MASSIVE NEUTRINOS

SM predicts massless neutrinos

What is the motivation for considering neutrino masses?

Is the question of the existence of neutrino masses an isolatedone, or is connected to other outstanding questions of particlephysics?

What sort of tests can be performed to know wheater theneutrinos have masses?

TOP-DOWN APPROACH:THEORETICAL MOTIVATIONS IN PARTICLE PHYSICS

nR ‟s are not introduced in the SM just because one wants topredict massless neutrinos.

Gauge symmetry of e.m. interactions massless photons

For massless n no such symmetry principle in SM

Masslessness of n unsatisfactory from a theoretical point of view

Many GUT‟s predict neutrino masses

BOTTOM-UP APPROACH:MOTIVATIONS FROM ASTROPHYSICS

1939: Bethe listed the chain of reactions responsible for burninghydrogen into helium in stellar cores. In these reactions, some electonneutrinos are produced. Since they interact weakly, they leave the starwithout any hindrance, bringing information about stellar core.

Solar neutrino problem: Experiments detect only ̴ 1/3 of the flux ofne expected from detailed calculations.

Neutrino oscillations?ne produced in weak process is not a mass eigenstates, but a

superposition of different mass eigenstates. One the passage from theSun to the Earth, the ne can partially oscillate to some other flavor,producing the solar ne deficit.

ALL NEUTRINOS CANNOT BE MASSLESS

EVIDENCE OF NEUTRINO OSCILLATIONS

Lot of effort since „60s

Finally convincing evidence for “neutrino oscillation”

Neutrinos appear to have tinybut finite mass

QUESTIONS RELATED TO NEUTRINO MASS

Neutrino mixing: gauge eigenstates would be a superposition of themass eigenstates. Non-trivial leptonic mixing matrix V

Generational lepton numbers Le , Lm , Lt , cannot remain globalsymmetries

Possible CP violation in the leptonic sector [the leptonic mixingmatrix V can be complex]

Neutrinos ≡ Antineutrinos? Dirac or Majorana particles?

Neutrino stability. Do neutrinos decay?

Neutrinos have mass

• They have mass. Can’t go at speed of light.

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• What is this right-handed particle?– New particle: right-handed neutrino (Dirac)

– Old anti-particle: right-handed anti-neutrino (Majorana)

With massive neutrinos one can mimic the electron situation,postulating two more states nR and nL . The boosted observator willsee a nR when we see a nL.

Dirac neutrino

Dirac field of the electron. 4 basic spinorseL, eR

eL, eR e+

e-

eL

eR , eR ??

By boosting to a different Lorentz frame, one cannot see a differentcharge on a particle! The boosted observer sees eR

Neutrino will be a Dirac particle with 4 complex degree offreedom

Majorana neutrino

Can we not do without postulating the two new spinor states?

Can‟t the boosted observed see the state ?nR

Unlike the electrons, nL and nR have both zero electric charge.

They differ only by lepton number (L)

But lepton number is not a global symmetry. It does not govern thedynamics. Nothing of sacred about it !

If it is broken, nL and nR can be the boosted counterparts of oneanother. These two spinors can thus constitute the left and right-handed projections of the same fermionic field.

Neutrinos will be two-components Majorana fields

A Majorana neutrino is its own antiparticle

Difference btw Majorana and Weyl neutrino

Both are two component spinors, but…

A Weyl neutrino is massless. nL moves at speed of light. No observercan overtake it and view if as a r.h. object. So a r.h. counterpart of nL

is not necessary to obtain a Lorentz covariant picture. Similarly a nR

does not require its l.h. counterpart. They could have different leptonnumber to distinguish themselves.

A Majorana neutrino has mass. But n≡ n . So the r.h. component of nL

can be nR or nR . Similarly, nL ≡ nL . That is why only nL and nR suffice.They can Lorentz transformed to each other. Neutrino cannot have anyadditive quantum number. The self-conjugacy is the reason why aMajorana particle has half as many degrees of freedom as a Diracparticle.

A Gedanken experiment to distinguish between a Dirac and a Majorananeutrino

Suppose that it were practically possible to put atrest a massive nm with spin-down in the middle ofthe room. If accelerated up to relativisticenergies in the up direction, when it hits the roofcan produce a m- through a CC interaction. Ifaccelerated up to relativistic energies in the downdirection, when it hits the floor it can produce am+ (if it is a Majorana particle) or have nointeraction (if it is a Dirac particle).

Coming to realistic experiments, we will show that oscillationexperiments cannot discriminate Majorana from Dirac neutrinos.The only realistic hope of experimentally discriminating Majoranafrom Dirac neutrino masses is based on the fact that Majoranamasses violate lepton number, maybe give a signal in the futureneutrinoless double beta decay experiments.

LEPTON NUMBER

The absence of a conserved lepton number is evident from the factthat Dirac neutrinos have L=+1 and Dirac antineutrinos have L=-1.Since in Majorana case neutrinos and antineutrinos are the sameobject, it is clear that there cannot be a conserved lepton number.However, since neutrino masses are very small, it is possible to assignto charged leptons and neutrinos an effective total lepton numberwhich is conserved in all the processes that are not sensitive to theMajorana mass of neutrinos. In these processes, neutrinos can beconsidered massless.We have that neutrinos with negative helicity have Leff=+1 andneutrinos with positive helicity Leff=-1, in agreement with theconvention of calling an antineutrino a neutrino with positive helicity.Conservation of effective lepton number in all interactions which arenot sensitive to neutrino mass.If Majorana mass term is considered as a perturbation of themassless Lagrangian, it generates transitions with

D Leff=±2

DIRAC MATRICES

To show the feautres that are characteristic of Dirac, Weyl andMajorana fields, it is convenient to introduce different Dirac matrixrepresentations that are related by unitary transformations. We adoptthe convention

Dirac representation

Weyl representation

Majorana representation

Because the field is real in nature, it is convenient to adopt therepresentation

so that all the components of the Dirac equation are also real.

MASSLESS NEUTRINOS

We work in the Weyl representation. The two-components Weylspinors are defined by

where y is the Dirac spinor

Charge conjugation is defined by

with the choice

Then charge conjugation is

The kinetic termin is written

The Majorana field is defined by the Majorana condition that

Imposed on a four-component spinor

Let us define the two fields by

These fields obviously satisfy the Majorana condition, and are taken as Majorana fields.

Conversely,

The kinetic term can be written

MASSIVE MAJORANA NEUTRINOS

The Lagrangian with a mass term for a Majorana field is given by

by omitting the term for the w field. We assume that m is real.

Defining

satisfies the Majorana condition in an extendend sense

The kinetic term reads

Writing

We obtain

Here, the mass M is complex, but its phase can be absorbed into thephase of yL

The second term breaks the lepton number carried by yL

MASSIVE DIRAC NEUTRINOS

If there are two Weyl fields we can construct a mass term as

If mii=0 the “lepton number” Li-Lj is conserved. If we define thetwo fields yL and yR by

We obtain the conventional Dirac mass term

for the Dirac field

The kinetic term is given by

MASSIVE NEUTRINOS IN WEINBERG-SALAM THEORY

We can introduce a Dirac mass term if nR exists in addition to nL

which is induced by giving the Higgs field f0 a vacuum-expectationvalue through the Yukawa coupling

If there is no nR , the Majorana mass term is the only mass termthat gives the neutrino mass.

Since is a SU(2) triplet (as T=T3=1), the simplest possible massterm is

and the neutrino mass is given by . The Lagrangian isnon-renormalizable, and M is an effective mass. The form of theLagrangian gives a hint to how the neutrino mass is realized,as

SEE-SAW MECHANISM

XnL nLnR nR

M

f f

When the heavy field (M>> < f >) is integrated out, this diagramgives

The seesaw mechanism is perhaps the simplest model that leads aneffective operator ll ff within a renormalisable class of interactions. Letus assume that the mass term is given by

Equivalently, this is obtained by diagonalising the mass matrix

nL nR

nL

nR

where the two rows (coloumn) refer to left- and right-handedneutrinos and the Dirac mass m = f < f > induces mixing betweenthe two sector

Then

(We have reversed the sign of mnL, using the degree of freedom forthe phase factor)

The attractive feature of this model is that the smallness of theneutrino mass can be understood in terms of a large-mass scale M,which often appears in higher unification theories.

SEE-SAW MODELS FOR NEUTRINO MASSES

n : light l.h. Majorana neutrinoN : heavy r.h. Majorana neutrino

Hierarchy problem: mantainingseparate the two mass scales

EFFECTIVE FIELD THEORY APPROACH

Generic new physics too heavy for being directly studied manifests atlow energy as non renormalizable operators (NRO), suppressed byheavy scales . NRO give small corrections, suppressed by powers of E=L, to physics at low energy E, that is therefore well described by arenormalizable SM theory. The introduction of NRO is how the Fermiscale made its first appearance.

History might repeat now. Adding NRO to the SM Lagrangian, Le; Lm;Lt;B are no longer accidentally conserved:

Dim 5 operator (n mass) Dim 6 operator (violate B-L, proton decay)

Rem: How to determine the dimension of an operator

The action

must be dimensionless to be Lorentz invariant

[x] = (energy)-1 [L] = (energy)4 D=4

Scalar field

Df = 1, each derivative introduces dimension 1

Vector field

DA = 1

Spinor field

Dy = 3/2

NEUTRINO MASS & DIM-5 OPERATORS

Let us contruct an SU(2)LX U(1)Y theory of neutrino mass. Since nL

resides inside the lepton doublet lL, without any detailed analysis wecan see that a dimension-5 operator is required: Schematically lL lLcontains the desired neutrino bilinear but it carries hypercharge Y = -1-1 =-2; on the other hands the Higgs doublet f carries hyperchargeY=+1, and so the lowest dimensional operator we can form is of theform llff with dimensions 3/2+3/2+1+1=5.

na nb

v v

lab depends on the model lab ~O(1) , M~MGUT , v=vEW → mn~10-3

D=5 operator violates lepton number → n must be Majorana

→

Many Od>4 op.s with SM fields but Od=5 is UNIQUE!

TREE-LEVEL REALIZATION OF THE SEE-SAW MECHANISM

Type I See-SawNR fermionic singlet

Type II See-SawD scalar triplet

Type III See-SawtR fermionic triplet

Minkowski, Gell-Mann, Ramond,Slansky, Yanagida, Glashow, Mohapatra, Senjanovic, …

Linearly prop to YD

suppressed by m/M2

Magg, Wetterich, Lazarides,Shafi, Mohapatra, Senjanovic, Schecter, Valle, …

Foot, Lew, He, Joshi, Ma, Roy, …, Bajc, Nemevsek, Senjanovic, Dorsner, Fileviez-Perez

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Suggests existence of high scales

• To obtain

m3~(Dm2atm)1/2

Λ~1015GeV!

• Hints at physics of very high scales

Neutrino mass may be probing unification

Λ

NEUTRINO MASS IN GRAND UNIFIED THEORIES

G SU(3)C x SU(2)L x U(1)Y SU(3)C x U(1)Q

energyMX MW

SO(10)

SU(4) x SU(2)L x SU(2)R

SU(5)

SU(4) x SU(2) x U(1)

SU(3) x SU(2) x U(1)

In the minimal version of SU(5) no nR and B-L conservation. Massless nR. Mass term “by hand” (like in SM)

SO(10)

B-L gauge symmetry to be broken at some scale

Room for nR !

n naturally acquire mass

NEUTRINOLESS DOUBLE BETA DECAY

If neutrinos are Majorana particles is possible the neutrinolessdouble beta decay (0n2b)

Violation of lepton number of two units (DL=2)