Natural emergence of neutrino masses and bi-linear "RPV" from R-symmetry

0.22 MB·

by  Sabyasachi Chakraborty

TIFR/TH/17-04
Natural emergence of neutrino masses and bi-linear “RPV” from
R-symmetry
SabyasachiChakrabortya, Joydeep Chakraborttyb
aDepartment of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai 400005, India
bDepartment of Physics, Indian Institute of Technology, Kanpur-208016, India
7
1
0
2 Abstract

n We propose a supersymmetric extension of the Standard Model (SM) with a continuous global U(1)
R
a
symmetry. The R-charges of the SM fields are identified with that of their lepton numbers. As an artifact,
J
a bi-linear “R-parity violating” term emerges at the superpotential. However, R-symmetry is not an exact
7
symmetryasitisbrokenbysupergravityeffects. Asaresult,sneutrinosacquireasmallvacuumexpectation
1
value inthis framework. Consequently,neutrino masses andmixing canbe explainedatthe tree levelitself.

] Thisis noticeablydifferentfromthe standardR-parityviolatingMinimalSupersymmetric StandardModel.
h
GravitinomassturnsouttobetheorderparameterofR-breakinganddirectlyrelatedtotheneutrinomass.
p
- To have thermal production of gravitinos, we require the gravitino mass to be > 200 keV. We show that
p
such a gravitino is an excellent dark matter candidate and is safe from all constraints. Finally, we looked
e
into the collider implications of our framework.
h
[

1
v
6 1. Introduction
6
5 Neutrino oscillation experiments have firmly established the existence of tiny non-zero masses of active
4 neutrinos and non-trivial mixing in the lepton sector. Neutrinos are massless within the paradigm of Stan-
0 dard Model (SM). Therefore, the oscillation data can naturally promote the query for physics beyond the
.
1 SM (BSM). Apart from CP-violating phases most of the oscillation parameters have been determined with
0 great accuracy [1]. Thus, it is expected that any BSM theory must provide a suitable mechanism for neu-
7 trinomassgeneration. Todoso,see-sawmechanismseemstobethenaturalonethatpredictstheMajorana
1
nature of the neutrino signifying lepton number violation. The basic idea behind the see-saw mechanism is
:
v tointegrateouttheheavymodes,leadingtohigher-dimensionalneutrinomassoperators. Dependingonthe
i
X choice of the heavy particles, one can classify variants of this mechanism, namely Type-I, -II, -III, Inverse,
Double see-saw etc. both in supersymmetric (SUSY) and non-SUSY scenarios. However, so far there is no
r
a such experimental data that can help us to pin down any particular mechanism belonging to any specific
model. Thus fromneutrino mass generationperspective, these mechanisms are more sacredthanparticular
models. Apart from neutrino masses and mixing the deviation from the galactic rotation curve and bullet
clusters provide concrete evidences in favour of dark matter (DM) which remains unexplained within the
SM framework. Cosmological observations has also measured the relic density of the DM with very high
precision [2, 3]. But unfortunately the DM characteristics, e.g. mass, spin, and its nature, i.e., cold, warm,
single or multi component are yet to be determined. Thus there must be a theory beyond the Standard
Model.
Keepingthesethingsinmindandalsofromatheoreticalperspective,SUSYdemandsspecialattentionas
apotentialBSMtheory. ApartfromprescribingasolutionforthehierarchyproblemwhichSMsuffersfrom,
Email addresses: [email protected] (Sabyasachi Chakraborty), [email protected] (Joydeep Chakrabortty)
Preprint submitted toElsevier January 18, 2017
it can also explain neutrino masses-mixing and provide a suitable DM candidate. For example, the lightest
supersymmetricparticle(LSP)isanexcellentDMcandidateintheparadigmoftheminimalsupersymmetric
standard model (MSSM) with R-parity conservation. On the other hand MSSM with R-parity violation
(RPV) is anintrinsic SUSY way to generate neutrino masses andmixing both at the tree level as wellas at
the one-loop level [4]. However, if R-parity is broken, the LSP becomes unstable and hence fails to explain
the observedrelicdensity ofthe Universe. Butthere couldbe potentialdarkmattercandidates,inthe form
of a gravitino, axino, axion and keV sterile neutrino [5].
Concerningthe experimentalverificationofproposedmodels, unfortunately,the early13TeVrunofthe
LHChasnotfoundanysignalsinfavourofSUSY[6,7]. Althoughthenon-observationofsuperpartnersdoes
not invalidate the idea of SUSY, but it certainly questions the ability of MSSM to resolve the naturalness
problem. Forexample,LHChasalreadyruledoutgluinoslighterthan2TeVwhenthegluinoandLSPmasses
arewellseparated. Thisinturnmakesthe wholecolouredsectorheavydue to the logarithmicsensitivityto
theultra-violet(UV)scalethroughrenormalisationgroupevolutionequations. Interestinglythiscorrelation
is not generic and can be avoided within the models with R-symmetry and Dirac gauginos. One needs to
extendthe gaugesectorofMSSMto N =2representationtoconstructaDiracgauginomass. Thisrequires
a singlet S, a triplet T under SU(2) and an octet O under SU(3) living in the adjoint representation of
L C
the SM gaugegroup. These fields couple with bino, wino andgluinos respectively,to generateDirac masses
for the gabuginos. Thebpresence of additionaladjointbscalarscancelthe UV logarithmicdivergence,resulting
in a finite correction [8] only. Hence, the Dirac gluino masses can be easily made heavy. An immediate
consequence of having heavy Dirac gluinos is the suppression of the cross-sectionfor gluino pair or squark-
gluino associated pair productions. In addition, the pair production of squarks are also suppressed as it
requires chirality flipping Majorana gaugino masses in the propagator which these frameworks are devoid
of. This invariably weakens [9] the stringent upper bound on the first two generation squark masses. The
added feature of R-symmetry, apart from excluding Majorana masses for gauginos, is to prohibit trilinear
scalarcouplings(A-terms) andthe traditionalhiggsinomass term,i.e., the µ term. The absenceofA-terms
result in the suppression of the flavor and CP violating interactions [10].
Motivatedbythesenicefeatures,weproposeaSUSYscenarioenvelopedbyR-symmetry. Ourprimeaim
in this paper is to generate neutrino masses within the R-symmetric Dirac gaugino framework. In fact the
R-chargesarenowidentifiedwiththeleptonnumberoftheSMfields. Ingeneral,wehavethefreedomtocast
these charges and one can look into other assignments in [11–19]. An artifact of our R-charge assignment
is the presence of a bi-linear “R-parity violating” term in the superpotential which is indeed R-symmetric.
This is a very striking feature as it allows the sneutrino to acquire vacuum expectation value (VEV) when
R-symmetry gets broken. This leads to a mixing between light neutrinos and Dirac gauginos which in turn
helps to generate light neutrino masses at the tree-level. This RPV coupling will allow the gravitino to
decay to neutrino and photons. Such a gravitino is an excellent decaying dark matter candidate provided
its lifetime is greater than the age of the universe.
We categorize our paper as follows. To start with, we propose the model, and discuss the basic features
of this scenario. We emphasize on the specific choice of R-charges which lead to the emergence of bi-linear
“RPV” terms in the superpotential, which are R-symmetric. Then in the following section, sec. (2.2), we
discuss soft SUSY breaking, R-symmetry breaking terms, and also the generationof sneutrino VEV. In the
next section we explore the neutral fermion mass matrix. There we show, how successfully the neutrino
masses and mixing can be generated in a simplistic scenario. We discuss the more generic methods in ap-
pendix(7). Whileconsideringnormalaswellasinvertedhierarchies,wenotetheconstraintsontherelevant
superpotential parameters. As it happens, the gravitino is the order parameter of R-symmetry breaking.
We also explorethe possibility ofgravitinodarkmatter in our modelsatisfying necessaryconstraints. Alto-
gether, we come up with a conclusion that after successfully defending relevant constraints, our model can
describe the neutrino mass and mixing parameters. We conclude by providing a direction to adjudge this
scenario in collider experiments.
2
2. The Model
Our scenario is based on a SUSY framework with an added global R-symmetry where the gauginos are
Dirac type unlike the MSSM. Enjoying the freedom of R-charge assignments, we tag the chiral superfields
withveryspecificvaluesasdepictedintable1. Itisratherstraightforwardtoshowthatthescalarssharethe
Superfields SM rep U(1) Superfields SM rep U(1)
R R
H (1,2,1) 0 R (1,2, 1) 2
u d
−
H (1,2, 1) 0 R (1,2,1) 2
d u
−
Qb (3,2,1) 1 bS (1,1,0) 0
i 3
Ubc (¯3,1, 4) 1 bT (1,3,0) 0
i −3
Dbc (¯3,1,2) 1 b (8,1,0) 0
i 3 O
Lb (1,2, 1) 2 b
i
−
Ebc (1,1,2) 0 b
i
b
Table 1: The gauge quantumbnumbers under the SM gauge group SU(3)c SU(2)L U(1)Y as well as the U(1)R charge
× ×
assignmentsofthechiralsuperfieldsresidinginthemodel.
same R-charges with their corresponding superfields whereas for fermions they are one less. Similarly, the
gauginos have R-charge one and the corresponding gauge bosons have zero R-charge. The lepton number
of the SM particles are identified with their R-charges as shown in table 1. The lepton number of the
superpartners can be non-standard. The definition of our R-charge assignment can also be understood as,
our choice of R equals R +L [13] where R is the traditional R-charge assignment in MRSSM [20–23] 1
0 0
and L stands for the lepton number. As we know, an invariant action demands the superpotential to have
R-charge of two units. Hence, the allowed terms in superpotential are:
W = yijH Q Uc+yijH Q Dc+yijL H Ec+yiR H Ec,
MSSM u u i j d d i j e i d j r d d i
W = λuSH R +λuH TR +λdSH R +λdH TR +ξ SH L +η H TL ,
adj S b ub bd T bu bdb S bdbub T db bu b i u i i u i
W = ǫ H L ,
“bi-RPV” i u i
bb b b bb bb b b bb bb b b bb
W = µ H R +µ H R . (1)
µ u u d d d u
b b
The total superpotential ofbourbscenaribo ibs W = WMSSM +Wadj +W“bi-RPV” +Wµ. The triplet T under
SU(2) is parametrised as T = T(a), where T(a) = T σa/2, σa’s being the Pauli matrices and we
L a=1,2,3 a
also denote the components of the triplet field as T = T , T = (T iT )/√2 and T = (T +iTb )/√2.
H ,H ,L ,Ec,Q ,Uc,Dc arbe thPe prototybpical MSS3Mbfie0lds.+ λu,λ1d,−ξ ,η2,y ,y ,y a−re trilin1ear/Y2ukawa
u d i i i i i i i u d e
couplings and µ, ǫ are couplings with mass dimension. Notice that, the lack of a µH H term is a serious
u d
pbroblbembas bthebhiggbsinoblike chargino masses directly stem from this. The LEP experiment has ruled out
charginos below 100 GeV. Hence, to generate a µ term, it is mandatory to includebtwobadditional SU(2)-
doubletchiralsuperfieldsR andR carryingnon-zeroR-charges. The presenceofaR-chargeforthese two
u d
fields imply that R-symmetry cannot be broken in the visible sector spontaneously otherwise we have to
encounter massless R-axiobns. As abresult, these doublets are known as inert doublets in the literature.
WeliketodrawattentiontowardsW whichcontainsthebi-linear“RPV”butR-symmetricterms.
“bi-RPV”
ThesetermsareautomaticallyappearingasarelicofourR-chargeassignment. Itisworthytomentionthat
in our model, R-symmetry prohibits baryonic “RPV” terms in the superpotential and in the process the
stringent constraints from proton decay are taken care of. Before discussing the neutrino mass generation
we would like to first address soft SUSY breaking and R-symmetry breaking leading to the generation of
sneutrino VEV.
1WenoteinpassingthatR0 isonefortheleptonandquarksuperfields.
3
2.1. Soft (super-soft) SUSY breaking interactions
We choose to work in a scenario where SUSY (global) breaking is not associated with R-symmetry
breaking. This can be achieved through both D-and F-type spurions. For example, Dirac gaugino masses
canbe generatedwith the help ofa spurionsuperfieldW =λ +θ D whichappearsin the Lagrangianas
α′ ′α α ′
follows [24, 25]:
W
gaugino = d2θ α′ κ W S+κ Tr(W T)+κ Tr(W O) +h.c., (2)
LDirac Λ 1 1α 2 2α 3 3α
Z h i
where W and W have R-charge 1, i.e., R[λ b] = R[λ ] =b1 and R[D ] =b 0. The integration over
iα i′α iα ′iα ′
the Grassmann co-ordinates generate the Dirac gaugino masses MD κ D /Λ. Here, the i = 1,2,3 for
i ∼ ih ′i
U(1) ,SU(2) andSU(3) respectively,andΛrepresentsthemessengerscale. AfterthediscoveryofHiggs
Y L C
boson around 125 GeV it is important to address the status of Higgs mass within the given scenario along
with other scalar masses as well. In a purely supersoft scenario, the scalar masses are one-loop suppressed
as opposed to the gaugino masses. As an outcome, D-flatness is a natural direction [8] which leads to a
vanishinglysmalltreelevelquarticfortheHiggsfield. Thisisproblematicfromtheperspectiveoffittingthe
Higgsmassto125GeV.Thusinsteadofworkinginthegeneralisedsupersoftsupersymmetryframework[26],
we recall F-type breaking [13, 15]. In our model, the scalar masses can be generated through a F-type
spurion defined as X =x+θ2F which allows the following U(1) preserving operators [13]
X R
bd4θXΛ†2X Φ†iΦi+ HuHd+ǫΛS+S2+T2+ Λ1 ×cubic +h.c. ,
Z b bhXi n (cid:18) (cid:19) oi
X b b b b b b b
d2θ ST2+SO2+S3 +h.c., (3)
Λ
Z (cid:16) (cid:17)
and can automatically generbatbe thebfobllowibng U(1) preserving renormalizable soft SUSY breaking terms
R
1
Lsoft = m2iφ†iφi+ tSS+ 2bSS2+BµHuHd+.... . (4)
i (cid:20) (cid:21)
X
The soft mass squaredterms are proportionalto F 2/Λ2 M2 and we also consider same magnitude
h Xi ≡ SUSY
for the F- and D-type VEVs. In passing, we would like to note that such a mechanism also generates a
scalarsinglettadpolet S. However,aslongast <M3 ,suchatadpoleisnotexpectedtodestabilizethe
S S SUSY
hierarchy[27]. Nevertheless,due to the absence ofR-breakingtermsB H ℓ inthe scalarsector,sneutrinos
ǫ u i
cannotacquireVEVwhichisarelevantingredientforneutrinomassgeneration. Thishasbeenarequirement
for us to consider the possibility of R-symmetry breaking. e
2.2. R-symmetry breaking
It is well established that our Universe is associated with a vanishingly small vacuum energy or cosmo-
logical constant. To explain this from the perspective of spontaneously broken supergravity theory in the
hiddensector,thesuperpotentialneedstoacquireanon-zeroVEV.Sincethesuperpotentialcarriesnon-zero
R-charge,therefore, W =0 implies breaking ofR-symmetry. As a result, the gravitinowouldalsoacquire
h i6
amass. Here,theorderparameterofR-breakingisthegravitinomass. ThisR-breakinginformationiscom-
municated to the visible sector through anomaly mediation and in the process the following R-symmetry
breaking terms are generated
L✚R = M1bb+M2ww+M3gg
+ AuuRu∗LHu0+AddRd∗LHd0+AℓℓRℓ∗LHd0+h.c., (5)
ee ee ee
where the Majorana gaugino masses are generated through small R-breaking effects as
e e e e e e
g2
M = i b m (i=1,2,3), (6)
i 16π2 i 3/2
4
with beta functions
b =33/5, b =1, b = 3 . (7)
1 2 3
−
The small R symmetry-breaking trilinear scalar interactions are as follows
m 9
A = 3/2 g2 3g2+3Y2+4Y2 ,
τ 16π2 −5 1− 2 b τ
(cid:18) (cid:19)
m 13 16
A = 3/2 g2 3g2 g2+6Y2+Y2 ,
t 16π2 −15 1− 2− 3 3 t b
(cid:18) (cid:19)
m 7 16
A = 3/2 g2 3g2 g2+Y2+6Y2+Y2 . (8)
b 16π2 −15 1− 2− 3 3 t b τ
(cid:18) (cid:19)
It is alsoimportant to note that the naturalpresence of a conformalcompensator field, Σ=1+θ2m ,
3/2
invariably generates a Bǫ term in the superpotential through the following operator [28]:
i
= d2θ Σ3ǫ H L . (9)
i u i
L
Z
AfterscalingoutthiscompensatorfieldwithΦ =ΣΦ,whberbeΦisachiralsuperfield,wegenerateabi-linear
′
term (H ℓ ) in the scalar potential
u i
b b b
Bǫ =ǫ m . (10)
e i i 3/2
Hence, the Bǫ term is always aligned with the ǫ term. Such terms are R-breaking effects and proportional
to the gravitino mass. The presence of this small effect would invariably generate a tiny sneutrino VEV
which is appropriate for neutrino mass–mixing, as we discuss in the next section.
2.3. Sneutrino VEV
TocomputethesneutrinoVEV,onehastoincludethecontributionsfromF-,D-andsoftSUSYbreaking
terms. The additional pieces associated with SU(2) and U(1) in the D-terms are
L Y
D2a = g Hu†τaHu+ℓ†iτaℓi+T†λaT +√2 M2DTa+M2DTa† , (11)
(cid:16) (cid:17)
where τa and λa’s represent the SU(2) generators in the fundame(cid:0)ntal and adjoint re(cid:1)presentations respec-
e e
tively. Similarly, the weak hyper-charge contribution D is given by
Y
g
DY = 2′ Hu†Hu−ℓ†iℓi +√2M1D S+S† . (12)
(cid:16) (cid:17)
The tree level potential terms which participate in the sneutrino fiel(cid:0)d minim(cid:1)isation equations are
ee
V = ǫ 2 ν0 2,
F | i| | i|
V = m2 ν0 2+Bǫ H0ν ,
soft | i| i u i
(g2e+g2)
V = ′ ν0 2 √2g MDv +√2gMDv ν0 2. (13)
D e e8 | i| −e ′ 1 S 2 T | i|
(cid:20) (cid:21)
Inthe limit ofvanishing VEVofthe singlet(Se)andthe triplet(T),i.e., vS,vT e 0,the sneutrino VEVcan
→
be approximated as
Bǫ v
i u
ν = . (14)
h ii −m2+ǫ2
i i
Such a choice of the singlet and the triplet VEVs also ensure that these fields are very heavy. Assuming
e
H0 = v m i.e., at the electroweak scale, we find eν Bǫ /m ǫ m /m ǫ m 2. In the next
h ui u ∼ i h ii ∼ i ∼ i 3/2 ≡ i 3/2
section, we investigate the neutral fermion sector which is of main importance of this analysis.
e e e e e
2Offcourse,inthesamemannerthe inertscalars(Ru,Rd)wouldalsoacquireaVEVduetoR-symmetrybreakingandas
aresultwouldmixwiththeHiggsfields. However,thatmixingisalsosuppressedbytheR-breakingparametergravitinomass
anddoesnotplayanyimportantroleinthephenomenological description.
5
3. The neutral fermion sector
The Lagrangian corresponding to the neutral fermion sector after R-symmetry breaking contain the
following terms
Lf0 = M1DbS+M2Dw0T +µuHu0Rd0+µdHd0Ru0 +λuSvuSRd0−λdSvdSRu0 +λuTvuT0Rd0+λdTvdRu0T0
g v g v gv gv
+ M bb+M w0w0+M SS+M T0T0+ ′ ubH0 ′ dbH0 uw0H0+ dw0H0
1 ee 2 e e S e e T e √2 ue−e √2 de−e √2 ue e√2 de e
g v gv
′ eiebν +ξ ve Seν + iewe0ν +ηev Te0ν +(ǫe+eξ v +ηeve)H0ν . e e e e (15)
− √2 i i u i √2 i i u i i i S i T u i
Here, i stands foreee, µ andeτ respectiveeley. e e
The neutral fermion mass matrix in the basis f0 =(ν ,b,S,w0,T0,H0,R0,H0,R0) has the schematic form
i u d d u
1
mass = (fe0)eTMe ef0.e e e e (16)
Lf0 2 N
where
M = mfe0 mD , (17)
N mT 0
(cid:18) D (cid:19)
with
M MD 0 0 g′vu 0 g′vd 0
1 1 √2 − √2
MD M 0 0 ξ v λuv 0 λdv
 1 S i i S u S d 
0 0 M MD gvu 0 gvd 0
 2 2 −√2 √2 
mfe0 =  g′0vu ξ0v Mg2vDu ηMvT ηi0vi λuTµvu 00 λdT0vd , (18)
 √2 i i −√2 i i 
 0 λuv 0 λuv µ 0 0 0 
 S u T u 
 g′vd 0 gvd 0 0 0 0 µ 
 − √2 √2 
 0 λdv 0 λdv 0 0 µ 0 
 − S d T d 
 
and
g′v1 g′v2 g′v3
− √2 − √2 − √2
ξ v ξ v ξ v
 1 u 2 u 3 u 
gv1 gv2 gv3
 √2 √2 √2 
m =  η1vu η2vu η3vu . (19)
D  
 (ǫ1+ξ1vS +η1vT) (ǫ2+ξ2vS +η2vT) (ǫ3+ξ3vS +η3vT) 
 
 0 0 0 
 
 0 0 0 
 
 0 0 0 
 
 
Here, H0 = v and H0 = v , with the constraint that v = v2 +v2 and tanβ = v /v . We also
h ui u h di d u d u d
consider µ µ µ, and M M M M m /16π2. We note that an order one value
u ≡ d ≡ 1 ∼ 2 ∼ S ∼ T ∼ 3/2 p
of the superpotential coupling λu provides substantial one-loop correction to the up-type Higgs boson
S,T
mass m2 [29, 30]. These corrections are ‘stop-like’ and an 125 GeV Higgs boson can be obtained without
Hu
requiring too heavy top squarks. In the next section, to carry on our proposed analysis, we adopt a
simplified framework after integrating out some suitably chosen heavy fields. This helps us to understand
the phenomenologicalaspects in detail.
6
3.1. Neutrino mass and mixing
In the limit when both the wino and the higgsinos are heavy and effectively decoupled, the neutral
fermion mass matrix can be written in the basis (ν ,b,S) 3. We neglect the bino and singlino Majorana
i
masses which are proportional to m /16π2 as their inclusion does not alter our conclusion. The neutral
3/2
fermion mass matrix in such a limit can be depicted aes e
0 g′ ǫ m ξ v
−√2 i 3/2 i u
MN =  −√g′2ǫim3/2 0 MD . (20)
ξ v Me 0
 i u D 
 e 
In order to diagonalise this effective matrix we adopt the methodology of [31] and choose ξ =ξu, ǫ =ǫv.
i i
This results in
ξeiθ/2 e e
ξ = 1+ρ U + 1 ρ U ,
i √2 i3(i2) − i2(i1)
ǫe iθ/2hp p i
−
ǫ = 1+ρ U 1 ρ U , (21)
i √2 i∗3(i2)− − i∗2(i1)
e hp p i
whereρ= √√11++rr−+√√rr withr ≡e|∆∆mm1223|22. Theindices(intheparenthesis)denoteelementsofthePMNSmatrix
| |
for normal (inverted) hierarchy. Using this approachwe find out the suitable parameters for both cases.
3.1.1. Normal hierarchy
For simplicity, we choose the lightest neutrino mass (m ) to be zero. The other mass eigenvalues can be
1
readily obtained as m = g ǫm ξ(1 ρ)/√2 and m = g ǫm ξ(1+ρ)/√2. With the help of present
2 ′ 3/2 3 ′ 3/2
−
oscillation parameters [1] we find
e ǫ ξ m 10 10 GeVe. (22)
3/2 −
∼
Using eq. (21), the parameters should have the following structures to be consistent with the neutrino
e
oscillation data
0.487 0.1
ξ ǫ −
ξ = 1.165 , ǫ = 0.55 . (23)
i i
√2  √2 
0.635 1.29
e
  e  
Forsimplicity weassumethe CP violatingDiracphaseto be zeroandallthe R-conservingmasses(M ,v )
D u
to be at the electroweak scale.
3.1.2. Inverted hierarchy
Incaseofinvertedhierarchywefollowthesameprescriptionandfindm =0,m =g ǫm ξ(1+ρ)/√2
3 2 ′ 3/2
and m =g ǫm ξ(1 ρ)/√2. Again, using the central value of the oscillation parameters we obtain
1 ′ 3/2
−
e
ǫ m ξ 4 10 10 GeV. (24)
e 3/2 ∼ × −
Thisalsoresultsinasimilarexpressionobtainedineq.(22). Themodelparametersintermsoftheneutrino
e
oscillation parameters in the absence of any CP violating Dirac phase becomes
1.37 0.27
ξ ǫ −
ξ = 0.0264 ǫ = 0.922 . (25)
i i
√2  √2 
0.349 1.036
− e −
  e  
3Inref.[19]asimilarformoftheneutralfermionmassmatrixwasobtainedbutwithdifferentR-chargeassignment. Inthat
case, sneutrino VEV can be large and sneutrino can play the role of a down-type Higgs field. This feature is vastly different
fromour scenariowherethesneutrino VEVissmallbecause itisanR-breaking effect. Assortedphenomenology and collider
implicationsofthesetwoscenariosarealsocontrasting.
7
In this section we use the linear see-saw simplification where the (M )12 element is either much larger
N
or smaller compared to the (M )13 element in eq. (20). The latter limit also solves a major drawback of
N
prototypicalMSSM-RPV scenariowhere ǫ has to be very smallin order to fit neutrino massesand mixing.
i
Apart from the tree level contribution there are one-loop corrections to the neutrino masses. There
are two such dominant contributions from the BB- and the µB-loops [32], which in the decoupling regime
(m m and sin2(α β) 1) can be written in simplified forms as
A H
≈ − ≈
(m )BB g2 cos2(α−β) ǫiǫjm33/2 ,
ν ij ≃ 4(16π2)2 (cid:20) cos2β (cid:21)" m2 #
ee
(m )µB g2 cos(α−β) ǫiǫjme23/2 . (26)
ν ij ≃ 4(16π2)2 (cid:20) cosβ (cid:21)" m #
ee
Thus even for ǫ˜i’s (1) with gravitino mass around few hundred keVe, these loop contributions turn out
∼ O
to be negligible, i.e., m (10 13) GeV.
ν −
∼O
4. Gravitino as dark matter
Inorderto checkthe viabilityofgravitinoDM, firstwewilldiscuss the productionofgravitinoandthen
its possible decay mode to adjudge whether the decay life-time of the gravitino is large enough to be a DM
candidateornot. Thenwewillalsodiscussthe constraintsonphotonflux emittingoutofagravitinodecay.
4.1. Production of gravitinos
Inourscenario,gravitinosareproducedinthethermalplasmabyinteractingwithotherSUSYparticles,
and their thermal relic density can be written as [33, 34]
Ω h2 0.1 1 GeV TR meg 2, (27)
3/2 ∼ m 107 GeV 2 TeV
(cid:18) 3/2 (cid:19)(cid:18) (cid:19)(cid:16) (cid:17)
where Ω3/2h2 0.1199 [2]. The thermal production of gravitinos would require TR > meg 2 TeV 4
∼ ∼
otherwise the production of gravitinos would be exponentially (Boltzmann) suppressed [35]. Even though
this outcome heavily depends on the exact SUSY spectrum, the constraint on the reheating temperature
can be roughly satisfied with
m >200 keV. (28)
3/2
Hence,the gravitinowouldconstitute a coldDM [36]. If onerequiresmore lightergravitinos,the reheating
temperaturewouldnaturallybeloweredthanthepresentconsiderationandthermalproductionofgravitinos
becomes questionable. Nevertheless, it is always probable to have a late time entropy creation from the
decay of other relics which can washout the gravitino density [37], and may relax the stringent constraint
on the gravitino mass. Such low reheating temperature is also troublesome from the perspective of thermal
leptogenesis[38]. Nevertheless,ifthereheatingtemperatureisabovetheelectroweakphasetransition,itcan
leave room for electroweak baryogenesis [39–42]. Another viable option for baryogenesis is the Afleck-Dine
mechanism [43].
4In our framework, gluinos arevery heavy compared to the electro-weak gauginos. Thus a pair of gluinos would decay to
qq¯χe01,whereq’sarethefirsttwogenerationquarks. Forthiskindofscenario,LHCprovidesstringentconstraintsonthegluino
masswhichiscloseto2TeV[6,7].
8
4.2. Gravitino decay width and life-time
Broken R-parity implies that the gravitino would most certainly decay. A light gravitino with mass
(1 100) keV will dominantly decay to a neutrino and a photon. This decay width of gravitino is well
O −
approximated as [44]
1 m3
Γ(G ν γ) U 2 3/2, (29)
→ i ≃ 32π| γν| M2
P
where U 2 = cosθ Z +sienθ Z 2, and M 2.4 1018 GeV is the reduced Planck mass.
| γν| a=i+4| W a1 W a2| P ∼ ×
Inadecoupledwino-higgsinoscenario, U 2 =cos2θ N 2,thephysicallightneutrinostatecan
P | γν| W a=i+4| a1|
be approximated as
P
ν = |νii + −−g√′eǫi2mme3/2 |bi + ξMivDu |Si ,(30)
m
| i 1+ ξvu 2+ g′vi 2 1+ ξvu 2+ eg′vi 2 1+ ξvu 2+e g′vi 2
r MD √2MD r MD √2MD r MD √2MD
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
wherem=1,2,3. Usingtheabovebasis,thegravitinodecaywidthgivenineq.(29)canbefurthersimplified
as
3 g v 2 m3
Γ(G ν γ) ′ i 3/2 cos2θ , (31)
→ i ≃ 32π √2M M2 W
(cid:20) D(cid:21) P
which in turn leads to the lifetimee of gravitino decay
5 1019
τ × sec. (32)
3/2 ≃ ǫ2m5
3/2
Now from this analysis it is obvious that this decay lifetime is much larger than the age of the Universe
e
(4.32 1017 sec) unless the gravitino is heavy ( (10 GeV)) and ǫ (1), unlike our choice, and this
× ≥ O ∼ O
assures gravitino to be an excellent DM candidate.
e
4.3. Photon flux from gravitino decays
As we have noted, the dominant decay mode of gravitino contains photon in final state, the constraints
from the abundance of the photon production must be adjudged. This photon flux would most definitely
show a peak at the energy E =m /2 and the maximum flux associated with it [44] can be deduced as
γ 3/2
dF
F = E
γ, max γ
dEγdΩ(cid:12)(cid:12)Eγ=m32/2
(cid:12)
2.89 10 (cid:12)(cid:12)14 ǫ2 m3/2 2 Ω3/2 h (cm2.str.sec) 1. (33)
≃ × − i 200 keV 0.3 0.7 −
h i (cid:20) (cid:21)(cid:20) (cid:21)
Even for ǫ (1), i.e., ǫ m, which is thee maximum value that we have accounted for, the flux is well
∼ O ∼
within the bounds as analysed in [45].
e e
5. Collider Phenomenology
In our model, as an artifact of R-symmetry breaking gravitino is the LSP. With suitable choice of
parameters, we can choose a valid SUSY spectrum with lightest neutralino to be the next-to-minimal
supersymmetric particle (NLSP).
Thus we would be left with a scenario where all the supersymmetric particles decay to the lightest
neutralinowhichdecaystoagravitinoaccompaniedbyeitherphotonorZ-bosonorHiggs. Sincesuchinter-
actions are suppressed by the Planck scale, the resulting decay width will be very small, i.e., corresponding
9
lifetime is too large to decay within the collider. In addition, the NLSP also undergoes R-parity violating
decay modes primarily in the following channel
χ0 h ν , γ ν . (34)
1,2 → i i
The dominant decay width is noted down as [46]
e
2 2
Γ(χ0 hν) = αmχe0i ξicosα + g′ ǫ m3/2 N N 1 m2h , (35)
i → 16sin2θW (cid:12)(cid:12)(cid:12)( √2 √2e m ) i3 11(cid:12)(cid:12)(cid:12) " − m2χe0i#
In our case, the deominant decay mode of thi(cid:12)s light neutralino is to a neutrino(cid:12)and a Higgs boson leading to
(cid:12) e (cid:12)
an interesting di-Higgs signature at the colliders [47]. Charged lepton flavour violation constraints [48] put
a limit on the parameter ξ (10 2) or less [19]. This implies the decay length would be around 0.1 mm
−
∼O
to a few mm. As an example, we show two representative points in the parameter space. (i) ǫ, ξ 10 3
−
∼
andm 100keVand (ii) ǫ 1, m 0.1 GeV andξ 10 9. Such differentset ofparametersprovide
3/2 3/2 −
∼ ∼ ∼ ∼
distinct signatures at the colliders. The later consideration also solves the ‘small ǫ’ issue in traditional
e
bi-linear “RPV” scenarios. Aemeasurable decay gap is a distinctive feature of this scenario. In the wino-
higgsinodecoupledscenariothe lightestneutralino is primarilyanadmixture ofthe bino-singlinostatesand
hencethedecaymodestoZν andW ℓ whicharehighlysuppressed. Onecaneasilyhavelighthiggsinosin
± ∓
this framework without much modifications in the neutrino sector and would also have interesting collider
implications [49].
InpresenceofsmallR-breakingparameter,asmallmasssplittingbetweentheneutralinosandsneutrinos
are induced due to the pseudo-Dirac nature of neutralinos and mixing of sneutrinos with neutral Higgs
bosons [50] respectively. Thus we can have neutralino oscillations at the LHC [51] which can be identified
in the distribution of the displaced vertex lengths. There is also a possibility of having induced sneutrino-
antisneutrino oscillations [52] which are an excellent probe for the lepton number violation. The bi-linear
“RPV”violatingbutR-symmetrypreservingterminthe superpotentialwiththe sneutrinoVEVwouldalso
lead to trilinear couplings involving lepton (quark) and slepton (squark) fields which are also an interesting
channelto lookfor[53]. Allthese togethermakeourscenariophenomenologicallyrichandexplorableatthe
LHC.
6. Conclusion
Neutrinos have always been the elusive key to understand the physics beyond the Standard Model. We
believe the lack of our definiteness regarding neutrino mass generation mechanism is one of the hurdle to
propelthe searchfor BSM physics. Keeping truston theoretical constructionand motivations of supersym-
metry, in this paper we come up with an R-symmetric SUSY scenario where gauginos are Dirac type and
thereisa naturalemergenceofbi-linear“R-parityviolation”aswell. While constructingthe superpotential
withinourframeworkwenotethatthereareR-symmetryconservingtermswhichareinterestingly“R-parity
violating”. This is indeed an interesting outcome of our scenario. Our prime aim in this paper is to explain
how active light neutrino masses and mixing can be generated. In the process we also discuss one of the
important issue: the generation of a Higgs mass around 125 GeV. Then we motivate the requirement of
R-symmetry breaking and that also through the conformalcompensator field. These R-symmetry breaking
terms are instrumental to generate tiny VEV of sneutrinos which is suitable for neutrino mass and mixing
generation. Thoughwe havethe full neutralfermionmassmatrix,but forthe sakeofneutrino massgenera-
tionweworkwithwino-higgsinodecouplinglimitwithoutlossofgenerality. Thenwededucetheconstraints
onrelevantsuperpotentialparametersforboth normalandinvertedhierarchies. After successfulgeneration
ofneutrino massesusing treeleveloperators,wealsoshowthatthe loopinduced contributionsaresmallfor
our choice of parameter space and can be ignored. In our scenario gravitino is the LSP and a possible dark
matter candidate. We explore this possibility by considering the production and decays of gravitino. These
lead to the constraints on masses and some couplings relevant for neutrino masses as well. We then also
take care of the photon flux constraint. It is worthy to mention that all these constraints are compatible
10