Anomaly-Induced Inflaton Decay and Gravitino-Overproduction Problem

0.19 MB·English

by  Motoi Endo

DESY 07-001
Anomaly-Induced Inflaton Decay and Gravitino-Overproduction Problem
Motoi Endo1, Fuminobu Takahashi1, and T. T. Yanagida2,3
1 Deutsches Elektronen Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany
2Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
3Research Center for the Early Universe, University of Tokyo, Tokyo 113-0033, Japan
We point out that the inflaton spontaneously decays into any gauge bosons and gauginos via the
super-Weyl, K¨ahler and sigma-model anomalies in supergravity, once the inflaton acquires a non-
vanishing vacuum expectation value. In particular, in the dynamical supersymmetry breaking sce-
narios, theinflaton necessarily decays intothesupersymmetry breaking sector, if theinflaton mass
is larger than the dynamical scale. This generically causes the overproduction of the gravitinos,
which severely constrains theinflation models.
7
0 PACSnumbers: 98.80.Cq,11.30.Pb,04.65.+e
0
2
Inflation [1] not only solves basic problems in the big concentrate on the effect of the super-Weyl anomaly, for
n
bang cosmology such as the horizon and flatness prob- simplicity. TheotherK¨ahlerandsigma-modelanomalies
a
J lems, but also provides a natural mechanism to gener- can affect the inflaton decay at the same order of mag-
ate density fluctuations necessary to form the present nitudeb. However, the following discussion is essentially
1
1 structure of the universe. In fact, the standard slow- unchanged even if these are includedc.
roll inflation predicts almost scale-invariant power spec- In the dynamical supersymmetry (SUSY) breaking

3 trum,whichfitstherecentcosmicmicrowavebackground (DSB) scenarios, SUSY is broken as a result of strong
v
data[2]quitewell. Duringinflation,theuniverseisdom- dynamics in a gauge theory. The couplings induced by
2
4 inated by the potential energyofan inflatonfield, andit thesuper-Weylanomalymakesitunavoidableforthein-
0 expandsexponentially[1,3]. After inflation, the inflaton flaton to decay into the hidden gauge bosons and gaug-
1 transfersitsenergytoathermalplasmabythedecayand inos, which subsequently produce gravitinos. The grav-
0 reheatsthe universe. Itisofgreatimportancetounravel itinoproductionturnsouttobeprevalentingenericDSB
7
the reheating processes to have a successful thermal his- models, which tightly constrainsboth the inflationmod-
0
/ tory after inflation. Indeed, the reheating is subject to elsandSUSYbreakingscenarios. Inparticular,aswewill
h severalconstraints;the reheating temperature should be see, the gravity-mediation scenario is almost excluded,
p
highenoughtogeneratethebaryonasymmetry,whilelow and high-scale inflation models such as hybrid [11] and
-
p enough to avoid the overproductionof unwanted relics. smooth hybrid [12] inflation models are severely con-
e Oneusuallyintroducescouplingsofthe inflatontothe strained. We stress that the gravitino production from
h
standard-model particles to cause its decay and hence theinflatondecayisalmostunavoidable,andthatitcan-
:
v reheating. The stronger couplings result in the higher not be solved by taking the reheating temperature low
i
X reheating temperature, and so, the couplings must be so enough. On the contrary, the lower reheating tempera-
weaktoevadetheoverproductionoftheunwantedrelics. ture makes the problem even worse.
r
a Insupergravity,forinstance,gravitinosareoverproduced We recently pointed out that the inflaton decays into
by particle scatterings in thermal plasma, if the reheat- matterfieldsinthevisibleandthehiddenSUSY-breaking
ing temperature is too high [4]. So far, it has been con- sectors through supergravityeffects, even without direct
sidered that one can avoid the cosmological difficulties couplings between them in the Einstein frame of the su-
associated with the unwanted relics (e.g. gravitinos), by pergravityd. We call it as a spontaneous decay [7]. (See
setting the coupling of the inflaton to the visible sector also Ref. [13] for the non-SUSY case.) Here, we show
weak enougha. thattheinflatondecaysintoanygaugebosonsandgaug-
Inthisletterweshowthat,oncetheinflatonacquiresa inos via the super-Weyl anomaly in the supergravity in
finitevacuumexpectationvalue(VEV),itspontaneously
decaysintoanygaugebosonsandgauginosviathequan-
tumeffects,anomaliesinsupergravity. Amongthesuper-
Weyl, K¨ahler, and sigma-model anomalies [8, 9], we will b Counterterms[10]mayalsocontributetotheinflatondecay.
c ThetotaleffectdependsontheformoftheKa¨hlerpotential. For
instance, inthecaseoftheminimalKa¨hlerpotential, thedecay
rateisproportionalto(TG−TR)2insteadofb20. Inthecaseofthe
Ka¨hlerpotentialofthesequesteredtype,thecontributionsfrom
a Itshouldbenoted,however,thatRefs.[5,6,7]recentlypointed the Ka¨hler andsigmamodelanomalies cancel, and thedecay is
out that the inflaton can decay into the gravitinos, which puts dominantlyinducedbythesuper-Weylanomaly.
severe constraints on both the inflation models and the super- d Apartofthedecaysisthoughtofasarisingfromdirectcouplings
symmetrybreakingscenarios. intheconformalframe.
2
addition to the spontaneous decay. where N is the number of the generators of the gauge
Let us assume that the inflaton does not have any di- group,αisdefinedasg2/4π,andweassumethecanonical
rectcouplingstothegaugesector. Then,theLagrangian normalization of the inflaton and gauge fields. Here we
ofthegaugemultipletsisinvariantunderthesuper-Weyl notice that the half of the decay rate comes from the
transformation at the classical level, and hence the in- decay into the two gauge bosons and the other half from
flaton decay into the gauge sector is prohibited at the that into the gaugino pair.
tree level [14]. However the symmetry is anomalous at Itisinterestingtocomparetheanomaly-induceddecay
the quantum level. The anomaly not only mediates the tothe recentlyobservedspontaneousdecaywhichoccurs
SUSY-breaking effects to the visible sector [15] but also at the tree level [7]. It was shown that the inflaton de-
enables the inflaton to couple to the gauge supermulti- cays into the matter fields in the visible and/or hidden
plets. By using the superfield description of the super- sectors, if the inflaton acquires a finite VEV. The decay
gravity, the 1PI effective Lagrangian of the super-Weyl proceeds via both the Yukawa interactions (with 3-body
anomaly is [8, 9] final states) and the mass terms in the superpotential,
g2b 1 even when there are no direct interaction terms in the
∆L= 64π02 Z d2Θ2EWαWα∂2(D¯2−8R)R¯+h.c. (1) Einstein frame. Thus, for a generic K¨ahler potential (in-
cludingtheminimalone),theinflatondecaymaybedom-
intheconformalframeandinthePlanckunits: MP =1. inatedbythespontaneousdecayviathetopYukawacou-
Here g is a gauge coupling constant, b = 3T −T is
0 G R pling. However, the anomaly-induced decay rate can be
the beta function coefficient, and Wα is a field strength comparabletothatofthespontaneousdecay,althoughit
of corresponding gauge supermultiplet. A sum over all
arisesatthe 1-looplevel. This is becausethe latterrates
matter representationsis understood. The chiraldensity are suppressed either by the phase space of the 3-body
E,theΘvariableandthecovariantderivativeDarethose final states, or by the mass ratio squared(M/m )2 as in
φ
defined in the supergravity [16]. Note that the inflaton
the case of the decay into the right-handed (s)neutrinos
lKin∗e∂arφly∗)c,oanntrdibtuhteessutpoetrhsepaRc-ecucurrrevnattuarsebRa¯∼co2ni(tKainφs∂aR¯φ=− with a Majorana mass M satisfying 2M <mφ.
φ a In the DSB scenarios, SUSY is spontaneously broken
−1(M∗ + Θ2(−1R + iemD ba) + ···), where M is a
6 2 a m as a result of non-perturbative dynamics in a gauge the-
auxiliaryfieldofthesupergravitymultiplet,andRisthe
ory, in which case the beta function coefficient is posi-
Einsteincurvaturescalar. Also,thecombination(K φ+
φ tive, b > 0. The dynamical scale of the hidden gauge
∗ ∗ 0
K φ ) appears in the scalar component of the graviton.
φ interactions is related to the SUSY breaking scale as
Theinflatonfieldthencouplestothegaugebosonsinthe
Λ = x m M , where m denotes the gravitino
3/2 P 3/2
Einstein frame as mass,anpdx>∼1representsamodel-dependentnumerical
∆L= g2b0 K φ F Fmn−iF F˜mn +h.c.(2) factor. The gauge bosons and gauginos have masses of
192π2 φM mn mn O(Λ) due to the strong couplings below the DSB scale.
P (cid:16) (cid:17)
Therefore, when the inflaton mass, m , is larger than
where F is a field strength of the gauge field and φ
mn
F˜mn = ǫmnklF /2. In addition, noting that the the DSB scale,the inflaton decaysinto the SUSY break-
kl ing sector via the super-Weyl anomalyf, since the de-
F-term satisfies the equation of the motion, Fi =
−eK/2Kij∗(W + K W)∗, the inflaton couples to the cay is kinematically allowed and the mass of the hidden
j j
(s)quarksaremuchsmallerthanm atthe decayvertex.
gaugino λ as φ
Let us consider how the decay proceeds. First, the in-
g2b m
∆L= 0K φφ∗λλ+h.c. (3) flaton decays into a pair of the hidden gauge bosons or
96π2 φMP gauginos, flying away to the opposite directions. Then
in the Einstein frame. Here we assume that the inflaton each of them interacts with the hidden (s)quarks and
massm isdominatedbythesupersymmetricmassterm, hadronizes due to the strong coupling, followed by cas-
φ
and used Fφ ≃−m . Therefore the inflaton field cou- cadedecaysoftheheavyhiddenhadronsintolighterones.
,φ∗ φ
plestothegaugesectorthroughthesuper-Weylanomaly The number of the hidden hadrons produced from each
as long as the K¨ahler potential contains a linear term of jet, which we call here as the multiplicity NH, depends
the inflaton, which is roughly given by the inflaton VEV onthedetailedstructureofthehiddensectorsuchasthe
hφie. The decay rate becomes gauge groups, the number of the matter multiplets, and
the mass spectrum of the hidden hadrons. We expect,
Γ(φ→gauge) ≃ Nα2b20|K |2m3φ, (4) however,thatNH isintherangeofO(1−102). Thehid-
4608π3 φ M2
P
f In addition, there may exists a decay into a messenger sector,
e Theanomaly-induceddecayissuppressedifhKφi=0,whilethe duetoadditionalgaugegroupsintroducedtomediatetheSUSY
anomalyitselfdoesnotvanishinthiscase. breakingeffects.
3
denhadronsshouldeventuallydecayandreleasetheiren- connection Γkij ≡Gkℓ∗Gijℓ∗. The gravitino abundance is
ergyintothevisiblesector. Thegravitinosarelikelytobe then given by
producedinthedecaysofthehiddenhadrons[17,18,19]
as well as in the cascade decay processes in jetsg. We Y = 2Γp3/a2ir3Trh ≃7×10−11η hφi 2
denote the averaged number of the gravitinos produced 3/2 Γ 4m (cid:18)1015GeV(cid:19)
φ φ
per each jet as Ng. Here we assume each hidden hadron mφ 2 106GeV
produces one gravitino in the end, and use the relation × , (7)
N ∼N h. The gravitino abundance is thereforei (cid:16)1012GeV(cid:17) (cid:18) Trh (cid:19)
g H
where we have assumed the minimal K¨ahler potential in
ΓH 3Trh the last equality. Although we have neglected the VEV
Y = 2N ,
3/2 gΓ 4m of z in (6), including the finite VEV that arises below
φ φ
m 2 106GeV the dynamical scale can make the rate even higher [19].
≃ 3×10−7ξ φ , (5) Indeed, taking account of the mixings and couplings be-
(cid:16)1012GeV(cid:17) (cid:18) Trh (cid:19)
tween the φ and z which are radiatively induced for
m < Λ, the above gravitino abundance increases. To
whereΓ is the partialdecayrateinto the hidden gauge φ
H
putitconcretely,suchanoperatoras|φ|2zzintheK¨ahler
sector given by Eq. (4), and Γ denotes the total decay
φ
potential is radiatively induced, and it additionally con-
rate of the inflaton, related to the reheating tempera-
ture as Trh ≡ (π2g∗/10)−14 ΓφMP. Here g∗ counts the tributes to the gravitino production for mφ > mz. The
mixingsintheK¨ahlerpotentialbecomesimportantespe-
relativistic degrees of freedpom, and we have substituted
cially for m <m .
g∗ =228.75inthesecondequalityofEq.(5). Wealsode- φ z
finedξ ≡Nα2b2N ,whereN andb dependontheSUSY Using (5) and(7), we can constrainthe inflationmod-
0 g 0
els. The results are summarized in Fig. 1, in which we
breaking scenarios. For instance, in the IYIT model [21]
take the maximalvalue of T allowedby the cosmologi-
with a SU(2) gaugegroupand four doublet chiralsuper- rh
calconstraints. ForlowerT ,theboundsonhφibecome
fields, we have N =3, and b0 =4. rh
1/2
It should be noted that the gravitinoabundance (5) is severer as ∝ Trh for a fixed mφ. We also set x = 1,
∗
inversely proportional to the reheating temperature [5]. Ng = 10, N = 3, α = 0.1, b0 = 4, hKφi = hφ i, mz = Λ
That is, for the lowerreheating temperature, more grav- asreferencevalues. We considerthefollowingfourcases.
itinos are produced. This should be contrasted to the In the casesA and B, we set m3/2 = 1TeV, assuming
thermally produced gravitinos, whose abundance is pro- the gravitino is unstable. The hadronic branching ratio
portional to the reheating temperature. For the rest of is given by Bh = 1(10−3) for the caseA (B). The grav-
the paper,we regardthe reheating temperatureas a free itino abundance in these cases are severely constrained
parameterbyintroducingappropriatedirectcouplingsof byBBN[26]. InthecaseC,m3/2 =1GeV,andthegrav-
the inflaton to the visible sector. We will take the maxi- itino is stable. In the caseD, we take m3/2 = 100TeV
malvalueallowedbycosmologicalconstraintstogivethe withthewinoLSPofamassgivenby2.7×10−3m3/2[15].
mostconservativeestimates onthe gravitinoabundance. The constraints on Trh and Y3/2 come from the require-
ment that the abundance of the gravitino (or the winos
The inflaton does not decay into the SUSY breaking
sectorifmφ <∼Λ. However,thegravitinopairproduction produced by the gravitino decay) should not exceed the
present dark matter abundance [27]. From the figure,
then becomes important [5]. The gravitino pair produc-
one can see that the high-scale inflation models such as
tion rate is [22]
thehybridinflationmodelareseverelyconstrained,while
η m3 the new inflation models may escape the bounds if Bh is
Γp3/a2ir ≃ 96π|∇φGz|2Mφ2 (6) suppressed even for m3/2 =1TeV.
P In this letter we have shown that the inflaton decays
into any gauge bosons and gauginos via the super-Weyl
with η = (m /m )4 for m > m and η = 1 for m <
z φ φ z φ anomaly in supergravity, once the inflaton acquires a
m , where G ≡ K + ln|W|2, and m is the mass of
z z nonzero VEV. In particular, the inflaton necessarily de-
the SUSY breaking field z with non-vanishing F-term.
cays into the SUSY breaking sector when the inflaton
Also ∇ G is defined by ∇ G ≡G −Γk G with the
φ z φ z φz φz k mass is larger than the DSB scale. This subsequently
producesthegravitinos,andthereforethegravitinoover-
production problem prevails among the DSB scenarios
g In particular, this is the case if the SUSY breaking field is a and most inflation models.
boundstateofthehidden(s)quarks. Let us mention that the anomaly-induced decay pro-
h In a class of the gauge-mediation models of SUSY breaking, cess and the associated gravitino problem shown above
theparticlesinthehiddensectormaydominantlydecayintothe
can be avoided in the following cases. In the chaotic in-
standard-modelparticles[20].
i Iftheinflatonspontaneouslydecayintothehiddensectoratthe flationmodelwithanapproximateZ2 symmetry[25,28],
treelevel[7],moregravitinoswillbeproduced. the VEV of the inflaton is so suppressed that both the
4
[2] D. N. Spergel et al.,arXiv:astro-ph/0603449.
[3] K. Sato, Mon. Not. Roy.Astron. Soc. 195, 467 (1981).
[4] S. Weinberg, Phys. Rev. Lett. 48, 1303 (1982);
L. M. Krauss, Nucl. Phys.B 227, 556 (1983).
[5] M. Kawasaki, F. Takahashi and T. T. Yanagida, Phys.
Lett. B 638, 8 (2006); Phys. Rev.D 74, 043519 (2006).
[6] T. Asaka, S. Nakamura and M. Yamaguchi, Phys. Rev.
D 74, 023520 (2006).
[7] M. Endo, M. Kawasaki, F. Takahashi and
T. T. Yanagida, Phys.Lett. B 642, 518 (2006).
[8] I. L. Buchbinder and S. M. Kuzenko,Phys. Lett. B 202
(1988) 233.
[9] G. Lopes Cardoso and B. A. Ovrut, Nucl. Phys. B 392,
315 (1993); Nucl. Phys. B 418, 535 (1994).
[10] J.A.Bagger,T.MoroiandE.Poppitz,JHEP0004,009
(2000).
[11] E. J. Copeland, A. R. Liddle, D. H. Lyth, E. D. Stew-
art and D. Wands, Phys. Rev. D 49, 6410 (1994);
FIG. 1: Constraints from the gravitino production by the G. R. Dvali, Q. Shafi and R. K. Schaefer, Phys. Rev.
inflaton decay, for m3/2 = 1TeV with Bh = 1 (caseA), Lett. 73, 1886 (1994); A.D. Linde and A. Riotto, Phys.
m3/2 = 1TeV with Bh = 10−3 (caseB), m3/2 = 100TeV Rev. D 56, 1841 (1997).
(caseC), and m = 1GeV (caseD). The region above the [12] G. Lazarides and C. Panagiotakopoulos, Phys. Rev. D
3/2
solid (red) line is excluded for each case. For mφ >∼ Λ, 52, 559 (1995).
we used the anomaly-induced inflaton decay into the hid- [13] Y. Watanabe and E. Komatsu, arXiv:gr-qc/0612120.
dengauge/gauginos toestimatethegravitinoabundance(5), [14] M. Endo, F. Takahashi and T. T. Yanagida, in prepara-
whilethegravitinopairproduction(7)wasusedfor mφ <∼Λ. tion.
Thetypicalvaluesofhφiandmφ forthesingle-fieldnew[23], [15] G. F. Giudice, M. A. Luty, H. Murayama and R. Rat-
multi-field new [24], hybrid [11] and smooth hybrid [12], and tazzi, JHEP 9812, 027 (1998); L. Randall and R. Sun-
chaotic [25] inflation models are also shown. Note that we drum,Nucl.Phys.B557,79(1999);J.A.Bagger,T.Mo-
adopt the chaotic inflation model without discrete symme- roi and E. Poppitz, in Ref. [10]; M. K. Gaillard and
tries,inwhichcasehKφiisexpectedtobearoundthePlanck B. D.Nelson, Nucl.Phys. B 588, 197 (2000).
scale. [16] J.WessandJ.Bagger,SupersymmetryandSupergravity,
(Princeton Unversity Press, 1992).
[17] R. Casalbuoni, S. De Curtis, D. Dominici, F. Feruglio
and R.Gatto, Phys.Lett. B 215, 313 (1988).
anomaly-induced decay and the spontaneous decay are [18] M. Dine, R. Kitano, A. Morisse and Y. Shirman, Phys.
suppressed. Similar arguments also apply to inflation Rev. D 73, 123518 (2006);
[19] M. Endo, K. Hamaguchi and F. Takahashi, Phys. Rev.
models in the no-scale supergravity [29].
D 74, 023531 (2006).
An interesting application of the anomaly-induced in- [20] M.FujiiandT.Yanagida,Phys.Lett.B549,273(2002);
flaton decay can be found in the case with the K¨ahler
K. Jedamzik, M. Lemoine and G. Moultaka, Phys. Rev.
potential of the sequestered type: K =−3ln[1−(|φ|2+ D 73, 043514 (2006).
|Q|2)/3], where Q collectively denotes the matter multi- [21] K.I.IzawaandT.Yanagida,Prog.Theor.Phys.95,829
plets[14]. Sincetherearenodirectcouplingsoftheinfla- (1996); K.A.IntriligatorandS.D.Thomas,Nucl.Phys.
tontothematterfieldsintheconformalframe,thepossi- B 473, 121 (1996).
[22] M. Endo, K. Hamaguchi and F. Takahashi, Phys. Rev.
bledecayprocessesarethosemediatedbythesupergrav-
Lett. 96, 211301 (2006); S. Nakamura and M. Yam-
ity multiplet. Then, only such an operator that violates aguchi,Phys.Lett.B638,389(2006); seealsoRefs.[18,
the conformal symmetry induces the inflaton decay, and
19].
so,thedecayviatheYukawacouplingsdoesnotoccurat [23] K. I. Izawa and T. Yanagida, Phys. Lett. B 393, 331
the tree level. Onthe contrary,the anomaly-inducedde- (1997).
cayisnotsuppressedeveninthiscase. Also,theinflaton [24] T.Asaka,K.Hamaguchi,M.KawasakiandT.Yanagida,
decaysintotheright-handed(s)neutrinos,sincetheright- Phys. Rev. D 61, 083512 (2000); V. N. Senoguz and
Q. Shafi,Phys. Lett. B 596, 8 (2004).
handedMajoranamassviolatestheconformalinvariance.
[25] M. Kawasaki, M. Yamaguchi and T. Yanagida, Phys.
This may naturally generates the baryonasymmetry via Rev. Lett. 85, 3572 (2000); Phys. Rev. D 63, 103514
leptogenesis scenario [30].
(2001).
[26] M.Kawasaki,K.KohriandT.Moroi,Phys.Lett.B625,
7 (2005); Phys.Rev.D 71, 083502 (2005).
[27] T. Moroi, H. Murayama and M. Yamaguchi, Phys.
Lett. B 303, 289 (1993); M. Bolz, A. Brandenburg and
[1] A. H. Guth, Phys. Rev. D 23, 347 (1981); A. D. Linde, W. Buchmuller, Nucl. Phys. B 606, 518 (2001); see
Phys. Lett. B 108, 389 (1982); A. Albrecht and alsoJ.PradlerandF.D.Steffen,arXiv:hep-ph/0608344;
P.J. Steinhardt,Phys. Rev.Lett. 48, 1220 (1982). arXiv:hep-ph/0612291.
5
[28] M. Endo, F. Takahashi and T. T. Yanagida, Phys. Rev. (1986); see, for a review, W. Buchmuller, R. D. Peccei
D 74, 123523 (2006). and T. Yanagida, Ann. Rev. Nucl. Part. Sci. 55, 311
[29] M. Endo, K. Kadota, K. A. Olive, F. Takahashi and (2005).
T. T. Yanagida, arXiv:hep-ph/0612263.
[30] M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45