Leptogenesis via multiscalar coherent evolution with supersymmetric neutrino see-saw

4 Pages·0.16 MB·

by  Masato Senami

Leptogenesis via multiscalar coherent evolution
with supersymmetric neutrino see-saw
Masato Senami∗ and Katsuji Yamamoto†
Department of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan
(Dated: February 1, 2008)
A novel scenario of leptogenesis is investigated in the supersymmetric neutrino see-saw model.
Theright-handedsneutrinoN˜ andtheφfieldintheL˜HudirectionofthesleptonandHiggsdoublets
start together coherent evolution after the inflation with right-handed neutrino mass MN smaller
thantheHubbleparameterofinflation. Then,aftersomeperiodthemotionofN˜ andφisdrastically
changedbythecrosscouplingMNhνN˜∗φφfromtheMNNN andhνNLHuterms,andthesignificant
asymmetriesofN˜ andL˜ aregenerated. TheL˜ asymmetryisfixedlaterbythethermaleffectasthe
leptonnumberasymmetryforbaryogenesis, whiletheN˜ asymmetrydisappearsthroughthedecays
3 N˜ →L¯H¯˜u,L˜Hu with almost the same rate but opposite final lepton numbers.
0
0 PACSnumbers: 12.60Jv, 14.60.St,98.80.Cq
2

n Baryogenesis via leptogenesis is considered as one of mationsothatonlyL hastheYukawacouplingwithN .
1 1
a the promising scenarios to explain the baryon number Then, the right-handed sneutrino N˜ and the φ field in
J 1
asymmetry in the universe [1]. The leptogenesis is in- L˜1Hu starttogethercoherentevolutionwithlargeinitial
0
teresting particularly in the point that it may be related field values after the inflation in the manner of Affleck
3
to the neutrino mass generation. In the supersymmetric and Dine. The motions of N˜ and φ are linked through
1
2 standard model, as investigated fully so far, the lepto- the superpotential term hνN1L1Hu. (N˜2 = N˜3 = 0 due
v genesis may be realized via the Affleck-Dine mechanism to the large masses MN2,MN3 > Hinf.) Henceforth the
3 [2, 3] in the L˜H flat direction of the slepton and Higgs generationindicesaresuppressedbyconsideringonlythe
u
7 doublets,L˜ andH ,requiringthe verysmallmassofthe one generation for leptogenesis, and the relevant scalar
u
0
lightestordinaryneutrino[4]. Itisalsopossibletorealize fields are specified as
0
21 tthhee rleepsttroigcetinoensiosnonthtehleigflhattesmtnaneuiftorlidnoofmL˜a-sHsum-Hayd,bewhcoenre- N˜,L˜ =(cid:18)φ/0√2(cid:19),Hu =(cid:18)φ/0√2(cid:19). (2)
0 siderably moderated [5]. In this letter, we investigate
/ another novel scenario of leptogenesis in the supersym- If some of M ’s are smaller than H in general, the
h N inf
p metricsee-sawmodelforneutrinomasses[6]. Thelepton coherentevolutionafter the inflation may be much more
- number asymmetry is indeed generated via the coherent multidimensional involving the N˜’s, L˜’s and H . The
u
p
evolutionofthemultiscalarfields,theright-handedsneu- leptogenesis scenario is essentially valid even in such
e trino N˜ and the φ field in the L˜H direction. The po- cases, where the main source for asymmetry generation
h u
: tentialtermsprovidedwiththesupersymmetricneutrino is the cross coupling MNhνN˜∗φφ from the MNNN and
v
see-saw and also the thermal effect [7] play important h NLH terms of supersymmetric neutrino see-saw.
i ν u
X roles for leptogenesis in the respective epochs. The lep- The relevant superpotential is given by
togenesis is completed when the generated lepton num-
ar ber asymmetry is fixed to some significant value by the W = MNNN + eiδNNNNN +h NLH . (3)
ν u
thermal effect at the scale much higher than the grav- 2 4M
itino mass m3/2 103GeV. Hence, this scenario is not TheNNNN termmayoriginateinthephysicsofPlanck
∼
restricted by the low-energy electroweak physics. scale,andits phase factor eiδN is included here with real
In the present scenario of leptogenesis, it is supposed M . The NNN term is discarded for simplicity by re-
N
that some of the masses MN’s of the right-handed neu- quiring the R-parity. The LHuLHu term is not consid-
trinos N’s (antineutrinos strictly) are smaller than the ered either, since it does not provide significant effect
Hubble parameter Hinf ∼ 1013GeV during the inflation. if the Yukawa coupling hν > 0 is not extremely small.
Specifically,wedescribethegenerationofleptonnumber As seen later in Eqs. (6) and (7), the Yukawa cou-
asymmetry by considering the simple case that pling h 3 10−3 is relevant for the present scenario
ν
of leptoge∼nesi×s starting at the large scale 1015GeV.
MN1 <Hinf (1) Then,its valueatthe electroweakscaleM ∼is evaluated
W
foroneN whileM ,M >H fortheothersN ,N . as h (M ) 10−3 by considering the renormalization
1 N2 N3 inf 2 3 ν W ∼
The lepton doublets are arranged with unitary transfor- group effects mainly provided by the top quark loop for
the H field. The ordinary neutrino mass via see-saw
u
mechanism is roughly estimated as
∗E-mailaddress: [email protected] m 10−4eV hν(MW) 2 1011GeV (4)
†E-mailaddress: [email protected] ν ∼ (cid:18) 10−3 (cid:19) (cid:18) M (cid:19)
N
2
depending on h (M ) and M . (The neutrino mixing quarticcouplingsh2 φ4 andh2 N˜ 2 φ2 dominateinthis
ν W N ν| | ν| | | |
is present in general with matrix form of h .) Hence, epochwithh asgiveninEq. (7). Then,drivenbythese
ν ν
thisneutrinorelevantforleptogenesisshouldbeidentified quarticcouplings,the scalarfieldsoscillateinmagnitude
with the lightest one, being compatible with the data with scaling by redshift as
on the atmospheric and solar neutrino experiments [8,
9]. It is interesting that the lightest neutrino mass is N˜ φ (HinfM)1/2(H/Hinf)2/3 H2/3. (8)
| |∼| |∼ ∝
expectedtobem 10−4eVforthepresentleptogenesis
ν
with N˜ and φ, wh∼ile m . 10−8eV is required for the Thefieldphases,however,remainalmostconstantexcept
conventional Affleck-Dinνe leptogenesis in the L˜H flat for the vicinities of N˜ =0 and φ=0, and the significant
u asymmetries of N˜ and L˜ do not appear in this epoch.
direction [4].
The scalar potential is given with W in Eq. (3) as
(iii) Transition epoch: H &H >H
tr th
V = c H2 N˜ 2 c H2 φ2
− N | | − φ | | Since N˜ and φ decrease with H as given in Eq.
eiδN h 2 (8), the mass term M2 N˜ 2 and M -h cross coupling
+ M N˜ + N˜N˜N˜ + νφφ +h2 N˜ 2 φ2 N| | N ν
+ (cid:12)(cid:12)(cid:12)(cid:12)H Nb MNMN˜N˜ +a eiδN2N˜N˜(cid:12)(cid:12)(cid:12)(cid:12)N˜N˜ +ν|h.|c.| | MhH2νuN|bφhb|4νleNa˜pn∗adφrφahm2νb|eeN˜ctoe|2rm|φe|2cowmipthar|aN˜b|le∼to|φth|e∼qMuaNrt/ihcνcoaunpdlinthges
N N
(cid:18) 2 4M (cid:19)
+ H(cid:18)ahh2νN˜φφ+h.c.(cid:19)+Vth(φ). (5) Htr ∼1010GeV(cid:18)101M1GNeV(cid:19)3/2, (9)
Herethesoftsupersymmetrybreakingtermsareinduced
where h (H /M)1/2 with H = 1013GeV is taken
by the expansion of the universe with the Hubble pa- ν ∼ inf inf
from Eq. (7). The thermal mass term should also be
rameter H. The thermal terms [7] are also included in
consideredatH H ,whichisgivenby(yT )2 φ2 with
Vth(φ). The D2 terms are vanishing for the φ field. The relevant coupling∼y utrnder the condition y φp <| |T [7].
evolutionofthescalarfieldsisgovernedbytheequations | | p
ThetemperatureT ofthediluteplasmaofinflatondecay
of motion with this potential V and the redshift of H. p
products is given in terms of the reheating temperature
T of the universe after the inflaton decay is completed:
(i) Inflation epoch: H =H R
inf
Thescalarfieldssettleintooneofthe minima(N˜0,φ0) Tp ∼(TR2HMP)1/4, (10)
of V during the inflation with H = H , which are de-
inf where M = 2.4 1018GeV is the reduced Planck
termined as P ×
mass. The thermal mass is constrained at H H
tr
|N˜0|∼|φ0|∼3×1015GeV(cid:18)1013GHeVin1fM018GeV(cid:19)1/2 (6) aNs˜ yTp φ< Tp2M/|φN|/h∼ν. hHνTenp2c/eM,Nthefotrheyr|mφ|al<maTsps∼tweritmh
| | ∼ | | ∼
(yT )2 φ2 is smaller than the M2 N˜ 2 and M h N˜∗φφ
p | | N| | N ν
for the Yukawa coupling terms at H H for the right-handed neutrino mass
tr
∼
hν ∼3×10−3(cid:18)1013GHeVin/f/1M018GeV(cid:19)1/2. (7) MN &1010GeV(cid:18)3 h1ν0−3(cid:19)4/5(cid:18)109TGReV(cid:19)4/5. (11)
×
ct(hWooneusigdhheernictfoefrdoordteehsfintniatoketneerHsesqinutfhiree=caa1s0efi1nw3eGitehtVutnhtiniysgp.ricaanlgIlyfe.)hofνWhν<e, athseIwnmetlolhtaiisosnstihotuefaNh˜t2νioa|φnn|,d4tφtheeirsmMchdN2aon|mNg˜ei|nd2adatrneadfsotMricHaNllhy.ν.NH˜Sp∗tφre,cφisfiotcetarhlmlayst,
((HHiinnff//MM))11//22,, |oN˜n0|thaendot|hφe0r| thaakned,laφrg0er=va0lumesa.yIfbehνob>- tMhe2 NN˜˜ 2fiewldithoscNi˜llatesHm.aiTnlhyedmriovteionnboyftφhefomlloawsss taefrtemr
tained due to the h2ν|N˜|2|φ|2 term. The leptogenesis can N˜Nto|wa|rdthe n|ew| ∝stable configurationwith (hν/2)φφ
be realized even in these cases with some modifications M N˜ so as to make F 2 F 2 M2 N˜ 2 in V≃,
of scenario, which will be described elsewhere. − N | N| ∼ | φ| ≪ N| |
where F M N˜ +(h /2)φφ and F =h N˜φ. Conse-
N N ν φ ν
≃
(ii) Oscillation epoch: H >H >H quently, the scalar fields decrease roughly as
inf tr
After the inflation the Hubble parameter decreases as N˜ (MN/hν)(H/Htr) H, (12)
| |∼ ∝
H =(2/3)t−1 inthematterdominateduniverse,andthe φ (M /h )(H/H )1/2 H1/2 (13)
N ν tr
multiscalar coherent evolution of N˜ and φ starts with | |∼ ∝
the initial condition (N˜,φ) = (N˜ ,φ ) at t = t H−1, with F F H3/2. Throughthisdrasticchangein
0 0 0 ∼ inf | N|∼| φ|∝
as given in Eq. (6). The higher order potential terms the multiscalar coherent evolution, the significant asym-
suppressed by M are soon reduced by redshift, and the metries of N˜ and L˜ appear, which is really seen in the
3
rate equations since the L˜ violating sources in Eq. (15) decreases fast
enoughasH4/H2andH11/4/H2laterwithrapidlyvary-
d n 2
N˜ Im[b M HN˜N˜] ing phase of N˜∗φφ. This concludes that the thermal
dt(cid:16)H2(cid:17) ≃ −H2 N N effect plays the positive role for the completion of lep-
2 a h
Im M N˜F∗ + h νHN˜φφ ,(14) togenesis, which is in salient contrast to the conven-
− H2 (cid:20) N N 2 (cid:21) tional Affleck-Dine mechanism where the thermal effect
d n 2 h a h rathersuppressestheasymmetryseriously. Theresultant
L˜ Im νφφF∗ + h νHN˜φφ (15)
dt(cid:16)H2(cid:17) ≃ −H2 (cid:20) 2 N 2 (cid:21) lepton-to-entropy ratio after the reheating is estimated
with s 3H2M2/T as
≃ R P R
with n = n = n /2. The main sources are scaled as
L˜ Hu φ
wImit[h(h|νF/N2|)φ∝φHFN∗3/]2/,Ha2nd≃he−nIcme[tMheNaNs˜yFmN∗m]/eHtr2ies∝nHN˜5a/n2d/HnL˜2 nsL ∼10−10(cid:16)ǫ1L(cid:17)(cid:18)101M8GeV(cid:19)(cid:18)109TGReV(cid:19). (20)
oscillaterapidlybytheexchangeN˜ L˜. Thesumn +
↔ N˜ Here the reheating temperature is restricted as TR .
n ,however,variesrathermoderatelywiththeremaining
L˜ 108 1010GeVtoavoidthegravitinoproblem[10,11,12].
sources H3, since the main sources are cancelled as −
Im[F F∝∗]=0 with F M N˜ +(h /2)φφ. Theleptonnumberasymmetryisconvertedtothebaryon
N N N ≃ N ν number asymmetry through the electroweak anomalous
effect as n /s= (8/23)n /s [13]. Hence, the sufficient
(iv) Completion epoch: H &H m B L
th 3/2 −
≫ baryon-to-entropy ratio can be provided for the nucle-
After the transition epoch continues for some period, osynthesis with η =(2.6 6.2) 10−10 [14].
− ×
the thermal log term [7] eventually becomes significant The motion of N˜ after the decoupling from φ for
on the evolution of φ. It is mainly provided as H < H M is determined by the M2 N˜ 2 and
th ≪ N N| |
b HM N˜N˜ terms,andtheanalyticsolutionisobtained
a α2T4ln(φ2/T2) (16) N N
th s p | | p in a good approximation with H MN for the two
(ath = 9/8) through the modification of SU(3)C cou- eigenmodes ηR(t) and ηI(t) in N˜(t) ≪as
plingduetothedecouplingoftopquarkfromtheplasma
η (t) η¯ cos[M t+σ (b /3)lnt+δ ] (21)
with large mass ht φ/√2 > Tp. This thermal log term R,I ≃ R,I N R,I | N| R,I
| |
acts as the effective mass term for the φ field giving
with σ =+1, σ = 1, and
(a α2T4/φ2)φ H1/2 in ∂V/∂φ∗. It dominates over R I −
th s p | | ∝
thheN˜terφm) wFiNthhνtφh∗e H∝ubHbl2eipnar∂aVm/e∂teφr∗ (|FN| ∼ |Fφ| = N˜ ≡H(M/MN)1/2(bN/|bN|)−1/2(ηR+iηI). (22)
ν
| || | Theparametersη¯ andδ aredeterminedastheresult
R,I R,I
h 4/3 of N˜ motion from t = t H−1 through t > H−1
H 107GeV ν 0 ∼ inf th ≫
th ∼ (cid:18)3 10−3(cid:19) M−1. The N˜ asymmetry is evaluated with Eqs. (21)
× N
M −1/6 T 4/3 and (22) as
N R
, (17)
× (cid:18)1011GeV(cid:19) (cid:18)109GeV(cid:19) n (t) 2H2Mη¯ η¯ cos[(2b /3)lnt+δ δ ],(23)
N˜ ≃− R I | N| R− I
where Eqs. (10), (13) and (12) are considered. Then, where the rapid oscillations of η and η with M t in
R I N
the rotation of the φ field phase is accelerated by this Eq. (21) are cancelled.
thermal log term with the change of field scaling This N˜ asymmetry oscillates slowly in lnt for some
φ H1/2 H3/2 (18) while due to the bN term, as seen in Eq. (23). Then,
| |∝ → the incoherent decays of N˜ become significant with the
while keeping N˜ H. After a while the top quark en- dominant modes
| |∝
ters the plasma at H 0.1H with φ T (h 1).
Then,thethermalmas∼stermTth2 φ2 in|st|ea∼dbpecomte∼sim- N˜ L¯H¯˜u[L= 1], L˜Hu[L=+1], (24)
p| | → −
portant, and the φ field decreases as φ H7/8. In
| | ∝ where the decay products are ultra-relativistic with
theprecedingepoch,thesignificantexchangeofasymme- h N˜ M /2. The motion of N˜ is significantly de-
tarsiesse,ennN˜in↔EqnsL˜.,(t1o4o)kapnldac(e15th).roTughhestehceoNu˜p-lφincgosuaprleinagcs-, ceνl|era|te≪d byNthese N˜ decays at H ∼ ΓN˜ ≃ (h2ν/4π)MN
( 105GeV numerically), so that it is linked again to
tually turned off inthis epoch with the rapiddecreaseof ∼
φ, tracking the instantaneous minimum of V as N˜
|φ|∝H3/2 andH7/8 laterduetothethermalterms,and (h /2M )φφ with N˜ φ2 H7/4 in magnitud≃e
the φ and N˜ evolve almost independently. − ν N | | ∝ | | ∝
In this way, the L˜ asymmetry is fixed to some signifi- and dθN˜/dt ≃ 2dθφ/dt ∝ H1/4 in phase. Then, the N˜
cantvalueastheleptonnumberasymmetryfort>H−1, asymmetry remaining after the transition epoch dimin-
th ishes rapidly throughthe decays as n =2(dθ /dt)N˜ 2
N˜ N˜ | |
H15/4. It is the essential point that the decay modes
n n ǫ [(3/2)H2M], (19) (∝24) have almost the same rate Γ /2 but the opposite
L˜ ≃ L ≡ L N˜
4
serve the expected changes of the asymmetries through
H H H , resulting in the desired lepton num-
2:0 inf → tr → th
ber asymmetry ǫ 1. Particularly,the variations of ǫ
L ∼ L˜
and ǫ are separated for t > H−1; ǫ is fixed to some
1:0 N˜ th L˜
significantvaluewhileǫ oscillatesslowlyinlntasgiven
N~ N˜
in Eq. (23). It is also checked that the sum n +n
(cid:15)a 0 N˜ L˜
varies rather moderately in the transition epoch, while
the respective asymmetries oscillate rapidly.
(cid:0)1:0
In summary, we have investigated the leptogenesis via
L~
multiscalarcoherentevolutioninthesupersymmetricsee-
(cid:0)2:0
saw model. The right-handed sneutrino N˜ and the φ
Hinf Htr Hth field in L˜H of the slepton and Higgs doublets start to-
# # # u
1 101 102 103 104 105 106 107 gether coherent evolution after the inflation with M
N
t=t0 smaller than Hinf. Then, after some period the motion
of N˜ and φ is drastically changed by the cross coupling
FIG. 1: Typical time variations of the asymmetries ǫN˜(t) M h N˜∗φφ, and the significant asymmetries of N˜ and
(thin) and ǫL˜(t)≃ǫL(t) (bold) are depicted. L˜ Nareνgenerated. The L˜ asymmetry is fixed later by the
thermaleffect asthe leptonnumberasymmetry n . The
L
N˜ asymmetry, on the other hand, disappears through
final lepton numbers L = 1. This means that the N˜ the incoherent decays N˜ L¯H¯˜u,L˜Hu with almost the
± →
asymmetry does not leave any significant lepton number same rate but opposite final lepton numbers. The suf-
asymmetry. ficient amount of nL for baryogenesis can be obtained
The equations of motion for N˜ and φ are solved by with the lightest neutrino mass mν . 10−3eV given by
numericalcalculations to confirm the present scenarioof the see-saw mechanism with the right-handed neutrino
leptogenesis. The typical time variations of nN˜(t) and mass MN ∼1010−1014GeV.Hinf.
n (t) n (t)aredepictedinFig. 1intermsoftheasym-
mL˜etry≃fracLtions ǫ n /[(3/2)H2M]. Here the model
a a
≡
parameters are taken for example as H = 1013GeV, Acknowledgments
inf
M = 5 1018GeV, M = 1011GeV, eiδN = ei(3/10)π,
N
hν = 3 ×10−3, cN = 1.2, cφ = 0.8, bN = 1.3ei(2/3)π, This work is supported in part by Grant-in-Aid for
×
a = 1.5ei(5/4)π, a = 0.8ei(1/4)π, T = 109GeV. Scientific Research on Priority Areas B (No. 13135214)
N h R
The relevant scales, H , H and H , are marked to- fromtheMinistryofEducation,Culture,Sports,Science
inf tr th
gether specifying the respective epochs. We really ob- and Technology, Japan.
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