3) Exploring the consequences of finite neutrino mass:

1
3) Exploring the consequences of finite neutrino
mass Neutrino magnetic moments, muon decay, beta
decay. Scattered comments on particle physics
models and the possible path to the nSM.
2
From the neutrino oscillation experiments we know
that neutrinos are massive fermions. We also know
that their mass is small, mn lt a few eV. These
findings, by themselves represent New Physics
beyond the SM, since they imply the existence of
the right-handed neutrino nR. In this lecture we
will explore the consequences of these facts on
other observables that would also indicate
existence of New Physics beyond the SM.
3
Neutrino magnetic moment (or more generally
neutrino coupling to the electromagnetic field)
and the distinction between the Dirac and
Majorana neutrinos.
Neutrino mass and magnetic moment are intimately
related. In the orthodox SM with massless
neutrinos magnetic moments vanish. However, in
the minimally extended SM containing
gauge-singlet right-handed neutrinos one finds
that mn is nonvanishing, but unobservably small,
mn 3eGF/(21/2 p2 8) mn 3x10-19 mB mn/ 1 eV
4
Typically, magnetic moment could be observed in
n-e scattering using its characteristic electron
recoil kinetic energy T dependence
selm pa2m2/me2 (1-T/En)/T
However, a nonvanishing mn will be recognizable
only if the elm. cross section is comparable with
the well understood weak interaction
cross section. Thus, the magnitude of mn that can
be probed in this way is
Considering realistic values of T it would be
difficult to reach Sensitivities below 10-11
mB. The present limits are about 10-10mB.
5
Limits on mn can be also derived from bounds on
the unobserved energy loss in astrophysical
objects. For sufficiently large mn the rate
for plasmon electromagnetic decay into nn pairs
would conflict with such bounds. However,
plasmons can also decay weakly in nn pairs. Thus
the sensitivity of this probe is again limited by
the size of the weak rate, leading to
Where wP is the plamon frequency. Since in
practice (hwP)2 ltlt meT, this bound is stronger
(10-12mB) than the laboratory limits. In any
case one cannot expect to reach in foreseeable
future much better limits on mn.
6
The interest in mn and its relation to mn dates
from 1990 when it was suggested that there is an
anticorrelation between the neutrino flux
observed in the Cl (Davis) experiment, and the
solar activity (number of sunspots that follows a
11 year cycle). A possible explanation of this
was proposed by Voloshin, Vysotskij and Okun,
with mn 10-11 mB and its precession in solar
magnetic field. Even though the effect does not
exist, the possibility of a large mn and small
mass was widely discussed. I like to describe a
model independent constraint on the mn that
depends on the magnitude of mn and moreover
depends on the charge conjugation properties of
neutrinos, i.e. makes it possible, at least in
principle, to decide between Dirac and Majorana
nature of neutrinos. See Bell et al., Phys. Rev.
Lett. 95, 151802 (2005) Phys. Lett. B642, 377
(2006) and Davidson et al. Phys. Lett. B626, 151
(2005).
7
Clearly, it is not enough to minimally extend SM
by allowing nR, one needs other new physics.
It is difficult to reconcile small mn with large
mn
mn L2/2me mn/mB mn/10-18 mB L(TeV)2 eV
8
To overcome this difficulty Voloshin (88)
proposed existence of a SU(2)n symmetry in which
nL and (nR)c form a doublet. Under this symmetry
mn is forbidden but mn is allowed. For Dirac
neutrinos such symmetry is broken by weak
interactions, but for Majorana neutrinos it is
broken only by the Yukawa couplings. Note that
Majorana neutrinos can have only transitional in
flavor magnetic moments. Also note, that in
flavor basis the mass term for Majorana neutrinos
is symmetric but the magnetic moments are
antisymmetric. In the following I show that the
existence of nonvanishing mn leads through loop
effects to an addition to the neutrino mass dmn
that, naturally cannot exceed the magnitude of mn
.
9
The usual graph for mn can be expressed in a
gauge invariant form
H H
W B
g
CW
CB
m
m
m
One can now close the loop and obtain a
quadratically divergent contribution to the Dirac
mass
m
10
In Bell et al., Phys. Rev. Lett. 95,
151802 (2005) we evaluated the relation between
the scale L the associated neutrino mass dmn and
the magneticmoment mn for Dirac neutrinos more
carefully and got dmn a/(16p) L2/me
mn/mBhere dmn is the contribution to the 3x3
neutrino massmatrix arising from radiative
corrections at the one loop order. The L2
dependence arises from the quadraticdivergence
appearing in the renormalization of the
d4neutrino mass operator.Requiring that dmn
1 eV and L 1 TeV implies thatmn ? 10-14
mBThis is several orders of magnitude more
stringent than the experimental upper limits on
mn.
11
The case of Majorana neutrinos is more subtle due
to different symmetries of mab (mass is
symmetric) and mab (magnetic moment is
antisymmetric) in flavor indeces.
These two one-loop graphs add to zero for mab for
Majorana neutrinos
mab
mab
These graphs, with X for mass insertions,
contribute to dmab by (ma2 - mb2) mab
mab
mab
This is from Davidson et al. Phys. Lett. B626,
151 (2005).
12
This is from Bell et al., Phys. Lett. B642, 377
(2006)
Thus, for mab lt 0.3 eV
13

• Conclusions
• If mn gtgt 10-14 mB is discovered experimentally
• it would imply that neutrinos are Majorana.
• 2) mn gtgt 10-14 mB is impossible for Dirac
  neutrinos
• 3) If mn gtgt 10-14 mB is discovered the scale L of
  
• lepton number violation would be well below
  the
• conventional see-saw scale.

14
Note that the distinction between Dirac and
Majorana does not require processes that
violate lepton number in rates, just amplitudes.
For example the neutrino g decay
Angular distribution of photons in the lab system
with respect to the neutrino beam direction is
where a 0 for Majorana and a-1 for Dirac and
left handed couplings
15
Neutrino mass constraints on the parameters
that characterize new physics in the case of
nuclear beta decay and muon decay (Caltech PhD
theses of Peng Wang and Rebecca Erwin, 2007 and
Erwin et al, Phys. Rev. D 75, 033005 (2007))
See also simplified versions with analogous
considerations in Prezeau and Kurylov, Phys. Rev.
Lett. 95, 101802 (2005) and Ito and Prezeau,
Phys. Rev. Lett. 94, 161802 (2005)
16
In nuclear beta decay one can measure the energy
and angular distribution of the electrons and
neutrinos (through nuclear recoils) from the
decay of oriented nuclei. (this is a classic
field, first observation of parity violation was
reached this way, Wu, 1957).
If you wish to look for new physics you can use
a hamiltonian
In SM CV CV 1, CA CA -1.26 and all
other vanish
Alternatively, one can use
e,d L,R
In SM aLLV Vud cosqCabbibo , and all other
vanish
17
Here is the famous formula of Jackson, Treiman
and Wyld (1957) that describes the distribution
of electrons and neutrinos in energy and angle
from the beta decay of oriented nuclei. The
observables are A,a,B,b,D, etc.
18
Here are, as examples, two observables expressed
in terms of Ci
Fierz interference coeff
electron-neutrino angular correlation coeff.
19
Bottom line improved constraints on CS, CS, CT,
CT
can be constrained by considering neutrino
masses. Therefore CSCS and CTCT can be
constrained.
0.14
Plot of CS/CV vs. CS/CV. Grey circle
constraint from experiment on CS2
CS2 Thick band at 450 constraint from
experiment on CS - CS Thin band at -450
constraint from neutrino mass
0.14
Same but for of CT/CV vs. CT/CV.
Again thin band at -450 constraint from neutrino
mass
20
Brief description of the effective field theory
procedures. Below the weak scale we have only SM
particles nR and
Effective lagrangian
An example of the one-loop graph for matching of
O(6) to O(4)M (the mass term for Dirac
neutrinos) solid lines - fermions, dashed - Higgs.
O(6)
An example of the contribution of one of the O(6)
operators to the mass term
Coefficients of the operators in the
lagrangian are constrained by the magnitude of
neutrino mass
An example of the relation between the
parameter of b decay and the operator coefficient.
21
Constraints on aedg for the Dirac case (Majorana
are similar for this). They are in units of
(v/L)2 x (mn/1 eV) becoming stronger when
L increases or mn decreases.
Based on that CS CS/CV ? 4 x 10-6 and
CT CT/CA ? 8 x 10-5 This then
constrains these new physics couplings
significantly more than the b-decay observables
alone.
22
Re/m branching ratio for p decay into e and
m pb branching ratio for the pion beta decay
p ? p0 e ne CKM unitarity, requiring that
using experimental Vud and Vus and constraining
the new physics contribution to Vud.
23
Now analogous consideration for the purely
leptonic process of the muon decay, m- ? e- nm
ne or its analog for m The energy and
angular distribution of the decay electron is
Here x 2Ee/mm electron energy as a fraction
of its maximum value. The Michel parameters have
SM values r 3/4, d 3/4 and x 1. Deviations
from these values (as well as more subtle effects
involving electron polarization) would be signals
of new physics.
24
Effective Langrangian is parametrized
as where g S,V,T as before and a,b L,R
again. In SM only nonvanishing a parameter is
gVLL 1, all others vanish. By restricting the
other g parameters we could constrain
the observable Michel values, e.g.,
From experiments, deviations from the SM values
are not more than
25
Again, by considering all possible d6 operators,
evolving them and considering possible loop
graphs that contribute to the neutrino mass
operators, we arrive at constraints on the
operator coefficients and through them on ggab
One loop graphs for the contribution of the d6
operators (shaded boxes) to the d4 Dirac
neutrino mass operator.
26
Summary table Constraints on the gabg stemming
from different d6 operators and neutrino mass.
Indeces 1,2,D etc are flavor, only two flavors
are considered for simplicity.
12 four- fermion operators constrain gT,SLR,RL
4 two- fermion and 2 Higgs ope- rators
Note that the operators O(6)F,,112D and
O(6)F,,221D do not contribute to the neutrino
mass, so there are no naturalness bounds on their
coefficients.
27
As an example lets follow steps needed for the
first row in the Table The entries follow from
the inequalities
Where dm1Dn is the contribution to the neutrino
mass term matrix element 1,D which, naturally,
should not exceed the upper bound on neutrino
mass. This, in turn follows from the matching of
d6 operators and d4 mass operator
k 1/4 and 1/8 for S,T
Here fBB is the Yukawa coupling of the charged
lepton of flavor B, mB fBB v/?2
28
The bounds discussed above for b-decay and
m-decay can be avoided by fine tuning,
cancellation between individual dmn. That is,
however, unnatural. Also, the bounds typically
do not constrain the observables (A,B, a, b, etc
in b-decay or r, d, etc. in m-decay ) but
affect the parameters aabg or gabg in the general
lagrangian that are usually obtained in general
fits.
29
Lets finish this lecture with few remarks
concerning the mixing matrix, its possible
symmetries, the remaining undetermined
parameters, their significance, and how to
possibly relate them to other things.
30
For 3 neutrinos (or quarks) the mixing matrix
(without the Majorana phases) is usually
parametrized as
From recent global fits c12 0.8280.028-0.035
, s12 0.5600.049-0.043 c23 0.7420.058-0.115
, s23 0.6710.108-0.071 c13
0.9960.004-0.008 , s13 0.0890.108-0.089 d
totally unknown
31

• The mixing matrix therefore as of now looks like
  this
• n1 n2
  n3
• e 0.82 0.56
  0.0(0.15)
• U m -0.42 0.61
  0.67
• t 0.38 -0.56
  0.74
• Here the first entry is for q13 0 and the
  (second) for q13 0.15,
• i.e. the maximum allowed value. (The possible
  deviation of q23
• from 450 is neglected as well as error bars on
  all mixing angles,
• also, the CP phase d is assumed to vanish.)
• Note that the second column n2 looks like a
  constant
• made of 1/v3 0.58, i.e. as if n2 is maximally
  mixed. The m and t lines
• are almost identical suggesting another symmetry.

32
We can contrast this with the CKM matrix for
quarks that can be quite accurately parametrized
as
With l Vus 0.22 ltlt 1. The CKM is nearly
diagonal. Note also that the product of the first
and third line is made of terms l3 only, so it
is ideal for the unitarity triangle tests. The
neutrino mixing matrix is very different,
essentially democratic, perhaps with the
exception of the upper right corner, the angle
q13. Note that the Dirac CP phase d
always appears in the combination s13 e?id
33

• Once more what is known empirically?
• a) Two mass differences Dm212 8 x 10-5 eV2
• Dm322 ?
  Dm312 2.5 x 10-3 eV2
• their ratio, Dm212/ Dm322 0.03 is a
  small number.
• b) Two mixing angles, q23 450, q12 350 are
  large and
• reasonably well determined. The third mixing
  angle,
• q13 is only constrained from above, sinq13 lt
  0.15.
• Perhaps sinq13 is another small parameter.
• Maybe, one can try (many people do) to find
  symmetries in the
• mixing matrix and make expansion in these small
  parameters
• to estimate the magnitude of q13 and of the CP
  phase d

34

• In fact, the neutrino mixing matrix resembles the
  tri-bimaximal matrix, which can be a convenient
  zeroth order term of such
• expansions. (compared to the empirical
  matrix above the
• the last line and last column were
  multiplied by -1)
• n1
  n2 n3
• e (2/3)1/2 (1/3)1/2
  0
• U m -(1/6)1/2 (1/3)1/2
  -(1/2)1/2
• t -(1/6)1/2
  (1/3)1/2 (1/2)1/2

35
Model building
Basic idea a) Bottom-up, i.e., use the
experimental data and guess some underlying
symmetries. Based on them find values or
ranges for the so far unknown parameters. b)
Top-down, i.e., try to construct some more
fundamental theory that would agree with
the known facts and would also predict the
missing entries. There is no shortage of
attempts in both categories, with a wide range of
predictions.
36
The mixing angle q13 is restricted by experiment
to sin2q13 lt 0.05 (90 CL)
(figure from Chen 0706.2168(hep-ph)
37
Once the angle q13 is experimentally determined
the following beautiful quote will be applicable
to most of these models
The terrible tragedies of science are
the horrible murders of beautiful theories
by ugly facts.
quote borrowed from Gary Steigman
W. A. Fowler (after T. H. Huxley)
38
How many parameters we should eventually
determine CKM matrix for quarks In the quark
mass eigenstate basis one can make a phase
rotation of the u-type and d-type quarks, thus V
-gt eiF(u) V e-iF(d) , where F(u) diag(Fu ,
Fc , Ft) , etc. The N x N unitary matrix V has
N 2 parameters. There are N(N-1)/2 CP-even angles
and N(N1)/2 CP-odd phases. The rephasing
invariance above removes (2N-1) phases, thus
(N-1)(N-2)/2 CP-odd phases are left. So, for N
3 there are 3 angles and 1 CP phase The usual
convention is to have the angles qi in
0,p/2 and the phases di in 0,2p.
39
Now for neutrinos Consider N massive Majorana
neutrinos that belong to the weak doublets Li .
In addition there are (presumably) also N weak
singlet neutrinos, that in the see-saw mechanism
are heavy (above the electroweak scale). In the
low-energy effective theory there are only the
active neutrinos, with the mixing matrix U
invariant under U -gt e-iF(E) U hv Here F(E)
involves the free phases of the charged leptons
and hn is a diagonal matrix with allowed
eigenvalues 1 and -1. It takes into account the
allowed rephasing for Majorana fields. Thus U
contains N(N-1)/2 angles in 0,p/2,
(N-1)(N-2)/2 Dirac CP-odd phases and (N-1)
Majorana CP-odd phases. (N(N-1)/2 phases
altogether.) These phases are in 0,2p. The
matrix U (often called PMNS) is responsible for
neutrino oscillations in low-energy experiments.
40
At high-energy see-saw theory there are two
mixing matrices W (no analog for quarks) and V
(different from the PMNS matrix U). They
contain together N(N-1) angles and N(N-1)
phases. (i.e. for N 3 there are 6 angles a 6
phases) However, in processes that involve only
active n and charged leptons only V appears,
parameters in W are irrelevant. In processes
that involve active n and Nh only W
appears. Leptogenesis depends only on W and on
the eigenvalues of Nh , it is independent on V
and on the (N-1) relative phases between W and V.