A note on the moduli-induced gravitino problem

0.14 MB·English

by  Natalia Shuhmaher

hep-ph/0703319
March 2007
A note on the moduli-induced gravitino problem.
Natalia Shuhmaher∗
Department of Physics, McGill University, Montr´eal, QC, H3A 2T8, CANADA
D´epartement de Physique Th´eorique, Universit´e de Gen´eve,
24 Ernest Ansermet, 1211 Gen´eve 4, Switzerland
The cosmological moduli problem has been recently reconsidered. Papers [1, 2] show that even
heavymoduli(m >105 GeV) can beaproblem for cosmology if abranchingratio ofthemodulus
φ
9 into gravitini is large. In this paper, we discuss the tachyonic decay of moduli into the Standard
0 Model’s degrees of freedom, e.g. Higgs particles, as a resolution to the moduli-induced gravitino
0 problem. Roughestimatesonmodeldependentparameterssetalowerboundontheallowedmoduli
2 at around 108 ∼109 GeV.

n
PACSnumbers: 98.80.Cq.
a
J

8 I. INTRODUCTION lus mass. Moduli decay once the Hubble rate is of

4 the orderofΓall. Therefore,moduliofmassbelow
v The cosmological moduli problem is a disease 100TeVdecaynearorafterthe timeofnucleosyn-
9 of many supersymmetry/supergravity theories [3, thesis, when the universe is nearly 1 second old.
1 4, 5, 6]. Many supersymmetry/supergravity theo- If the mass is above 100 TeV then the moduli de-
3 ries containfields whichhave flat potentials in the cay before the time of Big Bang Nucleosynthesis
3 supersymetric limit and only Planck suppressed (BBN). Examples of scenarios with heavy moduli
0 couplings to Standard Model (SM) particles. We exist [15, 16, 17, 18, 19].
7
0 generically call them moduli. The cosmological The heavy moduli scenario as a solution of the
/ moduli problem arises whenever decays of mod- cosmological moduli problem has recently been
h uli are in conflict with cosmological observations. reconsidered starting with the papers [1, 2]. It
p
Masses of moduli depend on the type of super- was shown that the decay of moduli into grav-
-
p symmetry breaking. Moduli much lighter than itinos is unsuppressed (for an opposite example
e the Hubble scale during inflation acquire a vac- see [20]). The part of the Lagrangian describing
h uum expectation value (VEV) of order the Planck the gravitino-modulus couplings is
:
v scale[7,8]andevenexceeditifthemassofmoduli
1
Xi is not sufficiently high [9, 10]. In the last case,the e−1L = −8ǫµνρσ(Gφ∂ρφ+Gφ†∂ρφ†)ψ¯µγνψσ(2)
modulus field can become an inflaton. Later on, a
ar large abundance of moduli threatens to overclose −81eG/2(Gφφ+Gφ†φ†)ψ¯µ[γµ,γν]ψν(3)
the Universe or jeopardize the processes of nucle-
osynthesis. Several solutions of the moduli prob- where ψ stands for the gravitino andG is a non
µ φ
lem have been suggested, see e.g. [11, 12, 13, 14]. vanishing dimensionless auxiliary field with G =
The cosmological moduli problem is automati- K/M2+ln(W 2/M6). Thesubscriptidenotesthe
p | | p
cally avoidedin heavy moduli scenarios. A widely derivativewithrespecttothefieldi. K andW are
used estimate for the perturbative decay rate Γ K¨ahler potential and superpotential respectively.
all
of moduli is Based on these coupling, the perturbative decay
rate of moduli into gravitinos is
1 m3
Γall ∼ 4πMφ2 . (1) Gφ 2 m5φ
p Γ3/2 ≡Γ(φ→2ψ3/2)≈ |288|π m2 M2 . (4)
3/2 p
where φ is the modulus field and m is the modu-
φ
The auxiliary field of the modulus, G , in gen-
φ
eral, can be small to suppress Γ3/2 to the to-
tal decay rate Γ (1). However, suppressed G
all φ
∗Email: [email protected] is not a typical case, e.g. in the framework of
2
the 4D supergravity Gφ > m3/2/mφ. Perform- The equation of motion for χ field with switched
ing elaborate calculations,∼authors of [1, 2] have off the expansion of space is
shown that the typical branching ratio Br(φ
2 2
→ χ¨ + k +m χ = 0. (7)
2ψ3/2) (0.01 1). Thelargebranchingratioof k eff k
heavym∼oOduliint−ogravitinoscausesgravitinoover- where (cid:0) (cid:1)
production. Hence, even having a modulus mass
α
2 2 2 2
above 100 TeV does not resolve the cosmological m =m +λχ + m φ, (8)
eff χ M φ
moduli problem. A detailed re-analysisof the cos- p
mologicalmoduliproblemtakingintoaccountcon- The oscillations of the field φ induce a negative
straintsongravitinooverproductionpushesupthe massforthefieldχ. The modesofthefieldχ with
gravitino mass above 105 106 GeV [2]. This is k < m2 are excited,
the moduli-induced graviti−no problem. q− eff
moTdhueli-ipnrdeuvcioedusglyravpituinbolisphreodblelmitedroaetusrneotoinncluthdee χk ∝e√−m2eff−k2t (9)
nonperturbativedecaychannels. We proposeaso- and the energy is transferred from the oscillating
lution of the moduli-induced gravitinoproblemby φintoexcitationsofχinapreheating-likeprocess.
having most of the moduli energy decay into the The process has a name of tachyonic resonance or
SM degrees of freedom through a tachyonic decay tachyonic (p)reheating and is widely discussed in
into a boson pair, e.g. Higgs. The decay pro- the literaturestartingwith [21,22,23], inparticu-
cess moduli > bosons is rapid and occurs before lar, the implementation of tachyonic resonance in
−
moduli start to perturbatively decay into graviti- thecontextoftheresolutionofthemoduliproblem
nos. The scheme allows to find a range of scalar is discussed in [14]. Thus, we see that for a cer-
moduli masses (mφ >108 109 GeV) which does tainrangeofparameters,theenergydensitystored
∼
notsufferfromthemoduli-inducedgravitinoprob- in the moduli nonperturbatively transfers into ex-
lem. Making use of conservative approximations, citations of χ field much before moduli perturba-
we find a range of masses with no overproduction tively decay into gravitinos. The couplings of χ to
of gravitinos. Standard Model particles are assumed to be un-
suppressed and, as a result, the decay rate of χ
is much larger than 1 sec−1. Thus, the modulus
energyis convertedinto radiationmuchbefore the
II. BASIC IDEA
time of BBN.
To study the stability of the potential (6), we
The general idea can be introduced in the fol- findtheminimumoftheV(φ,χ)intheφdirection
lowing way. As was mentioned previously, mod- which occurs for
ulihaveonlyPlancksuppressedcouplingstoother
1 α
fields andduringinflationobtaina VEVofthe or- φ= χ2. (10)
−2M
der of the Planck scale. After inflation, the mod- p
ulus field slowly rolls preserving its energy. When Substituting (10) into V(φ,χ) leads to
theHubbleparameterreachesthevalueofm ,the
φ
modulus field starts to oscillate. In the following, 1 1 α2 2 4 1 2 2
V(φ,χ)= m λ χ + m χ ,
weassumethatmodulihaveatrilinearcouplingto −4(cid:18)2M2 φ− (cid:19) 2 χ
p
a scalar field χ, (11)
and, we see that the effective potential is unstable
2
φχ . (5) for
1 α2
2
The effective potential, V(φ,χ) is m >λ. (12)
2M2 φ
p
1 2 2 1 2 2 1 α 2 2 1 4 Thus,thepresenceofadditionaltermswithPlanck
V(φ,χ)= m φ + m χ + m φχ + λχ .
2 φ 2 χ 2M φ 4 suppressed couplings is important to stabilize the
p
(6) potential (6) at large values of the fields.
3
The efficiency of the tachyonic resonance must ied in [25]. The authors have shown that trilin-
be carefully checked against the effects of dilution ear terms lead to faster re-scatteringand thermal-
due to the expansion of space. For the tachyonic ization. As a bonus, trilinear terms allow com-
resonance to be effective, the growth of the mode plete decay of the moduli. In addition to positive
k(9)shalldominatethedilutionduetotheexpan- effects, enhanced resonance and fast subsequent
sionofspace. The appropriateconditionwouldbe thermalizationmayenlargethereheatingtempera-
ture beyond the allowedregionwhich threatens to
m2 k2 >H (13) overproduce gravitinos through re-scattering pro-
q− eff − cesses [26].
or The trilinear interaction term (5) may arise, for
α m2Φ> m2φΦ2 (14) eKx¨aahmleprlep,oftreonmtiatlh1e non-renormilizable term in the
M φ M2
p p
λ
whereΦisthe amplitude ofthe φfield. Inthe last = d4θ HφH∗H∗+h.c. (17)
step,wereplacedHwiththeappropriatecontribu- LH Z Mp u d
tion from the modulus field. Further, we make an
assumption that the energy density of the modu- where H and H are up-type and down-type
u d
lusisthesignificantcomponentofthetotalenergy Higgs supermultiplets or corresponding scalar
density. If this is notthe case,the moduli-induced fields, respectively. The φ field is the moduli su-
gravitino problem disappears. The reason is that permultiplet and, in the following, its scalar part.
the produced gravitino represent only small por- After integrating out the superspace coordinates,
tionofthetotalenergydensity. Thecondition(14) we obtain
is fulfilled once
λ
αM >Φ. (15) = (D DµφH∗H∗ (18)
p LH M µ u d
p
AttheonsetofoscillationsΦ<M ,thusforα 1 +F H∗F∗+F H∗F∗+c.c.+ )
p ≥ φ u d φ d u ···
we can neglect the expansion of space in our anal-
ysis. where F = M2eG/2(G−1)iG is the auxiliary
In addition to the growing mode (9), there is i − p j j
field of the i’th supermultiplet, D is the covari-
also the decaying mode µ
ant derivative. The process of energy transfer de-
scribedabovemakesuseofon-shelldegreesoffree-
χk e−√−m2eff−k2t. (16) dom. Hence, we make use of the equation of mo-
∝
tion for the φ field to replace D Dµφ with m2φ.
The decaying mode causes inference terms and µ φ
Asaresult,thefollowinginteractiontermisapart
may put further restrictions on the region of ap-
of the Lagrangian:
plicability of the tachyonic resonance. The equa-
tion(7)cantaketheformofthewellknownMath-
λ
ieuequation(seee.g.[24]). Infactasitcanbeseen m2φH∗H∗+h.c. (19)
fromtheinstabilitychartoftheMathieuequation, LH ⊃ Mp φ u d
the resonant production is terminated as soon as
q αΦ/Mp 1/2; hence we are interested only In the low energy effective Lagrangian, the
≡ ≤
in cases with α 1. Models where α has to be term (19) is responsible for the interaction (5),
≫
smaller than 1 can be of interest if many trilinear where χ is the neutral scalar component of the
interactions enhance the resonanteffect. The con- lightest Higgs field in the mass basis.
dition on α is the same as in (15) which means
that the resonance production is efficient once the
change in the scale factor is negligible.
Tachyonic preheating in the parameter range 1 Hereweprovideonlyoneexampleoftheoriginoftrilinear
corresponding to large α was extensively stud- terms. Largeαmightrequireother interactions.
4
III. ESTIMATES the value
m2 M
Inthefollowingwewouldliketoestimatethere- Φ = χ p . (22)
gionof moduli mass for which the moduli-induced min m2 α
φ
gravitino problem is resolved. Another glance at
the equation of motion of the χ field At this point, the remaining energy density in the
moduli is
α
2 2 2 2
χ¨ + k +m +λχ + m φ χ = 0,
k (cid:18) χ M φ (cid:19) k m4M2
p m2φΦ2min = α2χm2p ≡ρ3/2. (23)
revealsthatthe tachyonicprocessismoreeffective φ
for larger masses of the moduli. We assume that
thetachyonicresonanceworksaslongasm2 can Theenergydensitystoredinthe gravitino,ρ3/2,
eff
obtain negative values, allows us to determine the gravitino abundance.
m2 Φ n3/2
mχ2 <αM . (20) m3/2Y3/2 ≡ m3/2 s (24)
φ p ρ3/2
= (25)
All the energy converted into excitations of the χ s
4 2
field afterwards is transferred to SM degrees of mχMp
= (26)
freedom. Further, since Br3/2 = O(0.01 ∼ 1) α2m2φs
we assume that once the bound (20) is violated
all the energy is transferred to gravitinos. The
where Y3/2 is the gravitino yield, n3/2 is the num-
above assumptions allow us to estimate the grav-
ber density of gravitino particles and s is the en-
itinoabundanceneglectingtheeffectoftheexpan-
tropy of the ultra-relativistic particles.
sion of space. At the end, we insert the known
bounds onthe gravitinoabundanceandderivethe ρ+p 4ρ
lower bound on the gravitino mass. s= = rad (mφMp)3/2, (27)
T 3 T ≈
R R
We distinguish between two cases at the onset
of moduli field oscillations: in the first case, the
where T is the reheating temperature (temper-
universe is supercooled and χ2 0; or, in the R
ature of ultra-relativistic plasma at the moment
h i ∼
second case, the universe is dominated by radia-
it reaches thermal equilibrium). While the actual
tion and χ2 T2 = m M . The universe is
φ p reheatingtemperature depends onthe thermaliza-
h i ∼
supercooled if oscillationps of the moduli were pre- tion processes, the upper bound is
ceded by an inflationary period, and the energy is
still stored in the oscillations of an inflaton, or if
T < m Φ m M (28)
R φ in φ p
the modulus itself is the inflaton (see [27, 28] for ≤
p p
discussions on the moduli-induced gravitino prob-
where Φ is the amplitude of the field φ at the
lem in this case). In this paper, we primary con- in
onset of oscillations. Since we have neglected the
centrate on the first case. In this case, we omit
expansionofspacethroughoutthecalculations,we
the self interaction term to obtain order of magni-
have plugged T = m M to obtain the last
tudeestimatesfortheboundontheallowedmoduli R φ p
equality in (27). p
mass.
Thegravitinoabundanceisseverelyconstrained
While the tachyonic resonance is in effect, the
in order not to jeopardize the success of BBN or
energy density in φ is transferred to χ particles
fromthedangerofoverproducingoflightestsuper-
andthentoradiation. Neglectingtheexpansionof
symmetricparticles. Themoststringentconstraint
space,
comes from the overproduction of 3He [29, 30]
2 2 which yields
ρ =m M (21)
rad φ p
ThetachyonicresonanceendsassoonasΦreaches m3/2Y3/2 <O(10−14 10−11) GeV. (29)
∼
5
The limit (29) is equivalent to m2 > λT2, the estimates on moduli mass reduce
χ
to (21-31).
m4
m3/2Y3/2 = α2mχ4TR (30) abTunhdeandceecaoyf gorfavmitoindousli. Ldeiltutuessdtehneotperteh-eexiinsittiinagl
φ
3 m4 gravitino yield by Y3/2. The entropy produced in
= 4α2mχ4φpmφMp tthheedneecwaygroafvmitoindoulyiieinldtoisradiationsn ∝Tn3,hence,
< O(10−14 10−11) GeV
∼
where we have inserted the expression for s (27). Yn = n3/2 Yn Y3/2sf = Y3/2sf
Makingfurther assumptions: α O(1), mχ 100 3/2 sf +sn 3/2 ≈ sn sn
GeV, the moduli is safe from t∼he overprodu≈ction T3
f
of gravitinos in direct decay if = T3Y3/2. (36)
n
8 9
10 10 GeV m . (31)
φ
∼ ≤ where s and T stands for the values of the pre-
f f
The lower bound (31) is the main result of the existing entropy and temperature of radiation at
paper. Γ =H. Making use of (34), we deduce
all
In the second case, when the field χ is a part of
the thermal bath and the contribution of the self m
interactiontermtotheeffectivemasscanbelarge, Y3n/2 = MφY3/2 (37)
p
we have
α
2 2 2 2
m = m +λ χ + m φ
eff χ h i M φ
p
2 2 α 2 IV. CONCLUSIONS
= m +λT + m φ, (32)
χ M φ
p
In this paper, we have discussed the influence
where we have used the Hartree approximationto
2 of the tachyonic resonance on the moduli-induced
gofromthe firsttothe secondline. ThelargeλT
gravitino problem. We primarily have discussed
term threatens to preventthe tachyonicresonance
thecasewhenχisnotapartofthethermalbathat
from occurring. Particulary, if, at the onset of os-
theonsetofoscillationsofthemodulus fieldwhich
cillations, the condition
is a main contributor to the total energy density.
αmφ Inthiscase,theroughestimatesshowsthatmoduli
1< (33)
λM masses above 108 109 are free from overproduc-
p
∼
tionofgravitinosindirectdecayofmoduli. Thees-
is not satisfied, the effective mass (32) is positive.
timatesomitseveralmodeldependentpointswhich
In an expanding moduli-dominated universe, the mayeitherenhanceordiminishtheinfluenceofthe
temperature redshifts as
resonance. In particular, in the process of calcula-
tionswedidnottakeintoaccounttheexpansionof
4/3
Φ
2 space. In the case when χ is a part of the thermal
T =m M (34)
φ p
(cid:18)Mp(cid:19) bath at the onset of the oscillations of φ, we have
found that the tachyonic resonance is less likely
Hence, m2 remains positive during oscillations
eff to work. In any case, even if the tachyonic reso-
of the φ if
nanceisinefficient,thedecayofmodulidilutesthe
α3 m preexisting abundance of gravitinos. If the energy
1> φ (35) densityofthemoduliissufficientlysubdominantto
λ3 M
p the total energydensity, the moduli-induced grav-
m2 itino problem disappears. The reason is that the
where we have inserted Φf = Mφp - the value of Φ producedgravitinorepresentonly smallportionof
at the time of perturbative decay (1). In the case the total energy density.
6
Acknowledgments in the course of the project and proofreading our
manuscript. We would like to acknowledge sup-
We wish to thank Jean Dufaux, Masahiro Ibe port from a Carl Reinhardt McGill Major Fellow-
and Lev Kofman for useful discussions and to ship. ThisresearchisalsosupportedbyanNSERC
AlessioNotariforproofreading. Wearegratefulto Discovery Grant to RB.
Robert Brandenberger (RB) for many comments
[1] M. Endo, K. Hamaguchi and F. Taka- [20] M. Dine, R. Kitano, A. Morisse and Y. Shir-
hashi, Phys. Rev. Lett. 96, 211301 (2006) man, Phys. Rev. D 73, 123518 (2006)
[arXiv:hep-ph/0602061]. [arXiv:hep-ph/0604140].
[2] S. Nakamura and M. Yamaguchi, Phys. Lett. B [21] J. H. Traschen and R. H. Brandenberger, Phys.
638, 389 (2006) [arXiv:hep-ph/0602081]. Rev. D 42, 2491 (1990).
[3] G.D.Coughlan,W.Fischler,E.W.Kolb,S.Raby [22] B.R.Greene,T.ProkopecandT.G.Roos,Phys.
and G. G. Ross, Phys.Lett. B 131, 59 (1983). Rev. D 56, 6484 (1997) [arXiv:hep-ph/9705357].
[4] J.R.Ellis,D.V.NanopoulosandM.Quiros,Phys. [23] G. N. Felder, J. Garcia-Bellido, P. B. Greene,
Lett.B 174, 176 (1986). L. Kofman, A. D. Linde and I. Tkachev,
[5] B. de Carlos, J. A. Casas, F. Quevedo and Phys. Rev. Lett. 87, 011601 (2001)
E. Roulet, Phys. Lett. B 318, 447 (1993) [arXiv:hep-ph/0012142].
[arXiv:hep-ph/9308325]. [24] N. W. McLachlan, Theory and application of
[6] T. Banks, D. B. Kaplan and A. E. Nelson, Phys. Mathieu functions (Clarendon, Oxford, 1947).
Rev.D 49, 779 (1994) [arXiv:hep-ph/9308292]. [25] J. F. Dufaux, G. Felder, L. Kofman, M. Peloso
[7] A.D. Linde, Phys.Lett. B 116, 335 (1982). and D. Podolsky, JCAP 0607, 006 (2006)
[8] A.VilenkinandL.H.Ford,Phys.Rev.D26,1231 [arXiv:hep-ph/0602144].
(1982). [26] D. Lindley, Astrophys. J. 294, 1 (1985);
[9] A. A. Starobinsky, In *De Vega, H.j. ( Ed.), M. Y. Khlopov and A. D. Linde, Phys. Lett. B
Sanchez,N.(Ed.): FieldTheory,QuantumGrav- 138 (1984) 265; F.Balestra et all., Sov. J. Nucl.
ity and Strings*, 107-126 Phys. 39, 626 (1984);
[10] D. I. Podolsky and A. A. Starobinsky, M. Y. Khlopov, Yu. L. Levitan, E. V. Sedel-
Grav. Cosmol. Suppl. 8N1, 13 (2002) nikov and I. M. Sobol, Phys. Atom. Nucl. 57,
[arXiv:astro-ph/0204327]. 1393 (1994) [Yad. Fiz. 57, 1466 (1994)]; J. R. El-
[11] G.Lazarides, C.Panagiotakopoulos andQ.Shafi, lis, J. E. Kim and D. V. Nanopoulos, Phys.
Phys.Rev.Lett. 56, 432 (1986). Lett. B 145, 181 (1984); R. Juszkiewicz, J. Silk
[12] G.Lazarides, C.Panagiotakopoulos andQ.Shafi, and A. Stebbins, Phys. Lett. B 158, 463 (1985);
Phys.Rev.Lett. 56, 557 (1986). J.R.Ellis,D.V.NanopoulosandS.Sarkar,Nucl.
[13] M.Dine,L.RandallandS.D.Thomas,Phys.Rev. Phys. B 259, 175 (1985); J. Audouze, D. Lind-
Lett.75, 398 (1995) [arXiv:hep-ph/9503303]. ley and J. Silk, Astrophys. J. 293, L53 (1985);
[14] N.Shuhmaherand R.Brandenberger, Phys.Rev. M. Kawasaki and K. Sato, Phys. Lett. B 189, 23
D 73, 043519 (2006) [arXiv:hep-th/0507103]. (1987).
[15] T. Moroi, M. Yamaguchi and T. Yanagida, Phys. [27] M. Kawasaki, F. Takahashi and
Lett.B 342, 105 (1995) [arXiv:hep-ph/9409367]. T. T. Yanagida, Phys. Lett. B 638, 8 (2006)
[16] L.RandallandS.D.Thomas,Nucl.Phys.B449, [arXiv:hep-ph/0603265].
229 (1995) [arXiv:hep-ph/9407248]. [28] M. Kawasaki, F. Takahashi and T. T. Yanagida
[17] M. Kawasaki, T. Moroi and T. Yanagida, Phys. Phys. Rev. D 74, 043519 (2006)
Lett.B 370, 52 (1996) [arXiv:hep-ph/9509399]. [arXiv:hep-ph/0605297].
[18] T. Moroi and L.Randall, Nucl.Phys.B 570, 455 [29] M.Kawasaki,K.KohriandT.Moroi, Phys.Lett.
(2000) [arXiv:hep-ph/9906527]. B 625, 7 (2005) [arXiv:astro-ph/0402490].
[19] S. Kachru, R. Kallosh, A. Linde and [30] M. Kawasaki, K.Kohriand T. Moroi, Phys.Rev.
S. P. Trivedi, Phys. Rev. D 68, 046005 (2003) D 71, 083502 (2005) [arXiv:astro-ph/0408426].
[arXiv:hep-th/0301240].