Nonthermal $CP$ violation in soft leptogenesis

0.82 MB·English

by  Rathin Adhikari

Nonthermal CP violation in soft leptogenesis
Rathin Adhikari∗ and Arnab Dasgupta†
Centre for Theoretical Physics, Jamia Millia Islamia (Central University), Jamia Nagar, New Delhi 110025, India
Chee Sheng Fong‡
Instituto de Física, Universidade de São Paulo, C. P. 66.318, 05315-970 São Paulo, Brazil
Raghavan Rangarajan§
Theoretical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India
(Dated: May 5, 2015)
Soft leptogenesis is a mechanism which generates the matter-antimatter asymmetry of the Uni-
verse via the out-of-equilibrium decays of heavy sneutrinos in which soft supersymmetry breaking
5 terms play two important roles: they provide the required CP violation and give rise to the mass
1 splitting between otherwise degenerate sneutrino mass eigenstates within a single generation. This
0 mechanismisinterestingbecauseitcanbesuccessfulatthelowertemperatureregimeT (cid:46)109 GeV
2 inwhichtheconflictwiththeoverproductionofgravitinoscanpossiblybeavoided. Inearlierworks
the leading CP violation is found to be nonzero only if finite temperature effects are included. By
y
considering generic soft trilinear couplings, we find two interesting consequences: (1) the leading
a
CP violation can be nonzero even at zero temperature realizing nonthermal CP violation, and (2)
M
the CP violation is sufficient even far away from the resonant regime allowing soft supersymmetry
breaking parameters to assume natural values at around the TeV scale. We discuss phenomeno-
4
logical constraints on such scenarios and conclude that the contributions to charged lepton flavor

violating processes are close to the sensitivities of present and future experiments.
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p PACSnumbers: 11.30.Fs,11.30.Pb,12.60.-i,13.15.+g,14.60.Pq
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∗ [email protected]
† [email protected]
‡ [email protected]
§ [email protected]
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I. INTRODUCTION
Leptogenesis [1] is an attractive mechanism for generating the observed matter-antimatter asymmetry of the Uni-
versewhereinonefirstcreatesanasymmetryintheleptonsectorwhich, inturn, inducesanasymmetryinthebaryon
sector via anomalous B+L violating interactions. In standard type-I seesaw supersymmetric leptogenesis [2–5] in-
volving the out-of-equilibrium decays of heavy neutrinos and sneutrinos, the CP violation required to generate the
lepton number asymmetry comes from the neutrino Yukawa couplings. This scenario, with hierarchical right-handed
neutrinos (RHNs), faces a conflict as successful leptogenesis requires the mass of the lightest RHN to be at least
109GeV [6] while the simplest resolution of the gravitino problem [7, 8] requires the reheating temperature after
inflation to be less than 106–9GeV depending on the gravitino mass [9].1
One may avoid this conflict by incorporating new elements in leptogenesis. In models of soft leptogenesis [13, 14]
(forarecentreview,seeRef.[15])CP violationcomesfromsoftsupersymmetry(SUSY)breakingterms(hereonwards
wewillsimplyrefertothemassoftterms)withsoftparametersassumedtobeatthem TeVscale;i.e.,westill
SUSY
hopeSUSYisresponsibleforstabilizingthehierarchybetweentheweakandgrandunification∼scales. Oneinteresting
feature is that soft leptogenesis can proceed even with one generation of the RHN chiral superfield.2 Essentially, the
heavy sneutrino N and antisneutrino N∗ from the same chiral supermultiplet will mix due to the presence of the
soft terms. The decays of the mixed mass eigenstates violate both CP and lepton number and generate a matter-
antimatterasymm(cid:101)etry. AlthoughtheC(cid:101)P violationissuppressedbypowersofmSUSY/M 1withM thescaleofthe
lightest RHN, the mass splitting between these otherwise degenerate sneutrino mass eige(cid:28)nstates is also proportional
to m /M. Crucially, this small splitting also results in enhancement of the CP violation from mixing. Because
SUSY
of the suppression factor m /M in the CP violation, one cannot have very large M. Estimating the leading CP
SUSY
parameter as (cid:15) m /M and that successful leptogenesis generically requires (cid:15) (cid:38) 10−6, we obtain M (cid:46) 109 GeV
SUSY
assuming m ∼at the TeV scale. Hence soft leptogenesis occurs in the regime where the conflict with the bound on
SUSY
the reheating temperature from gravitino overproduction can be mitigated or even avoided.
In the original proposals of Refs. [13, 14], the authors showed that in the scenario of N N∗ mixing, the leading
CP violation in decays to fermions and scalars have opposite signs and cancel each other at−the order (m /M)
SUSY
at zero temperature T = 0. They further showed that once finite temperature effects a(cid:101)re ta(cid:101)ken intoOaccount, this
cancellationispartiallylifted, i.e. oneobtainsanasymmetryproportionaltoafactor[c (T) c (T)], wherec (T)
F B F,B
are phase space and statistical factors associated with fermion and boson final states, and−where the contributions
do not completely cancel each other at finite temperature. Working under the assumption of proportionality of soft
trilinear couplings A = AY where the Y ’s are the neutrino Yukawa couplings and α the lepton flavor index, they
α α α
showed that the resulting CP violation is of the order of (m /M) at the resonance which, however, requires an
SUSY
unconventionally small soft bilinear coupling B m O. Away from the resonance, the CP violation is of Y2
and, hence, too suppressed for successful leptog(cid:28)enesiSs.USOYn the other hand, assuming generic A couplings, ROef. [1α6]
showed that successful leptogenesis can be obtained with B m away from the resonant regime. (cid:0) (cid:1)
SUSY
Later in Ref. [17] it was argued that direct CP violation∼, i.e., from vertex corrections, due to gaugino exchange
in the loop, survives at the order m2 /M2 at T = 0. Since the neutrino Yukawa coupling is replaced by the
gauge coupling in the CP violationOparaSmUSeYter, a large CP violation can be obtained for M at the TeV scale. Further
study in Ref. [18], however, showed t(cid:0)hat in fact in(cid:1) this scenario, the cancellation still holds up to m2 /M2 at
T =0,anditwasconcludedthatfinitetemperatureeffectsarenecessarytopreventthecancellationO. TheSUcSaYncellation
is consistent with the result obtained in Ref. [19] which states that to have a nonvanishing total CP(cid:0)violation th(cid:1)ere
should be lepton number violation to the right of the “cut” in the loop diagram, and this requirement is not fulfilled
in these cases. More recently, in Ref. [20] it was shown that if finite temperature effects are taken into account
consistently, the cancellation of direct CP violation from the gaugino contribution still holds even at T =0.
Infact,insoftleptogenesisatfinitetemperature,thepartialcancellationintheresultingleptonandslep(cid:54) tonnumber
density asymmetries sourced by CP violation from mixing and the complete cancellation in the case of the gaugino
vertex correction [20] only hold under the assumption of equilibration between the chemical potentials of leptons and
sleptons (superequilibration) which is valid below T < 108 GeV for m TeV [5]. As shown in Ref. [21], in the
SUSY
nonsuperequilibration regime, the partial cancellatio∼n between lepton and∼slepton number density asymmetries in
the mixing scenario is avoided, resulting in an enhanced efficiency for soft leptogenesis. However, for reasons given
later, we shall below consider mixing and vertex scenarios in the superequilibration regime (and also find a case
where the lepton and slepton number density asymmetries do not partially cancel each other). On the other hand,
considering M (cid:38) 108 GeV and m 1 TeV, the CP violating parameter from the gaugino contribution in the
SUSY
∼
1 SeeRefs. [10–12]foranotherresolutionofthegravitinoproblemduetodelayedthermalizationoftheUniverseafterinflation.
2 In a realistic model, we need at least two RHNs to accommodate neutrino oscillations. Assuming RHNs to be hierarchical, soft
leptogenesisonlydependsontheparametersrelatedtothelightestRHNanddecouplesfromtheparametersrelatedtoheavierRHNs.
3
nonsuperequilibration regime is (cid:15) 10−1m2 /M2 (cid:46) 10−11 and, hence, is too small for successful leptogenesis.
Therefore processes involving gaugi∼nos will nSoUtSYbe considered further in this work.
In this article, we revisit soft leptogenesis by relaxing the assumption of the proportionality of the A couplings.
In Sec. II, we review the Lagrangian for soft leptogenesis with generic A terms and spell out the constraints from
α
out-of-equilibrium decays of heavy sneutrinos and the cosmological bound on the sum of neutrino masses. In Sec.
III, we obtain the CP violating parameter for both the self-energy corrections (mixing) and vertex corrections. We
showthatgenericA couplingsgiverisetotwointerestingconsequences: (1)theleadingCP violationcanbenonzero
α
even when thermal corrections are neglected implying a nonthermal CP violation, and (2) the mixing CP violation
away from the resonance is of the order of (Y ) and, hence, can be large enough for leptogenesis. Because of the
α
small mass splitting, the mixing CP violatioOn always dominates over the vertex CP violation even far away from the
resonant regime. In Ref. [22], it was shown that with A = AY , soft leptogenesis gives negligible contributions to
α α
theelectricdipolemomentofchargedleptonsandchargedleptonflavorviolatingprocesses. InSec. IV,werepeatthe
exercise and show that with generic A couplings, the contributions to charged lepton flavor violating processes are
α
close to the sensitivities of present and future experiments. Finally, in Sec. V, we conclude. This article is completed
with two appendixes. In Appendix A, we discuss the inclusion of thermal effects under the assumption of decaying
heavy sneutrinos at rest. In Appendix B, we review the two specific scenarios of A discussed in Ref. [16] and discuss
α
an interesting point missed by Ref. [16] which actually allows for nonzero leading CP violation at zero temperature.
II. THE LAGRANGIAN
The superpotential for the type-I seesaw is given by
1
W = M NˆcNˆc+Y Nˆc(cid:96)ˆ Hˆ , (1)
N 2 i i i iα i α u
where Nˆc, (cid:96)ˆ and Hˆ denote, respectively, the chiral superfields of the RHNs, the lepton doublet and the up-type
i α u
Higgs doublet, and i and α are the RHN family and lepton flavor indices, respectively. The SU(2) contraction
L
between (cid:96)ˆ and Hˆ is left implicit. In the following, we will assume that the RHNs are hierarchical such that only
α u
the lightest RHN N is relevant for soft leptogenesis. Henceforth, we will drop the family index of RHN, for example,
1
N N and Y Y . The corresponding soft terms are
1 α 1α
≡ ≡
1
=M2N∗N˜ + BMNN +A N(cid:96) H +H.c. . (2)
−Lsoft 2 α α u
(cid:18) (cid:19)
The mass and interaction terms involv(cid:102)ing(cid:101)the sneutrino N(cid:101)fr(cid:101)om W (cid:101)a(cid:101)re given by
N
−LN(cid:101) =|M|2N∗N˜ + M∗YαN∗(cid:101)(cid:96)αHu+YαHucPL(cid:96)αN +H.c. , (3)
(cid:16) (cid:17)
where PL,R = 12(1∓γ5). Through field re(cid:101)definitions, it ca(cid:101)n(cid:101)be shown t(cid:101)hat the t(cid:101)hree physical phases are
Φ =arg(A Y∗B∗). (4)
α α α
Without loss of generality, the phases can be assigned to A and all other parameters will be taken real and positive.
α
WewouldliketostressthatwedonotassumetheproportionalityofA totheneutrinoYukawacouplings(A =AY )
α α α
as has been done in Refs. [13, 14, 23] where there is only one physical phase Φ=arg(AB∗). As we will show in Sec.
III, by considering generic A couplings, the CP violation can be nonvanishing even at zero temperature.
α
Because of the bilinear B term, N and N∗ mix to form mass eigenstates
1
(cid:101) (cid:101) N+ = N +N∗ ,
√2
(cid:16) (cid:17)
i
N(cid:101) = (cid:101)N (cid:101)N∗ , (5)
−
−√2 −
(cid:16) (cid:17)
with the corresponding masses given by (cid:101) (cid:101) (cid:101)
M2 =M2+M2 BM. (6)
± ±
In order to avoid a tachyonic mass which implies an instability of the vacuum such that the sneutrino will develop a
(cid:102)
vacuum expectation value, we always assume B <M +M2/M.
(cid:102)
4
Rewriting the Lagrangian in terms of mass eigenstates N we have
±
=M2N∗N +M2N∗N
−LN(cid:101) −Lsoft + + + −(cid:101) − −
1
+ N Y HcP (cid:96) +(A +MY )(cid:96) H
√2(cid:101) (cid:101)+ α u(cid:101)L(cid:101)α α α α u
(cid:110) (cid:104) (cid:105)
+iN Y(cid:101) HcP(cid:101)(cid:96) +(A MY )(cid:96) H(cid:101) +H.c. . (7)
− α u L α α− α α u
(cid:104) (cid:105) (cid:111)
(cid:101) (cid:101) (cid:101)
A. General constraints
The total decay width for N is given by
±
M A 2 Y Re(A )
(cid:101) Γ Y2+ | α| α α , (8)
± (cid:39) 4π α 2M2 ± M
α (cid:20) (cid:21)
(cid:88)
where we have expanded up to (Y2,m2 /M2,Y m /M) and ignored the final state phase space factors. We
will impose the restriction thatOA α,B <SUMSYand Y α<S1UtSoYensure that we are always in the perturbative regime. In
α α
principle,m /M andY can|gou|pto4π beforeperturbativetheorybreaksdownbutwithourstrongerrestriction,
SUSY α
we are not anywhere near the nonperturbative regime.
The out-of-equilibrium condition for leptogenesis is
Γ (cid:46)H(T =M), (9)
±
wheretheHubbleexpansionrateisgivenbyH =1.66√g(cid:63)T2/MPlwithPlanckmassMPl =1.22 1019GeV.Assuming
minimal supersymmetric Standard Model relativistic degrees of freedom, we have g = 228.75.×The condition above
(cid:63)
translates to
A 2 Y Re(A ) M 1/2
Y2+ | α| α α (cid:46)1.6 10−5 . (10)
α 2M2 ± M × 107GeV
(cid:115) α (cid:20) (cid:21) (cid:18) (cid:19)
(cid:88)
From the condition above, we see that A is bounded from above depending on M. For example if M TeV, we
α
| | ∼
require A (cid:46) 10−4 GeV. At this low scale, the mass splitting between N and N is required to be of the order
α + −
of their|deca|y widths such that the CP violation is resonantly enhanced to yield successful leptogenesis [25, 26]. To
avoid excessive fine-tuning, if we consider Aα TeV, Eq. (10) implies M(cid:101)(cid:38)4 10(cid:101)7GeV.
In type-I seesaw, barring special cancell|atio|n∼, we have the upper bound on t×he sum of light neutrino masses from
cosmology [24]
Y2v2
α u (cid:46) m 0.23eV,
M νi (cid:39)
α i
(cid:88) (cid:88)
M 1/2 1 1/2
Y2 (cid:46)3 10−4 1+ , (11)
α × 107GeV tan2β
(cid:115) α (cid:18) (cid:19) (cid:18) (cid:19)
(cid:88)
where tanβ v /v and v = H are the up(down)-type Higgs vacuum expectation values. v2 + v2 =
≡ u d u(d) u(d) u d
√2G−1 (174GeV)2 with G the Fermi constant. For tanβ (cid:38) 1, the bound above is always less stringent than
Eq. (1F0),(cid:39)and, hence, the out-oFf-equi(cid:10)librium(cid:11)condition alone is sufficient.
III. CP VIOLATION
In this section we will study CP violation of the Lagrangian (7) from the interferences between tree-level and one-
loop diagrams shown in Figs. 1 and 3. We will take into account thermal corrections while approximating sneutrinos
N to always be at rest with respect to the thermal bath. Since we are in the regime where all three lepton flavors
±
can be distinguished (T (cid:46)109 GeV), we will not sum over the lepton flavor in the final states [23].
(cid:101)To quantify the CP violation, we define the CP asymmetry for the decays N a with a = (cid:96) H ,(cid:96) H as
± α α α u α u
→ { }
γ(N a ) γ(N a )
(cid:15)S,V ± → α − ± → α (cid:101), (cid:101) (cid:101) (12)
±α ≡ γ(N a )+γ(N a )
aβ;β (cid:101) ± → β (cid:101) ± → β
(cid:104) (cid:105)
(cid:80)
(cid:101) (cid:101)
5
ℓ ℓ ℓ
β ℓ β ℓ β ℓ
α α α
e e
N N N N e N N e
g± g∓ g± g∓ g± g∓
H H H
u u u
H H H
u g u u
(a) (b) (c)
g
ℓ ℓ ℓ
β ℓ β ℓ β ℓ
α α α
e e
N N N N e N N e
g± g∓ g± g∓ g± g∓
H H H
u u u
H H H
u g u u
(d) (e) (f)
g
Figure 1. One-loop self-energy diagrams for the decays N(cid:101)± → (cid:96)αH(cid:101)u [(a),(d)] and N(cid:101)± → (cid:96)(cid:101)αHu [(b),(c),(e),(f)]. The arrow
indicates the flow of lepton number. The dotted vertical lines indicate the corresponding intermediate states go on mass shell.
The diagram with fermionic loop and fermionic final states does not contribute to the CP violation since it does not involve
the soft couplings A .
α
wherethesuperscriptsS andV indicatetheCP violationcomingfromself-energyandvertexcorrections,respectively,
a indicates the CP conjugate of a , and γ(i j) is the thermal averaged reaction density for the process i j
α α
defined in Eq. (A1). In the following, we will in→clude the thermal effects associated with intermediate on-shell st→ates
which, as shown in Ref. [20], will result in the cancellation of vertex CP asymmetries from gaugino contributions
[17,18]. WewillalwaysapproximateN tobeatrestwithrespecttothethermalbathsothatwecanobtainanalytical
±
expressionsfortheCP asymmetries(seeAppendixA).Furthermore,wefocusonthesuperequilibrationregimewhich
falls in the temperature range T (cid:46)10(cid:101)8 GeV for mSUSY 1 TeV [5]. The advantage is that in this regime, lepton and
sleptons are not distinguished (they have the same chem∼ical potentials) and so the two Boltzmann equations for the
lepton asymmetry in particles and sparticles can be reduced to one equation for the net lepton asymmetry.3 Hence
we are allowed to sum over CP asymmetries of lepton and slepton final states as below.
A. CP violation from mixing
In this subsection, we discuss the mixing CP violation from self-energy corrections. There are two kinds of self-
energy diagrams as shown in Fig. 1: the diagrams with continuous flow of lepton number [Figs. 1(a)–1(c)] and the
diagrams with flow of lepton number inverted in the loop [Figs. 1(d)–1(f)]. Notice that diagrams with fermionic loop
andfermionicfinalstatesdonotcontributetotheCP violationsincetheydonotinvolvethesoftcouplingsA . From
α
Figs. 1(a)–1(c), we obtain the respective contributions to the CP asymmetries defined in Eq. (12) as follows4:
1 Im(A ) M2 B 2BM
(cid:15)S,(a) = Y2 Y β 1+ r (T)c (T),
±α 4πG±(T) α β β M (cid:32) M2 ± M(cid:33)4B2+Γ2∓ B F
(cid:88) (cid:102)
1 Im(A ) M2 B 2BM
(cid:15)S,(b) = Y2Y α 1+ r (T)c (T), (13)
±α −4πG±(T) α M (cid:32) M2 ± M(cid:33)4B2+Γ2∓ F B
(cid:102)
1 A 2 Im(A ) A 2 Im(A )
(cid:15)S,(c) = Y2 | β| Y α Y2 | α| Y β
±α 4πG (T) − M2  α M − α − M2 β M 
± β (cid:18) (cid:19) β
(cid:88) (cid:88)
2BM   
r (T)c (T),
×4B2+Γ2 B B
∓
3 We make the assumption of superequilibration also to highlight the positive effects of nonthermal CP violation in soft leptogenesis.
Includingnonsuperequilibrationeffects,theefficiencyofsoftleptogensisisexpectedtobefurtherenhanced,andthiseffectwasstudied
indetailinRef.[21]. Thevaliditywindowofsuperequilibrationcanbeenlargedbyincreasingthegauginomassesandµparameter[5]
and/ordecreasing|Aα|.
4 The absorptive parts which regularize the singularity in the N(cid:101)± propagators as M+ → M− are obtained by resumming self-energy
diagramsfollowingRefs.[25,26].
6
where we define Y2 Y2 and
≡ α α
(cid:80) A 2 2Y Re(A ) M2 B
G (T) Y2+ | α| α α c (T)+Y2 1+ c (T). (14)
± ≡(cid:34) α (cid:18) M2 ± M (cid:19)(cid:35) B (cid:32) M2 ± M(cid:33) F
(cid:88) (cid:102)
In the above r (T) and c (T) are temperature-dependent terms associated with intermediate on-shell and final
B,F B,F
states respectively, as given in Appendix A. We will also make use of the following identity
r (T)c (T)=r (T)c (T), (15)
F B B F
proveninAppendixA.NotethatifwesumovertheleptonflavorαanduseEq.(15),weobtain (cid:15)S,(a)+(cid:15)S,(b) =
α ±α ±α
α(cid:15)S±,α(c) =0, in agreement with the T =0 result of Ref. [19] that if there is no L violation to(cid:80)the(cid:16)right of the cu(cid:17)t in
the one-loop diagrams, the net CP violation on summing over all final states is zero.
(cid:80)From Figs. 1(d)–1(f), we have
1 Im(A ) M2 B 2BM
(cid:15)S,(d) = Y2 Y β 1+ r (T)c (T),
±α 4πG±(T) α β β M (cid:32) M2 ± M(cid:33)4B2+Γ2∓ B F
(cid:88) (cid:102)
1 Im(A ) M2 B 2BM
(cid:15)S,(e) = Y2Y α 1+ r (T)c (T), (16)
±α 4πG±(T) α M (cid:32) M2 ± M(cid:33)4B2+Γ2∓ F B
(cid:102)
1 A 2 Im(A ) A 2 Im(A )
(cid:15)S,(f) = Y2 | β| Y α Y2 | α| Y β
±α 4πG (T)− − M2  α M − α − M2 β M 
± β (cid:18) (cid:19) β
(cid:88) (cid:88)
2BM    
r (T)c (T).
×4B2+Γ2 B B
∓
Notice the leading contributions from N and N in Eqs. (13) and (16) come with the same sign and, hence, they
+ −
will contribute constructively to the lepton number asymmetry.
The total CP asymmetry from mixin(cid:101)g (cid:15)S (cid:101) (cid:15)S,(n) is given by
±α ≡ n={a,b,c,d,e,f} ±α
1 I(cid:80)m(A ) 4BM
(cid:15)S = Y2 Y β [c (T) c (T)]r (T)
±α 4πG (T) α β M 4B2+Γ2 F − B B
± β ∓
(cid:88)
1 A 2 Im(A ) 4BM
+ | α| Y β r (T)c (T) (17)
4πG (T) M2 β M 4B2+Γ2 B B
± β ∓
(cid:88)
1 Im(A ) M2 B 4BM
+ Y2 Y β r (T)c (T).
4πG±(T) α β β M (cid:32)M2 ± M(cid:33)4B2+Γ2∓ B F
(cid:88) (cid:102)
In the above, the first term vanishes in the zero temperature limit T 0 when c (T) 1 and r (T) 1, while
B,F B,F
the terms higher order in m /M survive. They remain nonzero→after summing over→the lepton flavor→α. In the
SUSY
following in order to make the dependence of thermal and nonthermal CP asymmetries in Eq. (17) on the model
parameters more transparent, it is instructive to look at two limiting cases (i) Y A /M and (ii) Y A /M
α α α α
where in case (i), the thermal CP violation dominates, while in case (ii), the nonthe(cid:29)rmal CP violation dom(cid:28)inates.5
• In the limit (i) Y A /M, we have
α α
(cid:29)
1 Im(A ) 4BM c (T) c (T)
(cid:15)S P Y β F − B r (T), (18)
±α (cid:39) 4π α β M 4B2+Γ2 c (T)+c (T) B
β Y F B
(cid:88)
where we define the flavor projector P Y2/Y2 with P = 1 and Γ Y2M, and we have dropped the
terms higher order in m /M. In thαis≡caseα, the CP asymαmαetry in Eq. (1Y8)≡is p4πroportional to c (T) c (T)
SUSY F B
which goes to zero as T 0, and, hence, the contributio(cid:80)n to the CP violation is the thermal one. −
→
5 Bythermal(nonthermal)CP violation,werefertothecasewhereCP violationdoes(not)vanishasT →0.
7
In the resonant regime where B Γ , we have (cid:15)S (A/M)/Y where we have suppressed the lepton flavor
∼ ± ± ∼ | |
index for an order of magnitude estimation. In this case, a large (cid:15)S can be obtained which allows TeV-scale
±
leptogenesis but at the cost of having unnaturally small A,B TeV.
| | (cid:28)
Away from the resonant regime when B Γ , the CP asymmetries go as (cid:15)S 10−1Y A/B assuming (1)
contribution from the CP phases of Eq. (4(cid:29)). T±aking A TeV(cid:38)B together w±it∼h the out|-o|f-equilibrium dOecay
condition (10) gives us sufficient CP asymmetries (cid:15)S|(cid:38)|∼10−6 for M (cid:38)107 GeV.
±
• In the other limit (ii) Y A /M, we have
α α
(cid:28)
1 A 2 Im(A ) 4BM
(cid:15)S | α| Y β r (T), (19)
±α (cid:39) 4π A 2 β M 4B2+Γ2 B
δ| δ| β A
(cid:88)
(cid:80)
where Γ |Aα|2. The CP asymmetries Eq. (19) clearly do not vanish at T = 0, and this represents
a nontheArm≡al CαP8vπiMolation. Of course thermal effects are always there but the fact that the CP violation is
nonvanishing(cid:80)at T =0 implies that it is less suppressed compared to case (i).
In the resonant regime B Γ , we have (cid:15)S Y/(A/M). In this case too a large (cid:15)S can be obtained which
allows TeV-scale leptogene∼sis b±ut at the cos±t o∼f havi|ng| unnaturally small A,B TeV±.
| | (cid:28)
AwayfromtheresonantregimewithB Γ ,theCP asymmetries,likeinthelimit(i),goas(cid:15)S 10−1Y A/B
assuming (1) contribution from the C(cid:29)P p±hases of Eq. (4). Hence taking A TeV (cid:38) B t±og∼ether wit|h|the
out-of-equOilibrium decay condition (10) gives us sufficient CP asymmetries (cid:15)|S|(cid:38)∼10−6 for M (cid:38)107 GeV.
±
To confirm our estimation of successful leptogenesis and also to illustrate the enhancing effects of nonthermal CP
violation,wenumericallysolvetheBoltzmannequationsusingtheexpressionfortheasymmetryparameterinEq.(17).
For simplicity, we consider only decays and inverse decays of N and N . We will also define the washout parameter
±
asK Γ /H(T =M)withΓ givenbyEq. (8). InFig. 2,weplottheabsolutevalueofthefinalbaryonasymmetry
± ±
Y∆B(≡ ) as a function of K for the following three scenarios: (cid:101)
| ∞ |
• NonThermaldominated(NTD):Inthisscenario,wechooseA/M =(10−4,10−2,1)wandY=(10−5,10−3,10−1)w.
• Thermal Dominated (TD): In this scenario, we choose A/M =(10−5,10−3,10−1)w and Y=(10−4,10−2,1)w.
• Mixed (MIX): In this scenario, we choose A/M =(10−4,10−2,1)w and Y=(10−4,10−2,1)w.
IntheaboveAandY representthecouplingswrittenas3-vectors. Inallthescenariosabove,wevarywbetween10−6
and 10−4 such that we scan through the parameter space from the weak washout (K = 0.1) to the strong washout
(K = 15) regime while still respecting the cosmological bound on the sum of light neutrino masses in Eq. (11). For
definiteness, we also fix M =5 107 GeV, tanβ =10, arg(A )= π/2, and B =1 TeV.
α
In Fig. 2, for the left plot, w×e solve from an initial time with T− M assuming zero initial number densities for
(cid:29)
N and N while for the right plot, we assume thermal initial number densities for N and N . For the observed
± ±
baryonasymmetry,weusetherecentcombinedPlanckandWMAPCMBmeasurementsofcosmicbaryonasymmetry
[24, 27] a(cid:101)t 2σ, (cid:101)
YCMB =(8.58 0.22) 10−11, (20)
∆B ± ×
which is plotted as the gray band in Fig. 2. From the plots, we see that in the NTD (blue dashed line) and MIX
(purple solid line) scenarios, Y ( ) falls off very slowly in the strong washout regime K > 1. The reason is that
∆B
the falloff in their efficiencies|is alm∞ost|completely compensated by the increases in their respective CP asymmetries
asoneincreasesA /M. Ontheotherhand,intheTD(reddottedline)scenario,intheK >1regime, Y ( ) falls
α ∆B
offmuchfasterduetotheadditionalsuppressionfromthepartialcancellationbetweentheCP asymme|triesf∞rom| the
decays of N to scalars and fermions.
±
In our study we are also interested in the situation when A and B are restricted to be around the TeV scale. In
α
Fig. 2, the(cid:101)regions to the left of the thick blue dashed and purple solid vertical lines correspond to K when Aα < 5
TeV for the NTD and MIX scenarios, respectively, while A < 5 TeV for the TD scenario in the entire range of K
α
considered in the plot. We find, for example, for the case of zero initial number densities of N and N , the correct
±
amount of baryon asymmetry can be obtained for NTD at K 0.8, MIX at K 0.6, and TD at K 4. Notice that
the appropriate sign baryon asymmetry can always be obtain∼ed by choosing th∼e appropriate phases∼of the complex
(cid:101)
couplings A . From this numerical exercise we conclude that generation of sufficient baryon asymmetry is possible
α
for TeV-scale A and B Γ , i.e., far away from the resonant regime. Besides, we also see that nonthermal CP
α ±
violation can significantly(cid:29)enhance the efficiency of soft leptogenesis.
Finally,inAppendixBwewilldiscusstwospecialcases,namely,(a)A =AY and(b)A =AY2/(3Y )considered
α α α α
in previous work.
8
B(cid:61)1 TeV, arg A (cid:61)(cid:45)Π 2, tanΒ(cid:61)10 B(cid:61)1 TeV, arg A (cid:61)(cid:45)Π 2, tanΒ(cid:61)10
Α Α
Y (cid:165) Y (cid:165)
(cid:68)B (cid:68)B
2(cid:180)10(cid:45)9 2(cid:180)10(cid:45)9
1(cid:180)10(cid:45)9 1(cid:180)10(cid:45)9
(cid:72) (cid:76) (cid:144) (cid:72) (cid:76) (cid:144)
5(cid:180)10(cid:200)(cid:45)10 (cid:72) (cid:76)(cid:200) 5(cid:180)10(cid:200)(cid:45)10 (cid:72) (cid:76)(cid:200)
2(cid:180)10(cid:45)10 2(cid:180)10(cid:45)10
1(cid:180)10(cid:45)10 1(cid:180)10(cid:45)10
5(cid:180)10(cid:45)11 5(cid:180)10(cid:45)11
2(cid:180)10(cid:45)11 2(cid:180)10(cid:45)11
1(cid:180)10(cid:45)11 1(cid:180)10(cid:45)11
K K
0.2 0.5 1.0 2.0 5.0 10.0 0.2 0.5 1.0 2.0 5.0 10.0
Figure2. Theabsolutevalueofthefinalbaryonasymmetry|Y (∞)|asafunctionofthewashoutparameterK ≡Γ /H(T =
∆B ±
M)forM =5×107 GeVforthethreescenariosdescribedinthetext: NTD(bluedashed),TD(reddotted)andMIX(purple
solid). The left plot corresponds to the case of zero initial number densities of N and N(cid:101)±, while the right plot corresponds to
the case of thermal initial number densities of N and N(cid:101)±. The regions to the left of the blue dashed and purple solid vertical
lines correspond to K values when A < 5 TeV for the NTD and MIX scenarios, respectively, while for the TD scenario we
α
always have A < 5 TeV in the range of the plot. The gray band represents the recent combined Planck and WMAP CMB
α
measurementsofcosmicbaryonasymmetry[24,27]at2σ. ThedipintheTDscenariointheleftplotreferstoachangeinthe
sign of the baryon asymmetry.
ℓ ℓ ℓ
H α H α H α
u u u
N N e N e
g
± N ± N ± N
g g g ∓
H H g H
ℓβ u ℓβ u ℓβ u
g
e (a) (b) e (c)
Figure 3. One-loop vertex diagrams for the decays N(cid:101)± →(cid:96)αH(cid:101)u [(a)] and N(cid:101)± →(cid:96)(cid:101)αHu [(b),(c)] with the conventions of Fig.
1. ThediagramswithfermionicloopandfermionicfinalstatesdonotcontributetotheCP violationsincetheydonotinvolve
the soft couplings A .
α
B. CP violation from vertex corrections
In this subsection, we discuss the CP violation from vertex corrections. From Figs. 3(a)–3(c), we obtain
1 Im(A ) M2 +M2
(cid:15)V,(a) = Y2 Y β ln ± r (T)c (T),
±α ∓8πG (T) α β M M2 B F
±
β
(cid:88)
1 Im(A ) M2 +M2
(cid:15)V,(b) = Y2Y α ln ± r (T)c (T), (21)
±α ∓8πG (T) α M M2 F B
±
1 A 2 Im(A ) A 2 Im(A )
(cid:15)V,(c) = Y2 | β| Y α + Y2 | α| Y β
±α ±8πG (T) − M2  α M α − M2 β M 
± β (cid:18) (cid:19) β
(cid:88) (cid:88)
M2 M2+M2  
ln ± ∓r (T)c (T).
×M2 M2 B B
± ∓
9
ℓ ℓ
H α H α
u u
N g N e
N N
± ±
g H g H
ℓβ u ℓβ u
g
e (a) (b)
Figure 4. One-loop vertex diagrams for the decays N →(cid:96)αHu [(a)] and N →(cid:96)(cid:101)αH(cid:101)u [(b)] with the conventions of Fig. 1.
Summing over the contributions above and expanding in B/M 1 in the numerators, we have
(cid:28)
(cid:15)V (cid:15)V,(a)+(cid:15)V,(b)+(cid:15)V,(c)
±α ≡ ±α ±α ±α
ln2 Im(A ) Im(A )
= Y2Y α +Y2 Y β [c (T) c (T)]r (T)
∓8πG (T) α M α β M  F − B B
±
β
(cid:88)
 
1 Im(A ) Im(A ) B c (T)
Y2Y α +Y2 Y β F +(ln2 1)c (T) r (T) (22)
−8πG (T) α M α β M M 2 − B B
± β (cid:20) (cid:21)
(cid:88)
 
ln2 A 2 Im(A ) A 2 Im(A )
| β| Y α + | α| Y β r (T)c (T)
∓8πG (T) M2 α M M2 β M  B B
±
β β
(cid:88) (cid:88)
 
1 A 2 Im(A ) A 2 Im(A ) B
+ | β| Y α + | α| Y β (ln2 1)r (T)c (T).
8πG (T) M2 α M M2 β M M − B B
±
β β
(cid:88) (cid:88)
 
The leading contributions from N [first and third lines of Eq. (22)] come at the order of (cid:15)V 10−1Y2 [taking
±
Y Im(A )/M],whicharetoosmallforsuccessfulleptogenesisfromEq. (10)forM > 107 GeV.O∼fcoursethesame
α α
con∼clusion holds also when Yα (cid:101)Im(Aα)/M or Yα Im(Aα)/M. Besides, notice also∼that the leading contributions
(cid:29) (cid:28)
from N come with the opposite signs, and, hence, they will contribute destructively to the total lepton number
±
asymmetry. Upon expanding G (T) terms also in B/M 1, we obtain an additional suppression factor B/M like
±
the terms in the second and fourth lines in Eq. (22). Henc(cid:28)e we conclude that the vertex CP violation is irrelevant for
(cid:101)
soft leptogenesis.
So far, we have been discussing the contributions of soft terms to CP violation in the decay of N . In fact the soft
±
terms also provide new sources of CP violation in the one-loop vertex diagrams for the decays of the heavy neutrino
N as shown in Fig. 4. Nevertheless the CP violation from these diagrams comes at the same ord(cid:101)er as Eq. (22) and,
hence, is too small for successful leptogenesis.
IV. PHENOMENOLOGICAL CONSTRAINTS
We are primarily concerned with scenarios with M > 107GeV for which the production of sneutrinos is beyond
the energy range of current colliders. However, even i∼f M TeV, the bound on the Yukawa couplings from the
±
∼
requirement of out-of-equilibrium decays of N [in Eq. (10)] makes N impossible to be produced at colliders [28].
± ±
On the other hand, the soft SUSY breaking parameters relevant for soft leptogenesis A , B, and M can contribute
α
to Electric Dipole Moments (EDM) of lepton(cid:101)s and to Charged Lepton(cid:101)Flavor Violating (CLFV) interactions though
the analysis of Ref. [22] under the assumption of universality soft trilinear couplings Aα = AYα(cid:102)showed that the
contributions to EDM and CLFV are much below the experimental bounds. Here we will repeat the analysis of
Ref.[22]consideringagenericA . InRef.[29],thephenomenologicalconsequencesofthesofttermsconsideringthree
α
generationsofRHNchiralsuperfieldshavebeendiscussedatlength. Clearly,thesesoftparametersareconnectedwith
the mechanism of SUSY breaking and as such are model dependent. Here we will remain agnostic about the SUSY
breaking mechanism and simply focus on the phenomenological constraints on these parameters and, in particular,
we will focus only on parameters related to N which are relevant for soft leptogenesis, i.e., B, M, A , Y , and M.
1 α α
Without fine-tuning, we consider the soft parameters B, M, and A to be similar or smaller than m TeV. On
α SUSY
∼
(cid:102)
(cid:102)
10
the other hand, the parameters Y and M are subject only to the out-of-equilibrium N decay constraint in Eq. (10)
α ±
and less stringently to the cosmological bound on the sum of neutrino masses in Eq. (11). The running of Y from
α
the high scale down to the weak scale gives some corrections at the level of 10% 20%(cid:101)(see Fig. 3 of Ref. [31]) which
we will ignore in the following. −
1. Electric dipole moment of the electron
Assuming (1) contribution of the phases and mixing angles in the chargino sectors, the contributions of A and
α
B to the EDMO of the electron are given by [22]
em tanβ m Y
d e χ α (A +BY ) , (23)
| e|≈ 16πm2 M2 | α| α
ν˜ (cid:12) (cid:12)
(cid:12) (cid:12)
whereme istheelectronmass,m2ν˜ isthesquaredmasso(cid:12)(cid:12)fthelig(cid:12)(cid:12)htsneutrinoandmχ themassofchargino. Forgeneric
A , the first term in in Eq. (23) dominates. Taking m =m =m and making use of Eq. (10), we have
α ν˜ χ SUSY
tanβ 107GeV 3/2 1TeV
d (cid:46)5 10−38 ecm, (24)
| e| × 10 M m
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) SUSY(cid:19)
which is much stronger than the current experimental bound d < 8.7 10−29ecm [30]. The contributions to µ
e exp
and τ EDM can be estimated by replacing m in Eq. (23) by m| |and m , r×espectively, but the current experimental
e µ τ
constraints on them are still a lot weaker: d <1.9 10−19ecm [32] and d <5.1 10−17ecm [33].
µ exp τ exp
| | × | | ×
2. Charged lepton flavor violating interactions
The branching ratio for charged lepton flavor violations due to nonvanishing off-diagonal elements of the soft mass
matrix of the doublet sleptons m2 is given by [22, 34]
(cid:96)˜
2
BR((cid:96) (cid:96) γ) α3 (m2(cid:96)˜)αβ tan2β, (25)
α → β ≈ G2 (cid:12)m8 (cid:12)
F (cid:12) SUSY(cid:12)
(cid:12) (cid:12)
where α is the fine structure constant. In general, the off-diagonal elements of m2 will induce too-large CLFV rates.
(cid:96)˜
The usual solution is to assume mSUGRA boundary conditions at the grand unified theory (GUT) scale where the
off-diagonal elements of m2 vanish. In this case, as m2 evolves from the GUT scale M to the RHN mass scale M,
(cid:96)˜ (cid:96)˜ GUT
the off-diagonal elements will be generated due to the renormalization effects as [35]
1 M
(m2) A∗A ln GUT (26)
(cid:96)˜ αβ ≈−8π2 α β M
(cid:18) (cid:19)
for α=β, and we have kept only the dominant contributions from A .
α
The(cid:54) most stringent constraint on the rare decay µ eγ comes from the nonobservation of the process from the
MEG experiment [36, 37] which has set the new bound→on the branching ratio for µ eγ,
→
BR(µ eγ) <5.7 10−13. (27)
exp
→ ×
Substituting Eq. (26) in Eq. (25) and applying the constraint Eq. (27), we obtain
m 4 10
A∗A (cid:46)5 103GeV2 SUSY , (28)
| µ e| × 1TeV tanβ
(cid:16) (cid:17) (cid:18) (cid:19)
where we have taken M =1016 GeV and M =107 GeV. Similarly using the experimental bounds on CLFV in τ
GUT
decays, BR(τ eγ) <3.3 10−8 and BR(τ µγ) <4.4 10−8 [38], we obtain
exp exp
→ × → ×
m 4 10
A∗A A∗A (cid:46)1 106GeV2 SUSY . (29)
| τ e|≈| τ µ| × 1TeV tanβ
(cid:16) (cid:17) (cid:18) (cid:19)