Neutrino Masses

1
Durham, 15 April '03
Neutrino Masses and Mixing
G. Altarelli CERN
Some recent work G.A., F. Feruglio, I. Masina,
hep-ph/0210342 a review G.A., F. Feruglio,
hep-ph/0206077
2
n Oscillations Imply Different n Masses
ne same weak isospin doublet as e-
ne
flavour
mass
ne nm nt
n1 n2 n3
e-
U
W-
U UP-MNS
U mixing matrix
Pontecorvo Maki, Nakagawa, Sakata
ne cosq n1 sinq n2 nm -sinq n1 cosq n2
e.g 2 flav.
Stationary source
Stodolsky
n1,2 different mass, different x-dep
pa2E2-ma2
na(x)eipax na
P(nelt-gt nm) lt nm(L) negt2sin2(2q).sin2(Dm2L/4
E)
At a distance L, nm from m- decay can produce e-
via charged weak interact's
3
Solid evidence for n oscillations (LSND
unclear)
Dm2atm 3 10-3 eV2, Dm2sol 7 10-5
eV2 (Dm2LSND 1 eV2)
4
In summary for solar n's
After Kamland
Before Kamland
Dm2 (eV2)
J. Bahcall et al
Note the change of scale
5
n Oscillations Summary of Exp. Facts
Homestake, Gallex, Sage, (Super)Kamiokande,
Macro...
GNO,K2K,..
New after SNO, KAMLAND
Atmospheric
Solar
Dm2atm 3.10-3 eV2 sin22q23gt0.92
?????????? solution selected Dm2 7 10-5 eV2,
sin22q12 0.8
nm-gt nt dominant nm-gt ne small (Chooz
U13lt0.2) nm-gt nsterile small
ne -gt nm,nt dominant ne -gt nsterile small
Solar
true or false? MINIBOONE
SolarSola
LSND
nm -gt ne,nsterile
Dm2 1 eV2, sin22q small
CPT violation?
6
n oscillations measure Dm2. What is m2?
Dm2atm 3 10-3 eV2 Dm2sunlt Dm2atm
Direct limits (PDG '02)
End-point tritium b decay (Mainz)
m"ne" lt 2.8 eV m"nm" lt 170 KeV m"nt" lt 18.2 MeV
Cosmology
Wn h2 Simi /94eV
(h21/2)
NEW!! WMAP
Simi 0.69 eV (95) Wn 0.014
Any n mass 0.23-1eV
Why n's so much lighter than quarks and leptons?
7
New powerful cosmological limit
Wnh2lt0.076 mnlt0.23 eV
3 degenerate ns
All i??? on the ???????? scale ???? mass is
very important!
Combined WMAP 2dFGRS ACBAR CBI..
Finding 0nbb would also prove Majorana ns
8
Neutrino masses are really special!
t
Log10m/eV
10
b
t
c
mt/(Dm2atm)1/21012
s
8
?
d
u
Massless ns? no nR L conserved
6
e
4
2
Small n?masses? nR very heavy L not conserved
????
0
Upper limit on mn
(Dm2atm)1/2
(D m2sol)1/2
-2
KamLAND
9
How to guarantee a massless neutrino?
1) nR does not exist
No Dirac mass
nLnR nRnL
and
2) Lepton Number is conserved
No Majorana mass
ncn-gtnTRCnR or nTLCnL
Cig0g2
10
Neutrinos
Dirac mass nLnR nRnL
(needs nR)
n's have no electric charge. Their only charge is
lepton number L
IF L is not conserved (not a good quantum
number) n and n are not really different
TCP, "Lorentz"
n, h -1/2gt
n, h 1/2gt
Majorana mass
nTR nR or nTL nL
(we omit the charge conj. matrix C)
Violates L, B-L by DL 2
11
Weak isospin I
nL gt I 1/2, I3 1/2 nR gt I 0, I3 0
Dirac Mass
nLnR nRnL
DI1/2
Can be obtained from Higgs doublets nLnRH
Majorana Mass
nTLnL
DI1
Non ren., dim. 5 operator nTL nLHH
Directly compatible with SU(2)xU(1)!
nTRnR
DI0
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B and L conservation in SM
"Accidental" symmetries in SM there is no
dim.4 gauge invariant operator that violates B
and/or L (if no nR, otherwise M nTR nR is dim-3
DL2) The same is true in SUSY with R-parity
cons.
u u -gt e d
e. g. for the DBDL -1 transition
all good quantum numbers are conserved e.g.
colour u3, d3 and 3x3 63 but
l
dcGu ecGu
dim-6
M2
SU(5) p-gt ep0
Once nR is introduced (Dirac mass) large Majorana
mass is naturally induced
see-saw
13
Yanagida Gell-Mann, Ramond ,
Slansky Mohapatra, Senjanovic..
See-Saw Mechanism
MnTRnR allowed by SU(2)xU(1)
Large Majorana mass M (as large as the cut-off)
mDnLnR
Dirac mass m from Higgs doublet(s)
nL nR
nL nR
0 mD mD M
MgtgtmD
Eigenvalues
-mD2
nlight
, nheavy M
M
sign conventional for fermions
14
In general n mass terms are
Majorana
Dirac
mDhv vlt0H0gt
More general see-saw mechanism
nL nR
nL nR
lv2/ML mD mD MR
mD2
lv2
mlight
and/or
ML
MR
mheavy MR
meff nTLmlightnL
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Neutrinos are (probably) Majorana particles
nLTmn?L
mD
mD
H
H
See-saw
mn??? mDTM-1 mD ?
nR
nL
mass M
nL
connection with mD
More in general non ren. O5 operator l/M
nLTHHT?L
H
H
?
e.g from
N new particle Iw0,1
nL
mass M
nL
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A very natural and appealing explanation
n's are nearly massless because they are Majorana
particles and get masses through L non
conserving interactions suppressed by a large
scale M MGUT
m2
m mt v 200 GeV M scale of L non cons.
mn
M
Note
mn (Dm2atm)1/2 0.05 eV m v 200 GeV
M 1015 GeV
Neutrino masses are a probe of physics at MGUT !
17
Most of the Universe is not made up of atoms
Wtot1, Wb0.044, Wm0.27 Most is Dark Matter and
Dark Energy
Dark Matter
WMAP
Most Dark Matter is Cold (non relativistic at
freeze out) Significant Hot Dark matter is
disfavoured Neutrinos are not much
cosmo-relevant Wnlt0.015 (WMAP)
SUSY has excellent DM candidates
Neutralinos Also Axions are still viable
For 3 neutrinos Wnlt 0.015 -gt mnlt 0.23 eV
5(Dm2atm)1/2
Degenerate models with m2 gtgt Dm2 are now
disfavoured
18
Baryogenesis
A most attractive possibility
BG via Leptogenesis near the GUT scale
T 10123
GeV (after inflation)
Buchmuller,Yanagida, Plumacher, Ellis, Lola,
Giudice et al, Fujii et al
Only survives if D(B-L)?0 (otherwise is washed
out at Tew by instantons)
Main candidate decay of lightest nR (M1012 GeV)
L non conserv. in nR out-of-equilibrium
decay B-L excess survives at Tew and gives the
observed B asymmetry.
Quantitative studies confirm that the range of mi
from n oscill's is compatible with BG via
(thermal) LG
??????????????????????? was derived
mi 10-1 eV
Close to WMAP
Buchmuller, Di Bari, Plumacher
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The scale of the cosmological constant is a big
mystery.
WL 0.65
rL (2 10-3 eV)4 (0.1mm)-4
In Quantum Field Theory rL (Lcutoff)4
Similar to mn!?
If Lcutoff MPl
rL 10123 robs
Exact SUSY would solve the problem rL 0
But SUSY is broken rL (LSUSY)4 1059 robs
It is interesting that the correct order is
(rL)1/4 (LEW)2/MPl
Other problem Why now?
So far no solution A modification of gravity
at 0.1mm?(large extra dim.) Leak of vac.
energy to other universes (wormholes)?
rad
Quintessence?
r
m
L
t
Now
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The current experimental situation is still
unclear
LSND true or false? what is the absolute scale
of n masses?
Different classes of models are possible
If LSND true sterile n(s)?? CPT violatn??
3-1
m21-2eV2
LSND
nsterile
If LSND false
3 light n's are OK
Degenerate (m2gtDm2)
m2o(1)eV2
m210-3 eV2
sol
Inverse hierarchy
atm
Normal hierarchy
m210-3 eV2
atm
sol
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nR is a heavy "sterile" neutrino
sterile no gauge interact's nR quantum
numbers colourTWQ0 nL is a light
"active" neutrino LEP Nnactive 3
Are there light sterile neutrinos?
If LSND is true 3 different oscill. frequencies
Dm2 LSND solar atm. at least 4 light n's
(assuming CPT conservation)
But LSND not confirmed by KARMEN The WMAP limit
Smnlt0.69 eV is no good news for LSND! Will be
double-checked by MiniBoone now taking
data Perhaps will fade away (so we assume here)
22
???????? limit makes LSND even less plausible
Pierce, Murayama Giunti Di Bari
??
23
3-n Models
ne
ne nm nt
n1 n2 n3
e-
U
W-
U UP-MNS
Pontecorvo Maki, Nakagawa, Sakata
flavour
mass
In basis where e-, m-, t- are diagonal
1 0 0 0 c23 s23 0 - s23 c23
c12 s12 0 -s12 c12 0 0 0 1
c13 0 s13e-id 0 1 0 -s13eid 0
c13
U

solar large
CHOOZ s13lt0.2
c13 c12 c13 s12 s13e-id ...
... c13 s23 ...
... c13 c23

atm. max
(some signs are conventional)
24
In general 9 parameters 3 masses, 3 angles, 3
phases
eif1m1 0 0 0 eif2m2 0 0
0 m3
UT
mn U
LTmnL
For s13 0
0nbb
m1c2m2s2 (m1-m2)cs/V2 (m1-m2)cs/V2
... (m1s2m2c2m3)/2
(m1s2m2c2-m3)/2 ...
... (m1s2m2c2m3)/2
mn
Note mn is symmetric phases
included in mi
Relation between masses and frequencies
P(nelt-gtnm) P(nelt-gtnt)1/2 sin22q??.sin2Dsun P(nm
lt-gtnt)sin2Datm- 1/4 sin22q??.sin2Dsun
In our def. Dsungt0, Datmgt or lt 0
25
0nbb can tell degenerate, inverted or normal
hierarchy
meec132 m1c122eiam2s122m3eibs132
LA0.3-1
Degenerate m c122eias122
Strumia Pascoli et al
mee m (0.3 -1) 0.23 eV
IH (Dm2atm)1/2c122eias122
mee (1.6-5) 10-2 eV
NH (Dm2sol)1/2s122 (Dm2atm)1/2eibs132
mee (few) 10-3 eV
Present exp. limit mee0.3-0.5 eV (and a hint of
signal?????)
26
Evidence for 0nbb?
Heidelberg-Moscow Klapdor-Kleingrothaus et al
Not at all compelling!!!! 1.5s?, 2.2s? 3.1s?
Iff true (WMAP ??)
mee/h0.390.11eVgtgt(Dm2atm)1/2
Feruglio, Strumia, Vissani
(h0.6-2.8 uncert. matrix element)
would clearly point to degenerate models
27
Degenerate n's
m2gtgt Dm2
Apriori compatible with hot dark matter (m1-2
eV) was considered by many Limits on mee from
0nbb then imply large mixing also for solar
oscillations (Vissani Georgi,Glashow)
(Exp)
mee0.3-0.5 eV
mee c213 (m1c212 m2s212)s213m3 m1c212 m2s212
If m1 m2 m21-2 eV
m1 -m2 and c212s212
LA solution tg2q0.5 cos2q?sin2q0.3
A factor of 3 compensation would be OK
Trusting WMAP m lt 0.23 eV, only a moderate
degeneracy is allowed for LA, m/(Dm2atm)1/2 5,
m/(Dm2sol)1/2 30. Less constraints from
0nbb?????? m1m2 allowed)
Recall l??????????????????? m lt 0.1 eV
28
????????????? and WMAP not too much hierarchy is
needed for n masses
rDm2sol/ Dm2atm1/40
or
?heaviestlt 0.23 eV mnextgt 7 10-3 eV
Anarchical or semi-anarchical models
29
Anarchy (or accidental hierarchy) No structure
in the leptonic sector
Hall, Murayama, Weiner
rDm2sol/ Dm2atm1/40
See-Saw mnm2/M produces hierarchy from random
m,M
r peaks at 0.1
could fit LA
But all mixing angles should be large
sin22q
marginal for LA predicts q13 near bound
30
????anarchy no structure in 23
l2 l l l 1 1 l 1 1
Note ?????? ??????
Consider a matrix like with coeff.s of o(1)
and det23o(1) ?l1 corresponds to anarchy
mn
l2 l ? l ? ? ? ? 1
mn
?????????????????????????
Normally two masses are of o(1) and ?????? But
if, accidentally, ???, then the solar angle is
also large.
?????????????????????????? is that ????is small,
but the hierarchy m23gtgtm22 is accidental
Ramond et al, Buchmuller et al
31
Inverted Hierarchy
2
m210-3 eV2
sol
1
Zee, Joshipura et al, Mohapatra et al, Jarlskog
et al, FramptonGlashow, Barbieri et al
atm
3
An interesting model for double maximal mixing
1st approximation
m 0 0 0 -m 0 0 0 0
0 m m m 0 0 m 0 0
mndiag
UmndiagUT 1/
V2
Can arise from see-saw or dim-5 LTHHTL e.g. by
approximate Le-Lm-Lt symmetry 1-2 degeneracy
stable under rad. corr.'s
32
1st approximation
m 0 0 0 -m 0 0 0 0
0 m m m 0 0 m 0 0
mndiag
UmndiagUT 1/
V2
LA? This texture prefers qsol closer to
maximal than qatm ????qsol-p/4 small for
(Dm2sol/Dm2atm)LA 1/40
0 m m 0
Pseudodirac ?12 maximal
0 0 0 0
?12 o(1)
?? fact ??-gt
23-gt
d 1 1 1 h h 1 h h
0 m m m 0 0 m 0 0
???? perturbations?
m
??? ?12 1 d h
(Dm2sol/Dm2atm)LA d h
In general more ?12 is close to maximal, more is
IH likely In principle one can use the charged
lepton mixing to go away from ?12 maximal
33
m210-3 eV2
Normal Hierarchy
3
atm
2
sol
1
Assume 3 widely split light neutrinos. For u,
d and l- Dirac matrices the 3rd generation
eigenvalue is dominant. May be this is also
true for mnD diag mnD(0,0,mD3). Assume
see-saw is dominant mnmTDM-1mD See-saw
quadratic in mD tends to enhance hierarchy
Maximally constraining GUT's relate q, l-, n
masses!
34
A crucial point in the 2-3 sector we need
both large m3-m2 splitting and large mixing.
m3 (Dm2atm)1/2 5 10-2 eV m2 (Dm2sol)1/2
8 10-3 eV for LA
The "theorem" that large Dm32 implies small
mixing (pert. th. qij 1/Ei-Ej) is not true
in general all we need is (sub)det230
Example
Det 0 Eigenvl's 0, 1x2 Mixing sin22q
4x2/(1x2)2
x2 x x 1
m23
So all we need are natural mechanisms for
det230
For x1 large splitting and large mixing!
35
Examples of mechanisms for Det230
see-saw mnmTDM-1mD
1) A nR is lightest and coupled to m and t King
Allanach Barbieri et al......
1/e 0 0 1
e 0 0 1
1/e 0 0 0

M-1
M

a b c d
a c b d
1/e 0 0 0
a2 ac ac c2
mn

1/e

0 0 x 1
2) M generic but mD "lopsided" Albright, Barr
GA, Feruglio, .....
mD
0 0 x 1
0 x 0 1
a b b c
x2 x x 1

mn
c
Caution if 0 -gt 0(e), det230 could be spoiled
by suitable 1/e terms in M-1
36
An important property of SU(5)
Left-handed quarks have small mixings (VCKM), but
right-handed quarks can have large mixings
(unknown).
In SU(5) LH for d quarks
RH for l- leptons
5
10
mddRdL
5 (d,d,d, n,e-)
L
R
10
5
meeReL
md meT
cannot be exact, but approx.
Most "lopsided" models are based on this
fact. Large atm. mixing arises from the charged
lepton sector.
37
Hierarchical n's and see-saw dominance
LTmnL -gt mnmTDM-1mD
allow to relate q, l, n masses and mixings in GUT
models. For dominance of dim-5 operators -gt less
constraints
1/M LTlTHHTlL -gt LTmnL
The correct pattern of masses and mixings, also
including n's, is obtained in simple models based
on
SU(5)xU(1)flavour
Ramond et al GA, FeruglioMasina Buchmuller et
al King et al Yanagida et al, Berezhiani et
al Lola et al.......
models could be more predictive, but
are often based on specific textures from a set
of special operators.

SO(10)
Albright, Barr Babu et al Buccella et al
Barbieri et al
38
The non trivial pattern of fermion masses and
mixing demands a flavour structure (symmetry)
(SUSY) SU(5)XU(1)F models offer a minimal
description of flavour symmetry
A flexible enough framework used to realize and
compare models with anarchy or hierarchy (direct
or inverse) in n sector, with see-saw dominance
or not.
On this basis we found that for LA there is
still a significant preference for hierarchy vs
anarchy
G.A., F. Feruglio, I. Masina, hep-ph/0210342
Previous related work Haba,Murayama
Hirsch,King Vissani Rosenfeld,Rosner Antonelli
et al.
39
Hierarchy for masses and mixings via horizontal
U(1) charges.
Froggatt, Nielsen '79
Principle
A generic mass term
q1, q2, qH U(1) charges of R1, L2, H
R1m12L2H
is forbidden by U(1) if q1q2qH?0
U(1) broken by vev of "flavon field q with U(1)
charge qq -1. The coupling is allowed if vev q
w, and w/Ml we get
Dcharge
R1m12L2H (q/M) q1q2qH
m12 -gt m12 lq1q2qH
Hierarchy More Dcharge -gt more suppression (l
small)
One can have more flavons (l, l', ...) with
different charges (gt0 or lt0)etc -gt many versions
40
2nd
3rd
1st fam.
With suitable charge assignments all
relevant patterns can be obtained
?10 (3, 2, 0) ?5 (2, 0, 0) ?1 (1,-1, 0)
Equal 2,3 ch. for lopsided
Recall u 10 10 deT 5 10 nD 5 1MRR 1 1
No structure for leptons
No automatic det23 0
all charges positive
Automatic det23 0
not all charges positive
41
l6 l5 l3 l5 l4 l2 l3 l2 1
All mass matrix entries are specified as a power
of l times a o(1) coefficient
m
We adopt a statistical approach and generate all
coeff.s as complex random numbers reif? with f
0,2p and
?? 0.5,2 (default) or 0.8,1.2, 0.95,1.05
or 0,1
(real numbers are also considered for comparison)
For each model we compute the success rate over
many trials for falling in the exp. allowed
windows
(boundaries close to 3s limits)
For each model the l, l values are optimised
0.01ltrlt0.2 Ue3lt0.2 0.24lttan2q12lt0.89 0.33lt
tan2q23 lt3.3
LA
42
Results for the LA solution with see-saw
dominance
Scale Srates100
A Anarchy SA Semi-anarchy H Normal
Hierarchy IH Inv. Hierarchy
1 or 2 refer to models with 1 or 2 flavons
of opposite ch.
With charges of both signs and 1 flavon some
entries are zero
Errors are linear comb. of stat. and syst. errors
(varying the extraction procedure interval of r,
real or complex)
Even for LA H, IH are better than SA that is
better than A
43
The trend is more pronounced with no see-saw mn
generated directly from LTmnL 5 5
With no-see-saw H coincide with SA
Note we always include the effect of
diagonalising charged leptons
44
Some distributions
ll0.25
We see that IH tends to predict maximal
solar mixing angle q12 Only compatible because
of ch. lepton diagonalisation
o(l2)
o(l2)
r
Ue3
1 o(l2)
tan2q12
tan2q23
If data drift away from maximal q12, IH will be
rapidly disfavoured
ch. lepton mixing small because me small
45
ll0.2
The main problem of Anarchy is Ue3 (as expected)
In all models the distr. for tan2q23 is flat
ll0.35
Ue3
r
tan2q12
tan2q13
46
l4 l2 l2 l2 1 1 l2 1 1
The main advantage of SA vs A is for Ue3
mn
Y5 (2,0,0)
Det230(1)
works when r is small by chance
Ue3 OK
ll0.2
r
Ue3
tan2q12
tan2q13
47
Summing up
n masses are consistent with the standard way
beyond the SM SUSY and GUTs
Recent exp progress Dm2sol went closer to
Dm2atm less hierarchy smaller
upper limit on absolute mass Smn?lt 0.69 eV
????? 6
Crucial clues s13 small (how small?)
disfavours anarchy s23 maximal, s12
large not maximal disfavours inv. hierarchy
0n???????????????????? degenerate
ns intermediate?? inverted
hierarchy small??? normal
hierarchy
48
Sterile n's from extra dimensions
LSND??
Large extra spacetime dimensions Gravity
propagates in all dim. (bulk) SM particles on a
4-dim brane
Context
dn4 (msR)n(MP/ms)2 msTeV
The Planck mass MP is actually at ms because
flux lines escape in extra dim
Assume 1 VERY large dim 1/R0.01eV (n-1)
smaller (1/rTeV)
(msR)(msr)n-1(MP/ms)2
(Dm2atm)1/2
nsterile SUSY partners of gravitational
moduli (string th.). Also propagate in the bulk
Arkani-Hamed et al Benakli,Smirnov
Dvali,Smirnov Faraggi,Pospelov Mohapatra et
al Ioannisian, Pilaftsis Ioannisian, Valle
Barbieri et al Lukas et al Dienes,Sarcevic

49
A "physical" picture for ns ns has KK
recurrences
Good Features
with mass
and mixes with L (lepton doublet)
The suppression factor (ms/MP) arises
automatically from the bulk volume!
No violation of L conservation? Too rigid one
introduces Majorana masses in bulk
Interference among a few KK states can help in
making the spectrum compatible with solar data.
Note L (cosm. const.) L(2 10-3 eV)4 from
WL0.6-0.7
But needs 1/Rlt0.01eV or R gt 0.02mm close to
existing limits!!
50
31 fit not improved going to 3n
Problems
GUT's? Connection with GUT's What forbids (on
the brane) mn 1/msnLTl?LHH ?? Recall that ms
is small ms TeV L conserv. on the brane must
be ad-hoc imposed ne, nm, nt ?? We need 3 ns
Only 1 extra dimensions has problems (linear
evolution of gauge couplings from 0.01eV to
TeV) Antoniadis, Bachas Arkani-Hamed et al but
for more large extra dim
high KK states do not decouple fast enough.
Mixing large. Compromise d2??