Eu . Phys. J. C (2012) 72:1859
DOI 10.1140/epjc/s10052-012-1859-7
Regula A icle - Theo e ical Physics
Neu ino mixing and masses in a le – igh model
wi h mi o e mions
R. Gai án2,a, A. He nández-Galeana1, J.M. Ri e a-Rebolledo1, P. Fe nández de Có doba3
1Depa amen o de Física, Escuela Supe io de Física y Ma emá ica, I.P.N., U.P. Adol o L. Ma eos, México D.F. 07738, Mexico
2Cen o de In es igaciones Teó icas, FES, UNAM, Apa ado Pos al 142, Cua i lán-Izcalli, México 54700, Mexico
3Ins i u o de Ma emá ica Pu a y Aplicada, Uni e sidad Poli écnica de Valencia, Valencia 46022, Spain
Recei ed: 1 July 2011 / Re ised: 18 No embe 2011 / Published online: 31 Janua y 2012
© Sp inge -Ve lag / Socie à I aliana di Fisica 2012
Abs ac In he amewo k o a le – igh model con ain-
ing mi o e mions wi h gauge g oup SU(3)C⊗SU(2)L⊗
SU(2)R⊗U(1)Y, we es ima e he neu ino masses, which
a e ound o be consis en wi h hei expe imen al bounds
and hie a chy. We e alua e he decay a es o he Lep on
Fla o Viola ion (LFV) p ocesses μ→eγ ,τ→μγ and
τ→eγ . We ob ain uppe limi s o he la o -changing
b anching a ios in ag eemen wi h hei p esen expe imen-
al bounds. We also es ima e he decay a es o hea y Ma-
jo ana neu inos in he channels N→W±l∓,N→Zνland
N→Hνl, which a e oughly equal o la ge alues o he
hea y neu ino mass. S a ing om he mos gene al Majo-
ana neu ino mass ma ix, he smallness o ac i e neu ino
masses u ns ou om he in e play o he hie a chy o he
in ol ed scales and he double applica ion o seesaw mech-
anism. An app op ia e pa ame e iza ion on he s uc u e o
he neu ino mass ma ix imposing a symme ic mixing o
elec on neu ino wi h muon and au neu inos leads o i-
bimaximal mixing ma ix o ligh neu inos.
1 In oduc ion
The e idence o neu ino oscilla ions ob ained in expe i-
men al esul s om a mosphe ic, sola , eac o and accele -
a o neu inos leads one o conclude ha he neu inos ha e
a mass di e en om ze o. The cu en neu ino expe imen-
al da a (Supe Kamiokande, SNO, Kamland, K2K, GNO,
CHOOZ) can be desc ibed by neu ino oscilla ions ia h ee
neu ino mixings [1–7]. The p esen da a gi e he sola neu-
ino lep on mixing angle an2θ12 =0.45 ±0.05, he a -
mosphe ic angle sin22θ23 =1.02 ±0.04 and sin22θ13 =
In e disciplina y Modeling G oup, In e Tech.
ae-mail: gai an@se ido .unam.mx
0±0.05 [8–10]. The complex phase has no ye been mea-
su ed.
The expe imen al in o ma ion on neu ino masses and
mixing poin s ou new physics beyond he S anda d Model
(SM) o pa icle physics, wi h a g ea ac i i y on he conse-
quences. Among he possible mechanisms o neu ino mass
gene a ion, he mos simple and a ac i e one is he see-
saw mechanism [11–20], which explains he smallness o
he obse ed ligh neu ino masses h ough he exchange o
supe hea y pa icles; an al e na i e explana ion is gi en by
ex a dimensions beyond he usual h ee ones [21]. I has
been sugges ed ha igh -handed (RH) neu inos expe ience
one o mo e o hese ex a dimensions, such ha hey only
spend pa o hei ime in ou wo ld, wi h appa en ly small
masses. A he p esen , i is no known whe he neu inos a e
Di ac o Majo ana e mions.
Models wi h hea y neu inos o mass o o de 1 TeV
can gi e ise o signi ican ligh -hea y mixing and de i-
a ion om uni a i y o he Pon eco o–Maki–Nakagawa–
Saka a (PMNS) ma ix [22–24]. The nonuni a i y na u e o
he neu ino mixing ma ix due o mixing wi h ields hea ie
han MZ
2can mani es in ee-le el p ocesses like π→μν,
Z→¯νν,W→lν o in cha ged lep on decays μ→eγ ,
τ→μγ , e c. which a e la o iola ing and a e and p o-
ceed a one loop le el [22–26]. The TeV scale seesaw mod-
els a e in e es ing because hey can ha e signa u es in he
CERN La ge Had on Collide (LHC) in he nea u u e [27–
29].
Neu inos also a e impo an in as ophysics and cosmol-
ogy [30] and p obably hey con ibu e o ho da k ma e in
he Uni e se and in i s e olu ion.
Pa i y P iola ion was one o he g ea es disco e ies o
pa icle physics [31–33]. Be o e his obse a ion, acco ding
o Fe mi’s hypo hesis i was belie ed ha weak in e ac ions
ha e pu ely ec o ial V o axial ec o ial (V–A) pa i y con-
se ing Lo en z s uc u e [34,35]. The heo y o Lee and
Page 2 o 8 Eu . Phys. J. C (2012) 72:1859
Yang in 1956 [36] p oposed a e mion cu en wi h V and
A s uc u e. I is known ha in he s anda d model (SM)
he elec oweak in e ac ions ha e a V–A o m, wi h only
le -handed (LH) (o dina y) e mions coupling o he weak
gauge boson W±. Bu one can include also mi o e mions
[37] wi h a V +A coupling, such ha P is conse ed. In his
sense, he e m “mi o e mion” is equi alen o “ ec o -
like e mion”, whe e o a heo y wi h gauge g oup G,ina
ep esen a ion Rone has se s o LH and RH e mions.
In he li e a u e a second meaning o ha e m is used.
Gis ex ended o a G×Ggauge heo y, and o e e y mul-
iple (R,1)a mi o pa ne (1,R)is added, such ha he e
is no gauge in a ian mass e m connec ing he LH and RH
mul iple s [38–41]. Thus i is na u al o conside he exis-
ence o mi o gene a ions.
Masses o mi o pa icles a ise om symme y b eaking;
o mi o gene a ion hey may lye below one TeV, and easi-
ble o be disco e ed in Fe milab Te a on Collide and LHC.
A solu ion o he s ong CP p oblem has been p oposed
wi hin a L–R symme ic con ex [46,47]. The elec oweak
g oup is ex ended o SU(2)L⊗SU(2)R⊗U(1)including
mi o e mions. These e mions a e conjuga ed o he o di-
na y ones wi h espec o he gauge symme y g oup such
ha a e mion ep esen a ion including bo h o hem is eal
and he cancela ion o anomalies is au oma ic [48].
In his pape we conside a L–R model wi h mi -
o e mions (LRMM) wi h gauge g oup G≡SU(3)C⊗
SU(2)L⊗SU(2)R⊗U(1)Y. We discuss in Sec . 2 he
o malism o mixing be ween s anda d and new exo ic
e mions In Sec . 3we p esen he model and discuss he
symme y b eaking p ocess wi h wo scala double s.
In Sec . 4we w i e he gauge in a ian Yukawa couplings
which a e spon aneous symme y b eaking gi e he mos
gene al Majo ana neu ino mass ma ix. Wi h a double ap-
plica ion o he ype I seesaw app oxima ion we es ima e
he ligh neu ino masses in e ms o ee Yukawa couplings
assuming ex u es o he ligh and mi o ma ices, ob ain-
ing consis en no mal hie a chical alues o masses and
a i-bimaximal mixing o ligh neu inos. We discuss in
Sec . 4 he mixing be ween s anda d and mi o e mions. In
Sec . 5we include he adia i e decays μ→eγ ,τ→μγ
and τ→eγ and es ima e bounds o hei b anching a ios.
Finally, we calcula e such a ios o he hea y Majo ana neu-
inos decays N→W+l−,N→Zνland N→Hνl, ge ing
a smoo h a ia ion wi h he hea y neu ino mass, e en when
i is much la ge han any o he in ol ed masses.
2 Fe mion mixing and la o iola ion
To conside he mixing o e mions, we shall ollow [22–
24], g ouping all e mions o elec ic cha ge qand helici y
a=L,R in o na+ma ec o column o nao dina y (o) and
maexo ic (e) gauge eigens a es, i.e. ψo
a=(ψo
na,ψo
me)T
a.The
o dina y e mions include he SM ones, whe eas he exo ics
include any new e mion wi h sequen ial (mi o o single )
p ope ies beyond he SM.
The ela ion be ween he gauge eigens a es and he co -
esponding ligh (l) and hea y (h) cha ged mass eigens a es
ψa=(ψl,ψh)T
a,a=L,R is gi en by he ans o ma ion
ψ0
a=Vaψa,a=L,R, (1)
whe e
Va=AaEa
FaGa.(2)
In (2), Aais a ma ix ela ing he o dina y weak s a es and
he ligh -mass eigens a es, while Ga ela es he exo ic and
hea y s a es. Eaand Fadesc ibe he mixing be ween he
wo sec o s.
F om he uni a y o V
VaV+
a=1,a=L,R (3)
i ollows ha he subma ix Aais no uni a y. The e m
F+
aFa, which is second o de in he small ligh –hea y
e mion mixing, will induce la o -changing ansi ions in
he ligh –ligh sec o .
The acuum expec a ion alues (VEV) o he neu al
scala s p oduce he SM e mion mass e ms, which oge he
wi h he exo ic mass and mixing ma ices lead o he mass
ma ix Mwhich akes he o m
M=Kˆμ
μˆ
K,(4)
whe e Kdeno es he SM e mion mass ma ix and ˆ
Kco -
esponds o he e mion mass ma ices associa ed wi h he
exo ic sec o , while μ,ˆμco espond o he mixing e ms
be ween o dina y and exo ic e mions.
The diagonal mass ma ix Mdcan be ob ained h ough a
biuni a y o a ion ac ing on he Land Rsec o s, namely
Md=V+
LMVR=ml0
0Mh,(5)
whe e ml,mhdeno e he ligh and hea y diagonal mass ma-
ices, espec i ely. The o m o he mass ma ix will depend
on he ype o exo ic e mion conside ed.
The scala - e mion couplings wi hin some speci ic Higgs
sec o a e no diagonal in gene al, and one can see ha he
couplings a e no diagonal in gene al; hus new phenomena
associa ed wi h la o -changing neu al cu en s (FCNC)
will be p esen in such model.
Eu . Phys. J. C (2012) 72:1859 Page 3 o 8
3 The model
In his and nex sec ions we ollow closely [42–45]. The
LRMM o mula ion is based on he gauge g oup SU(2)L⊗
SU(2)R⊗U(1)Y. In o de o sol e di e en p oblems such
as he hie a chy o qua k and lep on masses o he s ong CP
p oblem, di e en au ho s ha e enla ged he e mion con en
o he o m
l0
iL =ν0
i
e0
iL
,e
0
iR,ν
0
iR,
ˆ
l0
iR =ˆν0
i
ˆe0
iR
,ˆe0
iL ,ˆν0
iL,
Q0
iL =u0
i
d0
iL
,u
0
iR,d
0
iR,
ˆ
Q0
iR =ˆu0
i
ˆ
d0
iR
,ˆu0
iL,ˆ
d0
iL,
(6)
whe e he index i uns o e he h ee e mion amilies and
he supe sc ip s 0deno e gauge eigens a es. The quan um
numbe s o hese e mions unde he gauge g oup Gde ined
abo e a e gi en by
l0
iL ∼(1,2,1,−1)iL,ν
0
iR ∼(1,1,1,0)iR,
e0
iR ∼(1,1,1,−2)iR,ˆν0
iL ∼(1,1,1,0)iL,
ˆe0
iL ∼(1,1,1,−2)iL,ˆ
l0
iR ∼(1,1,2,−1)iR
u0
iR ∼3,1,1,4
3iR
,d
0
iR ∼3,1,1,2
3iR
ˆu0
iL ∼3,1,1,4
3iL
,ˆ
d0
iL ∼3,1,1,2
3iL
Q0
iL ∼3,2,1,1
3iL
,ˆ
Q0
iR ∼3,1,2,1
3iR
,
espec i ely, and he las en y co esponds o he hype -
cha ge (Y) wi h he elec ic cha ge de ined as Q=T3L+
T3R+Y
2.
A model wi h gauge g oup SU(2)L×SU(2)R×U(1)V×
SU(3)Hand he e mion con en (6) was o iginally sug-
ges ed in Z.G. Be ezhiani [49] as he “uni e sal seesaw”
model which gene a ed masses o cha ged e mions as well
as o he neu inos. He also wo ked on a SU(5)×SU(3)H
model o ex ension o SO(10)o Pa i–Salam [50,51], p e-
dic ing o ins ance mνe=O(10)eV. A low (elec oweak
scale) ene gies he model simula es he s anda d SU(3)C×
SU(2)L×U(1)Ymodel, and FCNC a e supp essed na u-
ally.
3.1 Symme y b eaking
The “Spon aneous Symme y B eaking” (SSB) is achie ed
ollowing he s ages:
G−→ GSM −→ SU(3)C⊗U(1)Q,(7)
whe e GSM =SU(3)C⊗SU(2)L⊗U(1)Yis he “S anda d
Model” g oup symme y, and Y
2=T3R+Y
2. The Higgs sec-
o o induce he SSB in (7) in ol es wo double s o scala
ields:
Φ=(1,2,1,1), ˆ
Φ=(1,1,2,1), (8)
whe e he en ies co espond o he ans o ma ion p ope -
ies unde he symme ies o he g oup G, wi h he “Vacuum
Expec a ion Values” (VEV’s)
Φ= 1
√20
,ˆ
Φ=1
√20
ˆ .(9)
The mos gene al po en ial ha de elops his pa e n o
VEV’s is
V=−μΦ†Φ+ˆμˆ
Φ†ˆ
Φ+λ1
2Φ†Φ2+ˆ
Φ†ˆ
Φ2
+λ2Φ†Φˆ
Φ†ˆ
Φ.(10)
In he las exp ession he e ms wi h μ,ˆμa e included
so ha he pa i y symme y (P)is b oken so ly, i.e., only
h ough he dimension- wo mass e ms o Higgs po en ial.
The scala Lag angian o he model is w i en as
Lsc =(DμΦ)+(DμΦ)+ˆ
Dμˆ
Φ+ˆ
Dμˆ
Φ,(11)
whe e Dμand ˆ
Dμa e he co a ian de i a i es o he SM
and he mi o pa s, espec i ely. The gauge in e ac ions o
qua ks and lep ons can be ob ained om he Lag angian
Lin =¯
ψiγμDμψ+¯
ˆ
ψiγμˆ
Dμˆ
ψ. (12)
The VEV’s and ˆ a e ela ed o he masses o he
cha ged gauge bosons Wand ˆ
Wby MW=1
2gL and
Mˆ
W=1
2gRˆ , whe e gLand gRa e he coupling cons an s o
SU(2)Land SU(2)R, and gL=gRi we equi e L–R sym-
me y.
4 Gene ic Majo ana neu ino mass ma ix
Wi h he ields o e mions in oduced in he model, we may
w i e he gauge in a ian Yukawa couplings o he neu al
sec o :1
hij ¯
ˆνiLνjR +λij ¯
liL ˜
ΦνjR +ηij ¯
ˆ
liR ˜
ˆ
ΦˆνjL
1To simpli y no a ion we d op he “0” supe sc ip .
Page 4 o 8 Eu . Phys. J. C (2012) 72:1859
+ˆ
Mij ¯
ˆνiL(ˆνjL)c+σij ¯
liL(ˆνjL)c˜
Φ
+χij ¯νiR(νjR)c+πij ¯
ˆ
liR(νjR)c˜
ˆ
Φ+h.c.,(13)
whe e i,j =1,2,3, ˜
Φ=iσ2Φ∗,˜
ˆ
Φ=iσ2ˆ
Φ∗,hij ,ˆ
Mij ,χij
ha e dimensions o mass, and σij ,ηij ,λij and πij a e di-
mensionless Yukawa coupling cons an s. When Φand ˆ
Φ
acqui e VEV’s we ge he neu ino mass e ms
hij ¯
ˆνiLνjR +
√2λij ¯νiLνjR +ˆ
√2ηij ¯
ˆνiR ˆνjL
+ˆ
Mij ¯
ˆνiLˆνjLc+
√2σij ¯νiLˆνjLc
+χij ¯νiR(νjR)c+ˆ
√2πij ¯
ˆνiR(νjR)c+h.c.,(14)
which a e w i en in he gene ic Majo ana ma ix o m
ΨνL, Ψ cνLMLMD
MT
DMR(Ψ c
ν)R
(Ψν)R(15)
whe e
(Ψν)L,R =νi
ˆνiL,R
,Ψc
νL,R =(νc
i)
(ˆνc
i)L,R
,(16)
ML=0
√2σ
√2σTˆ
M,M
R=χˆ
√2π
ˆ
√2πT0,
(17)
MD=
√2λ0
hˆ
√2η,(18)
wi h h,ˆ
M,χ,σ,η,λand πunknown ma ices o 3 ×3
dimension. By assuming he na u al hie a chy |(ML)ij |
|(MD)ij ||(MR)ij | o he mass e ms, he mass ma ix in
(15) can app oxima ely be diagonalized, yielding
ΨνL, Ψ cνLMν0
0MR(Ψ c
ν)R
(Ψ ν)R,(19)
whe e, neglec ing O(MDM−1
R) e ms, we may w i e in good
app oxima ion [52]ΨνL,R ≈ΨνL,R, and ΨcνL,R ≈Ψc
νL,R.
The Majo ana mass ma ix o he le -handed neu inos
may be w i en in his seesaw app oxima ion as
Mν≈ML−MDM−1
RMT
D.(20)
We assume a scena io whe e he dominan con ibu ion o
he ac i e known neu inos comes om he MLma ix ha -
ing he same s uc u e o a Type I seesaw. Then in his sce-
na io he eigen alues o he ligh neu inos may be ob ained
by applying again he seesaw app oxima ion, ha is,
Mligh =−
√2σˆ
M−1
√2σT
.(21)
Taking ad an age o he ac ha all σij and ˆ
Mij en ies
in (21) a e ee pa ame e s, we p opose he ollowing pa-
ame e iza ions o ˆ
Mand Mligh neu ino mass ma ices:
Mligh =Y2 2
2ˆm⎛
⎝
1+bb b
b1+b+cb−c
bb−c1+b+c⎞
⎠,
ˆ
M=ˆmDiag(Y1,Y2,Y3),
(22)
whe e Y,Y1,Y2,Y3,b,ca e dimensionless coupling con-
s an s and ˆm ep esen s he mi o scale. This pa ame e iza-
ion o he ligh neu inos mass ma ix imposes a symme ic
mixing o elec on neu ino wi h muon and au neu inos in
he i s ow and column o (Mligh )ij , and he 2 ×2 subma-
ix i,j =2,3 gene a e maximal mixing o muon and au
neu inos. This s uc u e o Mligh makes possible he diag-
onaliza ion o ligh neu inos by he so called “ i-bimaximal
mixing ma ix” [57], i.e.
UT
TBMligh VTB =−UT
TB
√2σˆ
M−1
√2σT
UTB
=Diag(m1,m2,m3), (23)
wi h
UTB =⎛
⎜
⎜
⎜
⎜
⎝
2
√6
1
√30
−1
√6
1
√3−1
√2
−1
√6
1
√3
1
√2
⎞
⎟
⎟
⎟
⎟
⎠
(24)
and he ligh neu ino mass eigen alues
(m1,m2,m3)=Y2 2
2ˆm(1,1+3b,1+2c). (25)
The supp ession by he mi o scale ˆmin (25) p o ides a na -
u al explana ion o he smallness o neu ino masses. The
allowed ange o alues o he squa e neu ino mass di e -
ences epo ed in PDG [56]:
m2
2−m2
1≈7.6×10−5eV2,
m2
3−m2
2≈2.43 ×10−3eV2,
(26)
wi h he inpu o no mal hie a chy o he neu ino masses
(m1,m2,m3)=(0.0865,0.0870,.1)eV,(27)
ix he pa ame e alues as b=0.00168 and c=0.07757.
These neu ino masses a e consis en wi h he bounds
mν<2eV[56], and se he mass di e ences
m2
3−m2
1≈2.5×10−3eV2.(28)
So, om (25), (27)
Y2 2
2ˆm≈8.65 ×10−2eV.(29)
Eu . Phys. J. C (2012) 72:1859 Page 5 o 8
The e o e, assuming ˆm=mˆν=100 GeV and =246 GeV
we ob ain
Y≈5.34 ×10−7.(30)
The ma ix MLin (17) may be diagonalized by using a uni-
a y ans o ma ion,
U†MLU=Diag(m1,m2,m3,ˆm1,ˆm2,ˆm3), (31)
whe e he mixing ma ix Ucompa ible wi h ou amewo k
is w i en in good app oxima ion as
U6×6≈UTB
√2σˆ
M−1
−(
√2σˆ
M−1)TI3×3.(32)
The pa icula nume ical solu ion cong uen wi h he abo e
scena io o he neu ino masses and mixing is
√2σ≈93041.9eV⎛
⎝−1.2001 0.6355 1.2952
0.6355 −1.2702 1.3006
1.2952 1.3006 0.5389⎞
⎠,
(33)
ˆ
M=100 GeV Diag(3.4918,3.2643,3.6043), (34)
and
√2σˆ
M−1≈9.3×10−7⎛
⎝−0.3437 0.1946 0.3593
0.1819 −0.3891 0.3608
0.3709 0.3984 0.1495⎞
⎠
(35)
o ligh ν–mi o mixing. Since he ligh –mi o mixing is
e y small, he mixing ma ix o ligh neu inos beha es in
good app oxima ion as he UTB,(24). I is wo h o men-
ion he e ha in he limi o e y small ligh –mi o cha ged
lep on mixing, (F †
LFL)ij ,(E†
LEL)ij 1, we may app oach
UTB as he usual UPMNS lep on mixing ma ix o h ee gen-
e a ions. Then, we ob ain (UPMNS)e2≃1
√3,(UPMNS)e3≃0,
and (UPMNS)μ3≃1
√2, which gi e o he sola and he a mo-
sphe ic neu ino mixing angles θ12 ≃35.20and θ23 ≃450,
wi h θ13 ≃0 in good ag eemen wi h cu en da a, al hough
ecen e idence [58,59] shows ha θ13 may ha e a alue
di e en om ze o.
In ea lie pape s on he s udy o neu inos and le –
igh symme y [60–63] appea simila ep esen a ions o
he e mions and mass ma ices as ou in (18), bu hese
au ho s ob ain masses o he s anda d and mi o neu i-
nos some o de s o magni ude di e en om ou s. On he
o he hand, he mass gene a ion in he LRMM he e consid-
e ed is achie ed wi h he scala ields Φand ˆ
Φ,(3), (4),
ans o ming as double s unde SU(2)Land SU(2)R, espec-
i ely, wi h a mi o scale much lowe han 1012–1013 GeV.
5 Radia i e decays
In his sec ion we analyze he lep on la o iola ion p o-
cesses μ→eγ ,τ→μγ and τ→eγ a ising in he model
by he exis ence o gauge in a ian mixing e ms be ween
o dina y lep ons and wi h he mi o coun e pa s. The
lowe o de con ibu ion o heses decays media ed by he
neu al scala ields comes om he Feynman diag ams
whe e he pho on is adia ed om an in e nal line. The
co esponding ampli ude is p opo ional o he ope a o
u(p2)σμνqνμu(p1), whe e q=p1−p2and μis he pho-
on pola iza ion [53–55].
In he limi memμmτ he a e decay is gi en by
Γ(l
i→lj+γ)=α
512π4GFm2
li2m5
li
M4
Hln M2
H
m2
li−4
3ij
−
k
xνkVL,jkV+
R,ki
2
,(36)
whe e xνk≡m2
νk
M2
W
,ij =|A+
LAR|ij ep esen s he la o -
changing couplings, and he second e m is he e y small
con ibu ion om he ligh neu ino p opaga ing inside he
loop.
In he limi α1 and MHMˆ
H he b anching a ios
a e, espec i ely,
B1(μ →e+γ)=3αm4
μ
8M4
Hln M2
H
m2
μ−4
3eμ
−
k
xνkVL,ekV+
R,kμ
2
,(37)
B2(τ →μ+γ)=3αm4
τ
8M4
Hln M2
H
m2
τ−4
3μτ
−
k
xνkVL,μkV+
R,kτ
2
(38)
and
B3(τ →e+γ)=3αm4
τ
8M4
Hln M2
H
m2
τ−4
3eτ
−
k
xνkVL,ekV+
R,kτ
2
.(39)
By using he cons ain s ij <1,i =j o he pa ame e s in
(37), (39), equi ed by uni a i y o V,see(2), (3), one ge s
o he abo e b anching a ios
B1<2.2×10−13,B
2<5×10−9and B3<5×10−9
(40)

Page 6 o 8 Eu . Phys. J. C (2012) 72:1859
which is cong uen wi h he expe imen al bounds [56]
B(μ →e+γ)<1.2×10−11,B(τ →μ+γ)<4.4×10−8
and B(τ →e+γ)<3.3×10−8PDG [56].
6 Hea y neu ino signals
Possible new neu inos can be de ec ed in a ious ways in
collide s. I hese neu inos a e hea y hey will be uns able
and may be de ec ed di ec ly in hei decay p oduc s.
Nex gene a ion o la ge collide s will p obe Na u e up
o TeV scales wi h high p ecision, p obably disco e ing new
hea y pa icles. Thus, i will be a window o any new physics
nea he elec oweak scale which couples o he SM. Such
collide s can be used o p oduce new hea y neu inos a an
obse able le el o imp o e p esen limi s on hei masses
and mixings [64–67]. These e mions wi h new in e ac ions,
like in he le – igh models [68], can be p oduced by gauge
couplings supp essed by small mixing angles. Fo he anal-
ysis o he hea y neu inos signals i is necessa y o know
hei decay modes, which a e di e en in he Di ac and Ma-
jo ana cases.
Hea y Majo ana neu ino single s can be p oduced in he
p ocess [69,70]
q¯
q→W∗→l±H(41)
wi h l=e,μ,τ, which c oss sec ions depend on MNand he
small mixing VlN. Hea y Majo ana neu ino decays in he
channels N→W±l∓,N→Zνland N→Hνl. The pa ial
wid hs o he Ndecays a e
ΓN→W+l−
=ΓN→W−l+
=e2
64πs2
θw|UlN|2m3
N
M2
W1−M2
W
m2
N1+M2
W
m2
N−2M4
W
m4
N,
(42)
Γ(N→Zνl)
=e2
64πs2
θwc2
θw|UlN|2m3
N
M2
Z1−M2
Z
m2
N
×1+M2
Z
m2
N−2M4
Z
m4
N,(43)
Γ(N→Hνl)=e2
64πs2
θw|UlN|2m3
N
M2
W1−M2
H
m2
N2
,(44)
whe e UlN is he ligh –mi o neu ino mixing
√2σˆ
M−1,
(35). F om (32), (35) he con ibu ions come om e ms o
he o de |VlN|10−7. F om hese exp essions we can con-
clude ha he o al b anching o each o he ou channels is
independen o he hea y neu ino mixing, de e mined only
by mNand he gauge and Higgs boson masses.
Hea y neu ino signals a e limi ed by he small mixing
o he hea y neu ino equi ed by p ecision cons ain s [73]
and masses o o de 100 GeV a e accessible a LHC. Fo his
mass ange, SM backg ounds a e la ge and, since p oduc-
ion c oss sec ions a e ela i ely small, hea y neu ino sin-
gle s a e a he di icul o obse e. The b anching a ios o
di e en alues o mN ead as in Table 1(MH=130 GeV);
and in all hese cases Bi≈1. He e
BW±=B N→W±l∓,B
Z=B (N →Zνl),
BH=B (N →Hνl).
(45)
Table 1shows ha hese decays a e no so sensi i e o he
hea y neu ino mass, such ha o hea y neu ino signals i
is no necessa y o ha e cen e o mass ene gies much la ge
han a hund ed GeV.
Among he possible inal s a es gi en by (42)–(44), only
cha ged cu en decays gi e inal s a es which may in p in-
ciple be de ec ed. Fo mN<M
W hese wo body decays a e
no possible and Ndecays in o h ee e mions, media ed by
o -shell bosons.
O he simple p oduc ion p ocesses like
q¯
q→Z∗→νN, (46)
gg →H∗→νN (47)
gi e l±and l+l− inal s a es which a e unobse able due o
he huge backg ounds. Fo he pai p oduc ion
q¯q→Z∗→NN (48)
he c oss sec ion is supp essed by |VlN|4, phase space and
he Zp opaga o , and is hus negligible.
Th ee signals a e p oduced in he wo cha ged cu en
decay channels o he hea y neu ino
l+N→l+l−W+→l+l−l+¯ν, (49)
l+N→l+l+W−→l+l+l−ν(50)
and small addi ional con ibu ions om τlep onic decays.
Hea y neu ino signals in he inal s a e l±l±a e gi en
in he lep on numbe iola ing neu ino decay and subse-
quen had onic Wdecay, o lep onic decay when he lep-
Table 1 B anching a ios o di e en alues o mN
mN(GeV)B
W±BZBH
100 0.34 0.1 0.2
390 0.3 0.306 0.09
780 0.3 0.297 0.107
MW,MZ,MH0.293 0.3 0.111
Eu . Phys. J. C (2012) 72:1859 Page 7 o 8
on is missed. LHC p esen ene gies a e enough o disco e
hea y Majo ana neu ino wi h e y small VeN [71,72].
7 Conclusions
He e he LRMM wi h gauge g oup SU(3)C⊗SU(2)L⊗
SU(2)R⊗U(1)Yis applied in o de o ind close alues
o neu ino masses i ed o expe imen al da a. We ha e
wo ked wi h Majo ana neu inos, which mass ma ix was
w i en in e ms o blocks ha s and o s anda d and mi -
o mass e ms. The la ge numbe o pa ame e s in ol ed
induces o make some simpli ica ions on he s uc u e o he
ma ix. A double seesaw app oach me hod is used and diag-
onaliza ion is pe o med, and wi h he help o neu ino da a
we accommoda e neu ino masses wi h no mal hie a chy o
he o de o (m1,m2,m3)≈(0.0865,0.0870,0.1)eV. So,
we ha e ound a consis en smallness hie a chy o he neu-
ino masses. Wi h he LRMM we ha e also analyzed he
adia i e decays μ→e+γ,τ→e+γand τ→μ+γ
o a Higgs mass o 130 GeV, ob aining bounds o he
b anching a ios cong uen wi h he expe imen al ones. De-
cay a es o hea y neu inos Nwe e calcula ed o di e -
en channels, and we ound ha hei BR a e nea ly equal
o MNMW,MZ,MHand also ha hey do no change
oo much o o he alues o MN. To ind hea y Majo ana
neu inos one has only a ew pa ame e dependence ( o neu-
ino single s, he hea y neu ino mass and i s mixing angle)
and also he mass scale could be accessible a he LHC.
Acknowledgemen s The au ho R. Gai án wishes o hank o he
“Sis ema Nacional de In es igado es” (SNI) in Mexico o pa ial sup-
po . and also acknowledges suppo by PAPIIT p ojec IN117611.
A. He nandez-Galeana is hank ul o pa ial suppo om he “Ins i-
u o Poli écnico Nacional” (G an s om EDI and COFAA) and “Sis-
ema Nacional de In es igado es” (SNI) in Mexico, and J.M. Ri e a-
Rebolledo wishes o hank o EDD-IPN and he “Sis ema Nacional de
In es igado es” (SNI) in Mexico o pa ial suppo .
Re e ences
1. Y. Fukuda e al. (Supe -Kamiokande Collabo a ion), Phys. Re .
Le . 81, 1562 (1998)
2. Q.R. Ahmad e al. (The SNO Collabo a ion), Phys. Re . Le . 87,
71301 (2001)
3. K. Eguchi e al. (KamLAND Collabo a ion), Phys. Re . Le . 90,
021801 (2003)
4. E. Aliu, e al. (K2K Collabo a ion), Phys. Re . Le . 94, 081802
(2005)
5. Y. Ashie e al. (SK Collabo a ion), Phys. Re . D 71, 112005
(2005)
6. M. Al mann e al. (GNO Collabo a ion), Phys. Le . B 616, 174
(2005)
7. M. Apolonio e al. (CHOOZ Collabo a ion), Eu . Phys. J. C 27,
331 (2003)
8. M. Mal oni, T. Schwe z, M.A. To o la, J.W.F. Valle, Phys. Re . D
68, 113010 (2003)
9. G.L. Fogli, E. Lisi, A. Ma one, A. Palazzo, P og. Pa . Nucl.
Phys. 57, 742–795 (2006)
10. A. S umia, F. Vissani, a Xi :hep-ph/0606054
11. P. Minkowski, Phys. Le . B 67, 421 (1977)
12. T. Yanagida, in P oceedings o he Wo kshop on he Uni ied The-
o y and he Ba yon Numbe in he Uni e se, ed. by O. Sawada, A.
Sugamo o, KEK, Tsukuba, Japan (1979), p. 95
13. M. Gell-Mann, P. Ramond, S. Slansky, in Complex Spino s and
Uni ied Theo ies in Supe g a i y, ed. by P. an Nieuwenhuizen,
D.Z. F eedman (No h-Holland, Ams e dam, 1979), p. 315
14. S.L. Glashow, in Qua ks and Lep ons (Plenum, New Yo k, 1979),
p. 687
15. R.N. Mohapa a, G. Senjano ic, Phys. Re . Le . 44, 912 (1980)
16. M. Magg, C. We e ich, Phys. Le . B 94, 61 (1980)
17. G. Laza ides, Q. Sha i, C. We e ich, Nucl. Phys. B 181, 287
(1981)
18. J. Schech e , J.W.F. Valle, Phys. Re . D 22, 2227 (1980)
19. R.N. Mohapa a, G. Senjano ic, Phys. Re . D 23, 165 (1981)
20. J. Schech e , J.W.F. Valle, Phys. Re . D, Pa . Fields 25, 774
(1982)
21. A. A anda, J.L. Díaz-C uz, Mod. Phys. Le . A 20, 203 (2005)
22. P. Langacke , D. London, Phys. Re . D 38, 907 (1988)
23. S. An usch, C. Biggio, E. Fe nández-Ma ínez, M.B. Ga ela, J.
López-Pa on, J. High Ene gy Phys. 10, 084 (2006)
24. A. Abada, C. Biggio, F. Bonne , M.B. Ga ela, T. Hambye, J. High
Ene gy Phys. 12, 061 (2007)
25. E. Na di, E. Roule , D. Tommmasini, Phys. Le . B 344, 225
(1995)
26. D. Tommasini, G. Ba enboin, J. Be nabeu, C. Ja lskog, Nucl. In-
s um. Me hods Phys. Res., Sec . B, Beam In e ac . Ma e . A oms
444, 451 (1995)
27. J. Ke s en, A.Y. Smi no , Phys. Re . D 76, 073005 (2007)
28. S. B ay, J.S. Lee, A. Pila sis, Nucl. Phys. B 786, 95 (2007)
29. F. del Aguila, J.A. Aguila -Saa ed a, R. Pi au, J. High Ene gy
Phys. 10, 047 (2007)
30. M. Fukugi a, T. Yanagida, Phys. Le . B 174, 45 (1986)
31. C.S. Wu, E. Amble , R.W. Haywa d, D.D. Hopes, R.P. Hudson,
Phys. Re . 105, 1413 (1957)
32. R. Ga win, L. Lede man, M. Wein ich, Phys. Re . 105, 1415
(1957)
33. J. F iedman, V. Telegdi, Phys. Re . 105, 1681 (1957)
34. E. Fe mi, Nuo o Cimen o 11, 1 (1934)
35. E. Fe mi, Z. Phys. 88, 161 (1934)
36. T.D. Lee, C.N. Yang, Phys. Re . 104, 254 (1956)
37. J. Maalampi, M. Roos, Phys. Rep. 186, 53 (1990)
38. Yu. Kobza e , L.B. Okun, I.Ya. Pome anchuk, So . J. Nucl. Phys.
3, 837 (1966)
39. L.B. Okun, hep-ph/0606202 1, 19 jun 2006
40. R. Foo , R.R. Volkas, Phys. Re . D 52, 6595 (1995)
41. Z.G. Be ezhiani, R.N. Mohapa a, Phys. Re . D 52, 6607 (1995)
42. V.E. Ce on, U. Co i, J.L. Díaz-C uz, M. Maya, Phys. Re . D 57,
1998 (1934)
43. U. Co i, J.L. Díaz-C uz, R. Gai án, H. Gonzalez, A. He nández-
Galeana, Phys. Re . D 66, 015004 (2002)
44. R. Gai án, O.G. Mi anda, L.G. Cab al-Rose i, Phys. Re . D, Pa .
Fields 72, 034018 (2005)
45. R. Gai án, S. Rod iguez-Romo, A. He nández-Galeana, J.M.
Ri e a-Rebolledo, P. Fe nández de Có doba, In . J. Mod. Phys.
A22, 2935 (2007)
46. S.M. Ba , D. Chang, G. Senjano ic, Phys. Re . Le . 67, 2765
(1991)
47. K.S. Babu, R.N. Mohapa a, Phys. Re . D 41, 1286 (1990)
48. J.C. Pa i, A. Salam, Phys. Re . D 11, 1137 (1975)
49. Z.G. Be ezhiani, Phys. Le . B 129, 99 (1983)
50. Z.G. Be ezhiani, Phys. Le . B 150, 177 (1985)
51. Z.G. Be ezhiani, So . J. Nucl. Phys. 42, 825 (1985)
Page 8 o 8 Eu . Phys. J. C (2012) 72:1859
52. S. An usch, S.F. King, Nucl. Phys. B 705, 239 (2005)
53. B.W. Lee, R.E. Sch ock, Phys. Re . D 16, 1444 (1977)
54. A. Abada, C. Biggio, F. Bonne , M.B. Ga ela, T. Hambye, Phys.
Re . D, Pa . Fields 78, 033007 (2008)
55. P.Q. Hung, Phys. Le . B 659, 585 (2008)
56. K. Nakamu a e al. (Pa icle Da a G oup), J. Phys. G, Nucl. Pa .
Phys. 37, 075021 (2010)
57. F. Ha ison, D.H. Pe kins, W.G. Sco , Phys. Le . B 520, 167
(2002)
58. A.B. Balan ekin, a Xi :1106.5021 [hep-ph]
59. S.F. King, a Xi :1106.4239 [hep-ph]
60. C.H. Alb igh , J. Oliensis, Phys. Re . D 33, 2602 (1986)
61. J. Oliensis, C.H. Alb igh , Phys. Le . B 160, 121 (1985)
62. D. Chang, R.N. Mohapa a, Phys. Re . Le . 58, 1600 (1987)
63. S. Be olini, J. Liu, Nucl. Phys. B 297, 401 (1988)
64. A. Da a, M. Guchai, A. Pila sis, Phys. Re . D 50, 3195 (1994)
65. F.M.L. Almeida J ., Y.A. Cou inho, J.A. Ma ins Simoes, M.A.B.
do Vale, Phys. Re . D 62, 075004 (2000)
66. O. Panella, M. Cannoni, Y.N. S i as a a, Phys. Re . D 65, 035005
(2002)
67. T. Han, B. Zhang, Phys. Re . Le . 97, 171804 (2006)
68. P. Langacke , R.W. Robine , J.L. Rosne , Phys. Re . D 30, 1470
(1984)
69. F. del Aguila, J.A. Aguila -Saa ed a, Phys. Le . B 672, 158
(2009)
70. F. del Aguila, J.A. Aguila -Saa ed a, Nucl. Phys. B 813,22
(2008)
71. A. Da a, M. Guchai, A. Pila sis, Phys. Re . D 50, 3195 (1994)
72. F. del Aguila, J.A. Aguila -Saa ed a, R. Pi au, J. High Ene gy
Phys. 10, 047 (2007)
73. F. Del Aguila, J. de Blas, M. Pe es-Vic o ia, Phys. Re . D 78,
013010 (2008)