Smallness of Baryon Asymmetry from Split Supersymmetry

0.11 MB·English

by  Shinta Kasuya

Smallness of Baryon Asymmetry from Split Supersymmetry
Shinta Kasuyaa and Fuminobu Takahashib
a Department of Information Science, Kanagawa University, Kanagawa 259-1293, Japan
b Institute for Cosmic Ray Research, University of Tokyo, Chiba 277-8582, Japan
(Dated: January,2005)
The smallness of thebaryon asymmetry in ouruniverseis oneof thegreatest mysteries and may
originate from some profound physicsbeyond thestandard model. Weinvestigate theAffleck-Dine
baryogenesisinsplitsupersymmetry,andfindthatthesmallnessofthebaryonasymmetryisdirectly
related tothehierarchy between thesupersymmetrybreakingsquark/slepton masses andtheweak
scale. Put simply,the baryon asymmetry is small because of thesplit mass spectrum.
Introduction.— The hierarchies in the energy scales the split SUSY beautifully explains the smallness of the
5
pose naturalness issues. Cosmological constant is found baryon asymmetry of the universe.
0
to be of order (meV)4 [1], while the fundamental scale Inthe splitSUSY scheme,manyaspectsofbaryogene-
0
2 should be the Planck scale (or the weak scale). To keep sisaresignificantlyalteredcomparedtousualsupersym-
the higgs mass around O(100) GeV, a new physics such metric SM. For instance, the electroweak baryogenesis
n
as supersymmetry must appear at O(1) TeV, although [8] does not work both because the strongly first order
a
J precisionelectroweakmeasurementshavepushedupthis phase transition cannot be attained due to large stop

6 scalehigherthanO(5−7)TeV,threateningsupersymme- massandbecauseavailableCPviolationisalsoverylim-
2 try as a solution to naturalness problem associated with itedforsplitmassspectrum. Therefore,otherbaryogene-

the hierarchy problem [2]. sismechanismsshouldaccountforthebaryonasymmetry
1
These naturalness issues may not be a problem any ofthe universe. Here we concentrateonthe Affleck-Dine
v
0 more in the context of the anthropic landscape of string (AD) baryogenesis [9].
4 theory[3]: we mayliveinsuchavacuumthathasa very The AD mechanism utilizes the squark/slepton con-
2 small cosmologicalconstant and a very small higgs mass densate to generate the baryon asymmetry. Since the
1 comparedwithfundamentalscale. Thefinetuningofthe squark/sleptonmasses m˜ are much largerthan the weak
0
latter ledto split supersymmetry(SUSY) framework[4], scale in split SUSY, the resultant asymmetry is sup-
5
where the SUSY breaking scale in the visible sector is pressed compared to that in the SSM. As we will see
0
/ liberated from the weak scale. below, in the simplest case, the baryon-to-entropy ra-
h
The idea of split SUSY [4] is to abandon solving the tio is given by the ratio between the two split mass
p
- hierarchy problem in the standard model (SM) and al- scales, the weak scale and the heavy scalar masses m˜:
p lowallthescalarsexceptforahiggstogetaheavymass. n /s ∼ 0.1A/m˜, which predicts m˜ ∼ O(1012) GeV for
B
e Then we can avoid many problems in supersymmetric n /s ∼ 10−10 and A ∼ TeV. Thus the smallness of the
h B
: SM(SSM), suchasthe absenceofalighthiggsandmost present baryon asymmetry can be ascribed to the split
v
of the sparticles, too fast decay of the proton by the di- mass spectrum.
i
X mension five operators, SUSY flavor and CP problems, Affleck-DinemechanismandsplitSUSY.—Inthemin-
r and the cosmological gravitino and moduli problems. imal supersymmetric standard model (MSSM), flat di-
a
The scale of the SUSY breaking is arbitrary in gen- rections including squarks and/or sleptons have baryon
eral in split SUSY. The two guiding principles usually and/or lepton numbers, so it can be considered as the
takenarethegaugecouplingunificationatthegranduni- AD field, Φ. They acquire heavy SUSY breaking masses
fied theory scale and the lightest SUSY particle as dark of order m˜. In addition, let us assume that the AD
matter. In order to satisfy both requirements, gaugi- field is lifted by a non-renormalizable operator, W =
nos should be kept as light as TeV scale. The simplest Φn/nMn−3, where M is a cut-off scale. In order to pro-
mass spectrum is such that all the squarks and sleptons ducebaryonasymmetry,theADfieldmusthaveatorque,
are at m˜ presumably much higher than the weak scale, whose force comes from the baryon-number-violatingA-
while keeping all the gauginos and higgs very light. In term. As mentioned above, the same mechanism for ob-
this case,the gauginomassesm shouldbe suppressed taining the light gaugino masses makes A as small as
1/2
by some mechanism [5], which also keeps A-term small: m . The relevant scalar potential for Φ is
1/2
A∼m ∼TeV.
1/2 AΦn |Φ|2n−2
How to create the baryon asymmetry of the universe V(Φ)=m˜2|Φ|2+ +h.c. + . (1)
is one of the biggest issues in the modern cosmology. (cid:18)nMn−3 (cid:19) M2n−6
Through the observations of light nuclei [6] and cosmic
During inflation supergravity and K¨ahler couplings
microwave background anisotropies [7], its abundance is
with the inflaton lead to a Hubble-induced mass term,
known to be very small: the baryon-to-entropy ratio
should be around 10−10. In this Letter, we show that δV =cH2|Φ|2, (2)
I
2
wherec isa coefficientoforderunity andH isthe Hub- must be satisfied, where T is the reheating tempera-
I RH
ble parameter during inflation. If cH2 +m˜2 < 0, this ture after inflation. On the other hand, for m˜M >
I p
negativemassmakestheoriginunstableandsetstheini- TRH >m˜, we obtain a similar inequality, p
tial value for Φ away from the origin:
m˜
|Φinf|≃min HIMn−3 n−12 ,Mp , (3) |Φosc|>rTRHMp. (7)
h(cid:0) (cid:1) i
where M =2.4×1018 GeV is the reduced Planck mass, It is easy to see that (6) is satisfied for e.g., n = 6 and
p
and this upper bound reflects the fact that the scalar M >∼Mp.
potential becomes exponentially steep above |Φ| ∼ M . It should be emphasized that cosmological gravitino
p
It is worthy of note that H should be larger than m˜ in problem [10] can be avoided, since the gravitino mass
I
order to develop the large initial amplitude of Φ. In the m3/2 can be much larger than the TeV scale in split
usual case of low-scale soft SUSY breaking mass, m˜ ∼ SUSY. Thus, the reheating temperature can be as large
TeV, this conditionissatisfiedformostinflationmodels. as 1016 GeV. However it is still nontrivial whether the
However,ifm˜ ismuchlargerthantheTeVscaleasinthe gravitinos dominate the universe and produce extra en-
split SUSY scheme, this condition sets a nontrivial and tropy at the decay, destroying a simple relation Eq. (5).
severeconstraintontheenergyscalefortheinflation. We Letusthereforeclarifytheconditionforgravitinosnotto
will come back to this issue below. dominatethe universe. Form3/2 >∼m˜,the gravitinosare
After inflation the Hubble parameter starts to de- mainly produced through scattering processes, although
crease. The AD field tracks the instantaneous minimum onecaneasilyseethegravitinosproducedinthiswaygive
givenby Eq. (3) with H replacedby H, the Hubble pa- only negligible contribution to the energy density of the
I
rameter atthat time. When H ∼m˜, the AD field comes universe. On the other hand, for m˜ > m3/2, potentially
to oscillate, and the baryon number density is efficiently dangerous process is the decay of the squarks/sleptons
produced at the same time: into gravitinos, and the abundance of the gravitinos is
solely determined by this process [5]:
n ≃B δA|Φ |2, (4)
B Φ osc
swehnetrseaBCΦPispthhaeseb,aarnyodn|Φchar|g≡e o(fm˜ΦM, δn−∼3)O1/((n0−.12))riespares-- Y3/2 ∼10−5N(cid:18)1012m˜GeV(cid:19)3(cid:16)10m9G3/e2V(cid:17)−2(cid:18)g∗1(0m˜2)(cid:19)−3/2,
osc
(8)
sumed to be smaller than M . The resultant baryon-to-
p
entropyratiodependsonthesubsequentthermalhistory. where g∗(m˜) counts the relativistic degrees of freedom
at T = m˜, and N is the number of thermally populated
For simplicity, let us first consider the case that the AD
squarks/sleptons at that time. Note that since the de-
field dominates the energy density of the universe when
caytemperature of the AD fieldis ofthe orderofm˜, not
it decays. Then the baryon-to-entropyratio is
all the squarks and sleptons might be thermally popu-
nB nB BΦδATd m1/2 lated. Also,evenifnoneofthemarethermallyproduced,
≃ ∼ ∼0.1 , (5)
s ρtot/Td m˜2 m˜ there is still contribution from the decay of the AD field
(squark/slepton condensate) into gravitinos. Thus N
where T ∼m˜ is the decay temperature of the AD field,
d
variesfromafewtoafewtens,dependingonthedetailed
whichdecaysintothegauginosandquarks/leptons. Note
structureofthesfermionmassspectrumandcomposition
thatthedecaydoesnotproceeduntiltheeffectivemasses
of the flat direction. In order to keep the direct relation
of decay particles become smaller than the parent mass,
between the baryon asymmetry of the universe and the
m˜, resulting in T ∼m˜.
d
mass hierarchy in split SUSY, the gravitino should not
From Eq. (5) we can see that the baryon asymmetry
dominate the energy density of the universe when it de-
issuppressedbythe hierarchybetweenthe gauginomass
cays. Thus, we obtain the constraint on the gravitino
andthesquark/sleptonmass. Becauseofthelargehierar-
chyinsplitSUSY,adesiredvalueofn /s∼10−10canbe mass as
B
oisbrtaatihneerdaimfm˜aziisngarroeusunldt;1t0h1e2oGbesVervfoerdmsm1/a2ll=ba1ryToenVa.sTyhmis- m3/2 >∼ 109N(cid:18)g∗1(0m˜2)(cid:19)−3/5(cid:18)g∗(1T032/2)(cid:19)1/10
metry is simply determined by the split mass spectrum
6/5
in the SSM, independent of n or M as long as the AD m˜
× GeV. (9)
field dominates the universe. This should be contrasted (cid:18)1012GeV(cid:19)
to the usual case of m˜ ∼ TeV leading to n /s∼O(0.1).
B
The conditions for the AD field to dominate the uni- where T3/2 is the decay temperature of the gravitinos:
verse are easily achieved. If the reheating is completed
beforetheADfieldstartsoscillating,i.e.,TRH > m˜Mp, T =4×103 g∗(T3/2) −1/4 m3/2 3/2GeV.
1 p 3/2 (cid:18) 102 (cid:19) (cid:16)109GeV(cid:17)
|Φ |> m˜M3 4 (6) (10)
osc p
(cid:0) (cid:1)
3
When the gravitino mass is smaller than 109 GeV, coulddrivethe(chaotic)inflationwiththequarticpoten-
there is an extra entropy production, which dilutes the tial[15]oreventhe quadraticpotentialwiththe inflaton
baryon asymmetry. The dilution factor is estimated as mass m ∼m˜ ∼1013 GeV in split SUSY.
I
∼(m3/2/109 GeV)−5/2. Conclusion.— We have considered the AD baryogen-
In the above discussion, we have assumed that the Q esis in split SUSY model, in which all the scalars ex-
balls [11] are not formed. Q balls may delay the decay cept for a finely-tuned higgs get a heavy mass m˜, while
of the AD field and affect the direct relation between gauginos (and higgsinos)have a weak scale mass. It was
the baryon asymmetry and the mass hierarchy. Here found that the small baryon asymmetry of the universe
we show that Q balls are actually absent in the con- directly results from the mass hierarchy between them.
text of split SUSY. In order to have the Q-ball config- To be specific, as is clear from Eq. (5), the heavy scalar
uration, the effective potential of the AD field must be mass scalem˜ shouldbe around1012 GeV, to explainthe
shallower than |Φ|2. Including one-loop corrections to baryon-to-entropy ratio n /s ∼ 10−10 for m ∼ TeV.
B 1/2
the squark/slepton masses, the effective potential scales Itisinterestingthatthus predictedvalue ofm˜ is closeto
as |Φ|2+2K, where K is the coefficient of the one-loop theupperlimitm˜ <∼1013 GeV,whichcomesfromthere-
corrections. Hence the Q-ballsolutionexists, if and only quirement that the lifetime of gluinos should not exceed
if K is negative. If squark/slepton masses are compara- about 1016 sec [4].
ble to the gaugino masses, i.e., m˜ ∼ m1/2, the one-loop Inaddition, this scenariosuggestshigh reheatingtem-
correction is dominated by negative contribution from perature of the inflaton, which in turn implies high-scale
the gauginos, therefore Q-ball configuration exists, lead- inflation. Thus, the detailed observation of the tensor
ing to Q-ball formation. However, if m˜ ≫ m1/2 as in mode or BB mode of CMB polarization may favor or
split SUSY, this is not necessarily the case. In particu- refute our scenario.
lar, for m˜/m ∼109, K is always positive, irrespective
1/2 Acknowledgments.— F.T. would like to thank the
of whether or not the flat direction includes the third-
JapanSocietyforPromotionofScienceforfinancialsup-
generation fields. Therefore, there is no Q-ball configu-
port.
ration in the split SUSY.
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