The direction of the Goldstone boson is then identified with
Here a K~ihler potential K is a real function of {0 i} and {0*} and a superpotential W is a holomorphic function of {0i}. We have used the units of Mel/ x ~ = 1. A scalar potential is given as
A,-(A-A')/x/2,
V=exp(G) [Gi(G-I)~GJ-3],
The invariance of the total K~ihler potential (4) under the U( 1 )pQ transformation implies
(2)
where G~- OG/OO~and G ~- OG/00* etc. From eq. (2), we can evaluate scalar mass terms. A gravitino mass is m 3 / 2 = ( e x p ( G / 2 ) ) where ( A ) represents the vacuum expectation value of a quantity A. A mass term of a chiral fermion is calculated as
(3) In eq. (3), we have taken an effect of the super-Higgs mechanism into account. First we consider the minimal coupling to the hidden sector. We are concerned with a model which has a (spontaneously broken) global symmetry, that is, the total K~ihler potential is invariant under a certain global transformation. In this case, it is known that mass terms for the fermionic partners of the Goldstone bosons are induced when SUSY is broken in the hidden sector and their magnitudes are of the order of the gravitino mass [ 11 ]. Instead of repeating general arguments, we will give an explicit example of a linear realization of the U ( 1 ) pQ symmetry in the minimal supergravity [12]. Let A and A' be fields with the PQ charge -t-2 and - 2 , respectively. Let Y be a PQ singlet field. Consider the total K~hler potential of the following form G = Iz[2+ IAI2+ IA' 12-1- I YI 2 (4)
where z is a field in the hidden sector. The superpotential of the observable sector g is assumed to be
g=2(AA'-f2)Y. 104
and the field orthogonal to this direction is given by
A2=_(A+A' ) / x / 2 .
GAA--GA,A' --GAA* + GA'A'* =O .
(8)
(9)
Differentiating this identity with respect to A and taking the expectation value, we find
(GAA) ( A ) + ( G A ) -- (GAA,) ( A ' ) -- ( G A ) ( A * )
M o = ( e x p ( G / 2 ) [ G o + ~ G , @ - G k ( G - ~ ) lkG i jt] ) .
+lnlh(z)+g(A,A', Y) 12,
(7)
(5)
--0.
(10)
The conditions for the vacuum configuration which are written as ( V i ) = 0 and ( V ) = 0 imply that ( G A ) < < ( A ) in our case. Using eq. (6) and ( GA A ) = 1, we find
( G A A ) - - ( GAA, )>~I .
(11)
Similarly, we get
(GA'A')-- (GAA') ~ 1.
(12)
Then the axino mass can be calculated as
MA,A, = ½(M~A --2M~,, +MA'A' ) ½m3/2((GAA) --2(GAA, ) + ( GA,A, ) ) ,~ m 3 / 2 .
(13)
Therefore we see that the axino has a mass of the order of m3/2 in the case at hand. We can also see that the saxion has a mass of the order of the gravitino mass. The fact that the axino and the saxion have masses ~ m3/2 is a general feature and the same spectrum is obtained even when we consider a non-linear realization of the global symmetry instead of a linear realization described above.
Volume 276, number 1,2
PHYSICS LETTERS B
We should note that (GAA,) ~ 1 implies ( Y ) ~
m3/2. In the global SUSY theory with the superpotential (5), we get ( Y ) = 0, which results in a vanishing axino mass. The shift of the vacuum expectation value is inevitable in the supergravity model described above and we see that the masses of the fermionic partner of a Goldstone boson are different in the global SUSY and in the local case. Let us now explore the possibility of a light axino. In fact, the axino mass depends on how the axion (axino) sector is coupled to the hidden sector. Specifically, when we take a no-scale supergravity model [ 9 ] as the hidden sector, we get an axino with a vanishing mass at the tree-level. Consider a model with a total K~ihler potential G:
G= - 3 ln(z+ z* +aK) +lnl WI 2 ,
(14)
where K is a real function of scalar fields in the observable sector and W is the superpotential which is independent of a hidden sector field z. a is a negative constant which is set so that the kinetic terms for the fields in the observable sector are properly normalized: ( G~ ) = ~ . Here, the suffices indicate the fields in the observable sector only. In this case, the gravitino mass is obtained as m 3 / 2 = ( I Wl(z+z*+ otK) -3/2) and the scalar potential becomes 1
V=-- 3a(z+z, +otK) 2 W*i(K-I)~Wj.
(15)
Since a is negative, the scalar potential is positive semi-definite, therefore the vacuum configuration satisfies ( W i ) =0. Hence the fermion mass term in the observable sector becomes
M..__/_ ( W~/ W) 1/2 'J - - \ (Z+ Z* + OlK)3/2
× ( Wo- ~ W~Wj-K~(K-~ )~ Wt) ) = ( ( z(W~ / W )K/)~/ z Wij) .
(16)
We can see from eqs. ( 15 ) and ( 16 ) that the scalar potential and the fermion masses in the no-scale model have the same form as those in the unbroken global SUSY case despite that SUSY is spontaneously broken in the hidden sector. Thus, the axino and the saxion remain massless at the tree-level.
6 February 1992
As is well-known, the PQ symmetry is not an exact symmetry but is explicitly broken by an anomaly effect. Due to this effect, the axion gets a mass ma
f,~m,~ MpQ
10-
5(1012 GeV) ~ eV.
(17)
As we have seen above, the masses of the axino and the saxion are the same as the axion mass at the treelevel. Such a light scalar saxion causes a fifth force whose interaction range is ~ (MI,o/1012 GeV) cm with much stronger coupling than the gravitational interaction and therefore it is experimentally ruled out. This difficulty can be overcome by radiative corrections. The axino and the saxion can receive non-negligible radiative corrections when relevant diagrams pick up soft SUSY-breaking terms. In the no-scale model, the only soft terms at the tree-level are gauge fermion mass terms. The soft breaking masses for non-singlet scalars emerge at the one-loop level. One may expect the soft breaking mass parameters to be roughly of the order of 100 GeV, although the precise values depend on the detail of the model one considers. On the other hand, a gauge singlet scalar such as the Y field in the above example can have a soft breaking mass only through higher loop corrections and it is negligible. Therefore the radiative corrections to the masses of the axino and the saxion crucially depend on how the axion supermultiplet couples to gauge nonsinglet fields. When the axion very weakly interacts with the gauge non-singlets as in the DFSZ axion [4 ], the radiative corrections are negligible. Then the saxion remains too light and hence this case is excluded experimentally as explained above. Next consider the case where the axion supermultiplet not-so-weakly couples to gauge non-singlets. To be specific, we will consider a SUSY version [ 12 ] of the KSVZ axion model [ 3 ]. Generalizations to other models including a left-right symmetric theory are straightforward. In this model, two chiral supermultiplets Q and Q are introduced. Q and ~ transform under the S U ( 3 ) c × S U ( 2 ) L × U ( 1 ) y × U ( I ) p Q as (3, 1, 0, - 1 ) and (3, 1, 0, - 1 ), respectively. They couple to the A field in the superpotential g' = kA(~(~,
(18)
where k is a coupling constant whose magnitude will 105
Volume 276, number 1,2
PHYSICS LETTERS B
6 February 1992
be discussed below. After the U(1)pQ s y m m e t r y breaking, the quarks Q, O have a mass
mQ ~ kMpQ .
Q
(19)
The diagram depicted in fig. 1 gives a radiative correction to the axino mass [ 13 ]
"k.
+ s......>....@.....>.._.s.
"
"" S s....__>..':::...... .::_....>_ ...... k2
Q
k2 m ~ ~ (4n)2AQ ,
(20)
where Aom Q denotes an off-diagonal mass term of the scalar c o m p o n e n t s o f the multiplets:
AQmQQQ+h.c. ,
(21)
Fig. 2. Feynman diagrams for the radiative corrections which generate a saxion mass.
m s2 ~ ~
where 0 ( 0 ) is the scalar c o m p o n e n t o f 0 ( ~ ) . AQ can be estimated by using renormalization-group equations and it is roughly AQ=O(102) GeV.
(22)
Using eqs. (21) and (22), we find that the axino mass m~ ~ 1 keV, which is relevant for the axino dark matter [6], can be acquired by choosing the coupling k as
k ~ 1 0 -3 .
(23)
In this case, the heavy quark mass becomes 9 MpQ mQ ~ kMpQ ~ l O (-l O-~-~ev) GeV .
(24)
Let us now turn to the saxion mass. The diagrams which contribute to the mass at one-loop level are given in fig. 2. The radiative correction is logarithmically divergent and we introduce a cutoff A, which we should take as the Planck scale or the G U T scale. Using this cutoff, it is easy to see that the saxion mass ms becomes
k2 (4n) 2 (m~+mLft)ln
>
ik
k:
<
mQ Fig. 1. Feynman diagram for the radiative correction which generates an axino mass. 106
(
2 + m soft."~
re°A2
k2 - 2 c ( 4 n ) 2 msoft ,
]
(25)
where resort is the soft SUSY-breaking mass o f the scalar and c is some constant o f the order o f 10. Substituting k ~ 10- 3 and resort ~ 102 GeV, we find rns ~ 101-2 M e V .
(26)
The interaction length due to this scalar exchange is now 1~ms < l0 -12 cm. Before closing this paper, we would like to discuss cosmological implications o f the light scalar al. First consider the case where the saxion mass ms > 10 MeV. Such a saxion emits high energy photons when it decays, and thus it m a y destroy the light nuclei i f the lifetime is longer than 10 2 s. However, when the saxion mass is heavy enough to decay into a m u o n pair, the lifetime of the saxion becomes 1
AQ mQ
mQ2 in
3//
MpQ
2
2
"~ [ 10 MeV'~
tlO' Oev) t
.)s.
Therefore if m s > O ( 10 2) MeV and M v Q < O ( 1 0 1 ° ) GeV, then rs becomes much below 1 s and such a scalar does not give any contradiction with the nucleosynthesis scenario. On the other hand, the scalar with m s < 10 MeV is a dangerous source o f the distortion o f the cosmic background radiations [14]. The unobserved distortion o f the cosmic background ra~ We would like to thank T. Yanagida and M. Kawasaki for discussions on this issue.
Volume 276, number 1,2 diations requires that 10~2GeV.
PHYSICS LETTERS B
MpQ is
m u c h greater t h a n
A m o d e l - d e p e n d e n t analysis is necessary to evaluate the masses o f the a x i o n s u p e r m u l t i p l e t m o r e precisely. Such an analysis enables us to discuss t h e i r c o s m o l o g i c a l i m p l i c a t i o n s m o r e closely. We w o u l d like to t h a n k T. Y a n a g i d a for suggesting the subject to us a n d carefully r e a d i n g the m a n u script. W e are also grateful to K. I n o u e , M. K a w a s a k i , Y. O k a d a a n d T. Y a n a g i d a for helpful discussions. T h e w o r k o f T . G . is s u p p o r t e d in part by the G r a n t i n - A i d for Scientific R e s e a r c h f r o m the M i n i s t r y o f E d u c a t i o n , Science a n d C u l t u r e o f J a p a n N o . 0446 a n d the w o r k o f M . Y . by the G r a n t - i n - A i d for Scientific R e s e a r c h f r o m the M i n i s t r y o f E d u c a t i o n , Science a n d C u l t u r e o f J a p a n No. 029 52019.
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6 February 1992
[ 3 ] J.E. Kim, Phys. Rev. Lett. 43 ( 1979 ) 103; M.A. Shifman, V.I. Vainstein and V.I. Zakharov, Nucl. Phys. B 166 (1980) 493. [4] M. Dine, W. Fischler and M. Srednicki, Phys. Lett. B 104 (1981) 199; A.P. Zhitnitskii, Sov. J. Nucl. Phys. 31 (1980) 260. [5] For a review, see, J.E. Kim, Phys. Rep. 150 (1987) 1. [6] K. Rajagopal, M.S. Turner and F. Wilczek, Nucl. Phys. B 358 (1991) 447. [7] K. Tamvakis and D. Wyler, Phys. Lett. B 112 (1982) 451; J.F. Nieves, Phys. Rev. D 33 (1986) 1762. [ 8 ] H.P. Nilles, Phys. Rep. 110 ( 1984 ) 1. [ 9 ] A.B. Lahanas and D.V. Nanopoulos, Phys. Rep. 145 (1987) 1; J. Ellis, C. Kounnas and D.V. Nanopoulos, Nucl. Phys. B 247 (1984) 373. [ 10] E. Cremmer, S. Ferrara, L. Girardello and A. van Proyen, Phys. Lett. B 116 (1982) 231; Nucl. Phys. B 212 (1983) 413. [ I 1] K. Inoue, A. Kakuto and H. Takano, Prog. Theor. Phys. 75 (1986) 664; A.A. Anselm and A.A. Johansen, Phys. Lett. B 200 ( 1988 ) 331. [12] J.E. Kim, Phys. Lett. B 136 (1984) 378. [ 13 ] P. Moxhay and K. Yamamoto, Phys. Len. B 151 (1985) 363. [14]M. Fukugita, M. Kawasaki and T. Yanagida, Tohoku University preprint TU-347 (1989), unpublished.
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