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.
From the neutrino oscillation experiments we know that
neutrinos are massive fermions. We also know that their
mass is small, mn < `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.
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
mn = 3eGF/(21/2 p2 8) mn = 3x10-19 mB [mn/ 1 eV]
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.
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,
Where wP is the plamon frequency. Since in practice (hwP)2 << 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.
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).
Clearly, it is not enough to minimally extend SM by allowing nR, one needs
other `new physics’.
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It is difficult
small mn with
mn ~ L2/2me mn/mB ~ mn/10-18 mB [L(TeV)]2 large mn
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
Note that Majorana neutrinos can have only transitional in flavor
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 .
The usual graph for mn can be expressed in a gauge invariant form:
= CW + CB
m m m
One can now close the loop and obtain a quadratically divergent
contribution to the Dirac mass
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 magnetic
moment mn for Dirac neutrinos more carefully and got
dmn ~ a/(16p) L2/me mn/mB
here dmn is the contribution to the 3x3 neutrino mass
matrix arising from radiative corrections at the one loop
order. The L2 dependence arises from the quadratic
divergence appearing in the renormalization of the d=4
neutrino mass operator.
Requiring that dmn 1 eV and L ~ 1 TeV implies that
mn 10-14 mB
This is several orders of magnitude more stringent
than the experimental upper limits on mn.
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
b These graphs, with X for
mass insertions, contribute
to dmab by ~(ma2 - mb2) mab
This is from Davidson et al. Phys. Lett. B626, 151 (2005).
This is from Bell et al., Phys. Lett. B642, 377 (2006)
mab < 0.3 eV
1) If mn >> 10-14 mB is discovered experimentally
it would imply that neutrinos are Majorana.
2) mn >> 10-14 mB is impossible for Dirac neutrinos
3) If mn >> 10-14 mB is discovered the scale L of
lepton number violation would be well below the
conventional see-saw scale.
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
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)
In nuclear beta decay one can measure the energy and angular distribution
of the electrons and neutrinos (through nuclear recoils) from the decay of
(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
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.
Here are, as examples, two observables expressed in terms of Ci
Fierz interference coeff
Bottom line: improved constraints on CS, CS’, CT, CT’:
can be constrained by considering neutrino
masses. Therefore CS+CS’ and CT+CT’ can be
Plot of CS/CV vs. CS’/CV.
Grey circle: constraint from experiment on |CS|2 + |CS’|2
Thick band at 450: constraint from experiment on CS - CS’
Thin band at -450: constraint from neutrino mass
Same but for of CT/CV vs. CT’/CV.
Again thin band at -450: constraint from neutrino mass
Brief description of the effective field theory procedures.
Below the weak scale we have only SM particles + nR and
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.
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
An example of the relation between the parameter
of b decay and the operator coefficient.
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.
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.
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’.
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:
Again, by considering all possible d=6 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 d=6 operators (shaded boxes) to the
d=4 Dirac neutrino mass operator.
Summary table: Constraints on the gabg stemming from different
d=6 operators and neutrino mass. Indeces 1,2,D etc are flavor,
only two flavors are considered for simplicity.
2 Higgs ope-
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.
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 d=6 operators and d=4 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
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.
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.
For 3 neutrinos (or quarks) the mixing matrix (without the
Majorana phases) is usually parametrized as:
From recent global fits:
c12 = 0.828+0.028-0.035 , s12 = 0.560+0.049-0.043
c23 = 0.742=0.058-0.115 , s23 = 0.671+0.108-0.071
c13 = 0.996+0.004-0.008 , s13 = 0.089+0.108-0.089
d totally unknown
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/√3 = 0.58, i.e. as if n2 is maximally mixed. The m and t lines
are almost identical suggesting another symmetry.
We can contrast this with the CKM matrix for quarks that can be
quite accurately parametrized as
With l ~ Vus ~ 0.22 << 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 eid
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 < 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
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
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.
The mixing angle q13 is restricted by experiment to sin2q13 < 0.05 (90% CL)
(figure from Chen 0706.2168(hep-ph)
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. ”
W. A. Fowler (after T. H. Huxley)
quote borrowed from Gary Steigman
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 -> 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(N+1)/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].
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
In the low-energy effective theory there are only the active
neutrinos, with the mixing matrix U invariant under
U -> 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.
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.