Transcript Document

Lecture 2:
Is the total lepton number conserved? Are neutrinos Dirac
or Majorana fermions?
Here is then what we know today about the masses and mixing of
the elementary fermions. The neutrino sector is really strange, i.e.
very different compared to the charged fermions.
CKM matrix is nearly diagonal
PMNS matrix is `democratic’, i.e. matrix elements are
of similar magnitude, except Ue3 which is evaluated
here near the middle of its allowed range
Neutrinos are many (at least 6) orders of magnitude lighter than the charged fermions.
To solve the dilemma of `unnaturally’ small neutrino mass we can give
up on renormalizability and add operators of dimension d > 4 that are
suppressed by inverse powers of some scale L but are consistent with
the SM symmetries.
Weinberg already in 1979 (PLR 43, 1566) showed that there is only
one dimension d=5 gauge-invariant operator given the particle content
of the standard model:
L(5) = C(5)/L (LceH)(HTeL) +h.c.
Here Lc = LTC, where C is charge conjugation and e = -it2. This
operator clearly violates the lepton number by two units and
represents neutrino Majorana mass
L(M) = C(5)/L v2/2 (nLc nL) + h.c.
If L is larger than v, the Higgs vacuum expectation value, the
neutrinos will be `naturally’ lighter than the charged fermions.
The energy scale L is more or less the energy above which the
effective operator expressions above are no longer valid.
In order to estimate the magnitude of L suppose that
C(5) ~ O(1) and neutrino mass ~ 0.1 eV. Then
L~ v2/mn ~ 1015 GeV
It is remarkable, but perhaps a coincidence, that this scale L
is quite near the scale at which the running gauge coupling
constants meet MGUT ~ 1015-16 GeV.
*
A summary of various ways the mass terms can be constructed is shown
here following the lectures by S. Willenbrock, hep-ph/0410370
The most popular theory of why neutrinos are
so light is the —
See-Saw Mechanism
(Gell-Mann, Ramond, Slansky (1979), Yanagida(1979), Mohapatra, Senjanovic(1980))
n
Very
heavy
neutrino
}
{
Familiar
light
neutrino
NR
Here are some formulae using the formalism just shown trying to
explain what it is all about.
How can we tell whether the total lepton number is
conserved?
A partial list of processes where the lepton number would be violated:
Neutrinoless bb decay: (Z,A) -> (Z+-2,A) + 2e(-+), T1/2 > ~1025 y
Muon conversion: m- + (Z,A) -> e+ + (Z-2,A), BR < 10-12
Anomalous kaon decays: K+ -> p-m+m+ , BR < 10-9
Flux of ne from the Sun: BR < 10-4
Flux of ne from a nuclear reactor: BR < ?
Observing any of these processes would mean that the lepton
number is not conserved, and that neutrinos are massive
Majorana particles.
It turns out that the study of the 0nbb decay is by far the most
sensitive test of the total lepton number conservation, so we
restrict further discussion to this process.
APS Joint Study on the Future of
Neutrino Physics (2004)
(physics/0411216)
We recommend, as a high priority, a phased program
of sensitive searches for neutrinoless double beta
decay (first on the list of recommendations)
The answer to the question whether neutrinos are
their own antiparticles is of central importance, not
only to our understanding of neutrinos, but also to
our understanding of the origin of mass.
Whatever processes cause 0nbb, its observation
would imply the existence of a Majorana mass term:
Schechter and Valle,82
(n)R
e–
W
e–
0nbb
u
d d
u
nL
W
By adding only Standard model interactions we obtain
(n)R  (n)L Majorana mass term
Hence observing the 0nbb decay guaranties that n are massive Majorana
particles.
Double bb decay is observable because even-even nuclei are more
bound than the odd-odd ones ( due to the pairing interaction)
Atomic masses of A=136 nuclei
136Xe
and 136Ce are stable against b decay, but unstable
against bb decay (b-b- for 136Xe and b+b+ for 136Ce)
In double beta decay two neutrons imbedded in the
ground state of an even-even nucleus are simultaneously
transformed into two protons plus two electrons (plus
possibly something else).
The phenomenon exists thanks to the pairing interaction
between nucleons that makes even-even nuclei more
bound than the odd-odd ones.
The 2nbb decay, in which two electron plus two
antineutrinos are emitted, is allowed by the usual rules,
while the neutrinoless decay 0nbb violates lepton number
conservation and is forbidden in the Standard Model.
Symbolic representation of the two bb decay modes
virtual state of the intermediate nucleus
From G. Gratta
virtual state of the intermediate nucleus
Candidate Nuclei for Double Beta Decay
Q (MeV)
Abund.(%)
48Ca48Ti
4.271
0.187
76Ge
76Se
82Se82Kr
2.040
2.995
7.8
9.2
96Zr96Mo
3.350
3.034
2.013
2.8
9.6
11.8
136Xe136Ba
2.802
2.228
2.533
2.479
7.5
5.64
34.5
8.9
150Nd150Sm
3.367
5.6
100Mo100Ru
110Pd110Cd
116Cd116Sn
124Sn124Te
130Te130Xe
All candidate
nuclei on this
list have Q > 2MeV.
The ones with an
arrow are used
in the present
or planned large
mass experiments.
One can distinguish the two modes by measuring the sum electron energy.
Ultimately, though, the 2n decay is an unavoidable background to the 0nbb.
With 2% resolution:
ratio
1:106
ratio
1:100
from S. Elliott
If (or when) the 0nbb decay is observed two
problems must be resolved:
a)What is the mechanism of the decay,
i.e., what kind of virtual particle is
exchanged between the affected
nucleons (or quarks)?
b)How to relate the observed decay rate
to the fundamental parameters, i.e.,
what is the value of the corresponding
nuclear matrix element?
What is the
nature of the
`black box’?
In other words,
what is the
mechanism of
the 0nbb decay?
Light Majorana neutrino,
only Standard Model
weak interactions
Heavy Majorana neutrino
interacting with WR.
Model extended to include
right-handed current
interactions.
Light or heavy Majorana
neutrino. Model extended
to include right-handed WR.
Mixing extended between
the left and right-handed
neutrinos.
Supersymmetry with
R-parity violation.
Many new particles
invoked. Light
Majorana neutrinos exist
also.
If the first graph (exchange of light Majorana neutrinos, i.e.
the assumption that the `standard’ neutrinos are Majorana)
is responsible, we will be able to determine the neutrino mass
from the 0nbb rate. If any of the other graphs dominate,
we cannot do that.
The relative size of heavy (AH) vs. light particle (AL) exchange
to the decay amplitude is (a crude estimate):
AL ~ GF2 mbb/<k2>,
AH ~ GF2 MW4/L5 ,
where L is the heavy scale and k ~ 50 MeV is the virtual
neutrino momentum.
For L ~ 1 TeV and mbb ~ 0.1 – 0.5 eV AL/AH ~ 1, hence both
mechanism contribute equally. If L >> 1 TeV, the heavy particle
exchange results in unobservably small 0nbb rate.
A possible key in deciding which mechanism dominates is in linking
LNV to LFV violation.
In the standard model lepton flavor conservation (LFV) is a consequence
of vanishing neutrino masses. However, the observation of neutrino
oscillations shows that neutrinos are massive and that the flavor
is not conserved. Hence a more general theory must contain LFV of
charged leptons generated probably at some high scale.
There is a long history of searches for LFV with charged leptons,
like m -> e + g, muon conversion m- + (Z,A) -> e- + (Z,A),
or m+ -> e+ + e+ + e- .
Impressive limits for the branching ratios have been established:
< 1.2x10-11
< 8x10-13
There are ambitious new proposals with much better
sensitivities:
MECO (unfortunately dead now): Bm ->e < 5x10-17 on Al
MEG: Bm -> e+g < 5x 10-14
i.e. improvement by a factor of ~ 1000 - 10000.
The direct effect of neutrino mass is “GIM suppressed”
by a factor of (Dm2n/MW2)2 ~ 10-50 hence unobservable.
g
W
m
n
e
Linking LNV to LFV Summary:
- SM extensions with low ( TeV) scale LNV
**
Left-right symmetric model,
R-parity violating SUSY, etc.
possibly G0nbb unrelated to mbb2
- SM extensions with high (GUT) scale LNV [
** In
absence of fine-tuning or hierarchies
in flavor couplings. Important caveat!
See: V. Cirigliano et al., PRL93,231802(2004)
]
What is the relation of the deduced fundamental parameters and the
neutrino mixing matrix? Or, in other words, what is the relation
between the 0nbb decay rate and the absolute neutrino mass?
As long as the mass eigenstates ni are Majorana neutrinos, the 0nbb
decay will occur, with the rate
1/T1/2= G(Etot,Z) (M0n)2 <mbb>2,
where G(Etot,Z) is easily calculable phase space factor, M0n is the nuclear
matrix element, calculable with difficulties (and discussed later), and
<mbb> = Si |Uei|2 exp(iai) mi,
where ai are unknown Majorana phases (only two of them are relevant).
We can relate <mbb> to other observables related to the
absolute neutrino mass.
minimum mass,
not observable
from observational
cosmology,
M = m1+m2+m3
from b decay
<mbb> vs. the
absolute
mass scale
blue shading:
normal hierarchy,
Dm231 > 0.
red shading:
inverted hierarchy
Dm231 < 0
shading:best fit
parameters, lines
95% CL errors.
Thanks to A. Piepke
Moore’s law of 0nbb decay:
There is a steady progress
in the sensitivity of the
searches for 0nbb decay.
Several experiments that
are funded and almost
ready to go will reach
sensitivity to ~0.1 eV.
There is one (so far
unconfirmed) claim that
the 0nbb decay of 76Ge
was actually observed.
The deduced mass <mbb>
would be then 0.3-0.7 eV.
Determination of <mbb> can be only as accurate as the
accuracy of the nuclear matrix elements M0n.
There are two basic methods:
1) Nuclear shell model (SM) treats complicated configurations,
but only few single-particle states (one shell or less).
2) Quasiparticle Random Phase Approximation (QRPA) can
treat many single-particle states, but only simple
particle-hole configurations.
Most existing calculations are QRPA, only very few are SM.
The spread of calculated values is often used as a measure
of uncertainty. Often, however, it merely reflects a spread
in choice of adjustable parameters in QRPA or other issues
not directly related to nuclear structure (e.g. the treatment
of the short range nucleon-nucleon repulsion)
A provocative question: Do we know at all how large the matrix elements
for the 0nbb decay really are? Or, in other words, why there is so much
variation among the published calculated matrix elements?
from Bahcall et al
hep-ph/0403167 ,
spread of published
values of squared
nuclear matrix element
for 76Ge
This suggests an
uncertainty of as much as
a factor of 5. Is it really
so bad?
QRPA-RQRPA results of Rodin et al, erratum in Nucl. Phys. A793, 213 (2007). Shell model
results from Caurier et al, arXiv:0709.0277. In contrast, a reasonably narrow range.
ISM
ISM
Important source of uncertainty: Cancellation between the
J = 0 (pairing) contribution and the J 0 (broken pairs)
contribution:

The same effects are observed in shell model
and in QRPA.
Pairing part
Broken pairs parts

Due to the cancellation between
the pairing J =0 and broken pairs
J  0 contributions the 0n m.e.
gets most of its contributions
from r < 2fm. Hence the
treatment of short range
effects is important and
reduces the m.e.
This is for 76Ge, other nuclei look
similar. This is a general behavior.
(slide by F. Simkovic)
0nbb half-lives for <mbb> = 50 meV based
on the matrix elements of Rodin et al.
76Ge
82Se
100Mo
130Te
136Xe
(0.8 - 1.2) x 1027 y
(2.2 - 4.0) x 1026 y
(2.0 - 3.8) x 1026 y
(1.7 - 5.2) x 1026 y
(0.4 - 1.3) x 1027 y (no 2n observed yet)
Note: Calculated matrix elements decrease with
increasing A, but the phase-space factors usually
increase, particularly the Coulomb factor, hence
relatively little variation of T1/2 with A.
Note: The sensitivity to <mbb> scales as 1/(T1/2)1/2
The most sensitive double beta decay experiments
to date are based on 76-Germanium.
Heidelberg-Moscow (76Ge) energy spectrum
Q value
Half-life limit: 1.9 x 1025 years (H-M and IGEX)
Majorana neutrinos ruled out for masses greater than ~0.35-1.0 eV
41
bb0n discovery claim
HV. Klapdor-Kleingrothaus, et. al,
Nuclear Instruments and Methods A 553 (2004) 371-406 42
Halflife deduced: 1.50+7.55-0.71 x1025 y at 95%C.L.
What should happen next?
1) In a number of new experiments ( CUORE, EXO,
Majorana, MOON, SuperNEMO, etc) the amount
of source will be increased from the present ~10 kg
to ~100 kg, and the sensitivity from the ~1025 years
to ~1027 y, covering the `degenerate’ mass region.
2) This should open the door for ~ton 0nbb decay
experiments.
3) Next generation of experiments on LFV will extend
the sensitivity considerably. In parallel, running of
LHC will shed light on the existence of particles with
~TeV masses.
4) Hopefully, progress in the nuclear structure calculation
will remove some or most of the uncertainty in the
0nbb nuclear matrix elements.
Spare slides
0nbb candidate isotopes:
Q value and natural abundance
High Q value reduces radioactive backgrounds,
large abundance makes the experiment cheaper.
Nuclear matrix elements for the 2n
decay deduced from measured halflives.
Note the pronounced shell dependence.
1/T1/2 = G(E,Z) (MGT2n)2
easily calculable
phase space factor
From Rodin et al. arXiv:0706.4304. Assessment of the uncertainty in M0n
with QRPA and its variant RQRPA.
Comparison between QRPA and SM:
Note that the SM
results are typically
smaller than QRPA.
it is not yet clear
which is better.
QRPA, RQRPA: V.Rodin, A. Faessler, F. Š., P. Vogel, Erratum, nucl-th/07006.4304
shell model: E. Caurier at al. Rev. Mod. Phys. 77, 427 (2005).
Slide by F. Simkovic, ignore the bottom line on the graph
The two methods, QRPA and SM are very different (slide by Poves (SM)):
The recent results of two leading QRPA groups agree quite well.
The results
differ for
82Se, 96Zr
Is it a
problem of
choosing of
single
particle
energies?