1NASA Hubble Photo

Neutrino Masses and Majorana Neutrinos

Boris Kayser HRI

December, 2016

2

One of the most fundamental questions we ask in elementary particle physics is:

What is the origin of mass?

Most theorists strongly suspect that the origin of the neutrino masses is different from the origin

of the masses of all other known particles.

If true, this would make the neutrinos quite special.

3

As we know, the building blocks of matter are the quarks, the charged leptons, and the neutrinos.

The Standard-Model Higgs field is probably still involved, but there is probably something more —

something way outside the Standard Model —

The discovery and study of the Higgs boson at the LHC has provided strong evidence that the quarks

and charged leptons derive their masses from a coupling to the Higgs field.

But as for the origin of the neutrino masses:

Majorana masses.

4

So, what areMajorana masses?

5

Majorana Masses —First In Pictures

6

A Little Background

The neutrinos of specific flavor, ne, nµ, nt, are defined by W boson decays:

W ne

eW nµ

µW nt

t

7

neDetector

e

nµ

µ

nt

t

As far as we know, when a neutrino of given flavor interacts and creates a charged lepton, that charged

lepton will always be of the same flavor as the neutrino.

nµ

e

but not

Summary: The only charged lepton !to which να (α = e, μ, or τ) couples is !α.

8

ne ,nµ ,nt Are Not Mass Eigenstates

|na > = S U*ai |ni> .Neutrino of flavor Neutrino of definite mass mia = e, µ, or t Leptonic Mixing Matrix

i

Instead, ne , nµ , and nt are superpositionsof the mass eigenstates, n1, n2, and n3:

(Leptonic Mixing)

9

(Mass)2

n1

n2n3

or

n1

n2n3

Dm221 º m2

2 – m21 = 7.5 x 10–5 eV2, Dm2

32 = 2.4 x 10–3 eV2~ ~

Normal Inverted

The Three – Neutrino (Mass)2 Spectrum

(There may be more mass eigenstates.)

The mass eigenstates, like KShort and KLong, or n1, n2, n3, are linear combinations of the underlying states.

For any fermion mass eigenstate, e.g. n1, the action of its mass term is —

XMass

n1 n1

In quantum field theory, mass terms are often written in terms of underlying states, like K0 and K0,

or ne , nµ , and nt , that are not the mass eigenstates.

Majorana Masses —In Pictures

10

The mass eigenstate must be sent back into itself:

H ν1 = m1 ν1

11

But for an underlying state, e.g. K0, a mass term can potentially have the action —

XMass

K0

For any electrically charged underlying state, such a mass term is forbidden by charge conservation.

Thus, no such terms are allowed for the quarks or the charged leptons.

K0

12

But the neutrinos – alone among the building blocksof matter – are electrically neutral.

Thus, an underlying neutrino n can have a mass term whose action is —

XMass

n

XMass

n

and

Majorana mass term

n

n

Dirac Mass

13

A Dirac mass has the effect: X n

Dirac mass

or X

Dirac mass

nn

13

A Majorana mass has the effect: X

Majorana mass

or X

Majorana mass

Majorana Mass

n

The two kinds of underlying mass terms

n n

n

n

14

When the underlying mass is a Majorana mass,the mass eigenstate neutrino “n1” will be —

n1 = n + n

The Majorana mass term sends this neutrino back into itself, as required for any mass eigenstate particle:

n + nn + nX

MassConsequence: The neutrino mass eigenstates n1, n2, n3

are their own antiparticles.

Majorana neutrinos

ni = ni (for given helicity)

15

Dirac and MajoranaMasses — In Equations

16

A Little More Background

ψ c x( ) =d3 p2π( )3

12Ep

fp, sup, se−ip⋅x + fp, svp, se

ip⋅x( )s∑∫

By comparison, the charge conjugate ψc(x) = – i γ2ψ*(x) of the field ψ(x) is —

ψ x( ) =d3 p2π( )3

12Ep

fp, sup, se−ip⋅x + fp, svp, se

ip⋅x( )s∑∫

Absorbs particle Creates antiparticle

The plane wave expansion of the free field of a fermion ψ is —

ψc(x) = ψ(x; particle antiparticle)

17

If, for given p and s, the antiparticle happens to be identical to the particle,

ψ x( ) =d3 p2π( )3

12Ep

fp, sup, se−ip⋅x + fp, svp, se

ip⋅x( )s∑∫

Absorbs particle Creates particleantiparticle

Technically, ψc(x) = (phase factor) ψ(x) is allowed, but we can, and will, take this phase factor = 1.

When antiparticle and particle are identical, ψc(x) = – i γ2ψ*(x) implies that —

ψc(x) = ψ(x) = ψ(x; particle antiparticle)

18

Expressed in terms of chiral fields, any mass term connects only fields of opposite chirality:

€

g R fLChiral fermion fields

€

j LkL = j RkR = 0Chiral fermion fields

For example —

€

j LkL =1− γ5

2$

% &

'

( ) j

1− γ52

$

% &

'

( ) k = j 1+ γ5

2$

% &

'

( )

1− γ52

$

% &

'

( ) k = 0

Note: Charge conjugating a chiral field reverses its chirality.

The Chiral Structure of Mass Terms

19

Helicity and ChiralityBoth massless and massive spin-1/2 fermions can

be in a state of definite left-handed helicity, or a state of definite right-handed helicity.

However, a massive spin-1/2 fermion cannot be in a state of definite left-handed chirality, or a state of definite right-handed chirality.

Its quantum field can have chiral projections, obtained by applying the chiral projection operators.

But the particle itself cannot have a definite chirality.

20

We will sometimes write neutrino mass terms in terms of chiral projections of fields.

But when talking about a propagating massive neutrino, we will talk about its helicity. It doesn’t have a chirality.

For simplicity, let us treat a world with just one flavor, and correspondingly, just one neutrino mass eigenstate.

Neutrino Mass Terms

Dirac Mass TermThis is the neutrino analogue of a quark mass term.

Suppose and are underlying chiral fields in terms of which the SM, extended to include neutrino mass,

is written. The antineutrinos these fields create are distinct from the neutrinos they absorb.

νL νR

The Dirac mass term in the Lagrangian density is then —

LD = −mDνRνL + h.c. = −mD (νRνL +νLνR )

ν1 ≡νL +νR LD = −mDν1ν1

ν1ν1 = νL +νR( ) νL +νR( ) =νRνL +νLνR

21

In terms of , , since

Added to the SM

€ €

〈ν1 at rest Hm ν1 at rest〉=〈ν1 at rest mD d3x ν1ν1∫ ν1 at rest 〉 = mD

22

LD = −mDν1ν1 is the mass term in the Lagrangian density for a Dirac neutrino mass eigenstate n1 of mass mD.

XMass mD

n1 n1

This mass term has the effect —

Normalization check

23

Majorana Mass Term

ν1ν1 = νR + νR( )c!

"#$%& νR + νR( )

c!"#

$%&= νR( )

cνR +νR νR( )

c

a right-handed Majorana mass term in the Lagrangian density is —

In terms of the underlying field introduced earlier, νR

Notice that . Antineutrino = neutrino. ν1( )c=ν1

In terms of , , sinceν1 ≡νR + νR( )c

LR = −mR

2ν1ν1

Xn n

LR = −mR

2νR( )c

νR + h.c. = −mR

2νR( )c

νR +νR νR( )c"

#$%&'

24

for a Majorana neutrino mass eigenstate n1 of mass mR.LR = −

mR

2ν1ν1

XMass mR

n1 n1

is the mass term in the Lagrangian density

〈ν1 at rest Hm ν1 at rest〉=〈ν1 at rest mR

2d3x ν1ν1∫ ν1 at rest 〉 = mR

The matrix element of is doubled in the Majoranacase, because ν1 can either absorb or create a neutrino, and so can .

ν1ν1

ν1

Normalization check

This mass term has the effect —

25

Lepton Number LThe lepton number L (not lepton flavor) is defined by —

L(n) = L(!–) = – L(n) = – L(!+) = 1.

The SM weak interactions conserve L:

So do Dirac masses. Absent any Majorana masses or non-SM interactions, L is conserved.

but notν ν

e– e+

W W

26

We then have for the neutrino mass eigenstates νi , for given helicity h:

ni(h) ≠ ni(h) Dirac neutrinos

The difference between ni and ni is that —

L(ni) = + 1 .L(ni) = – 1 while

Xn nMajorana masses mix n and n.Thus, they do not conserve L.

There is then nothing to distinguish a ni from a ni.

We then have for the neutrino mass eigenstates νi , for given helicity h:

ni(h) = ni(h) Majorana neutrinos

27

If there are no visibly large non-SM interactions that violate lepton number L, any violation of L

that we might discover would have to come from Majorana neutrino masses.

28

The Standard Model Higgs mechanism for Dirac fermion masses

A coupling constant this much smaller than unity leaves many theorists skeptical.

SM Higgs field

= yHSMνRνL ⇒ y HSM 0νRνL ≡mννRνLLSM

Vacuum expectation value

Coupling constant Must add to the SM

HSM 0≡ v =174 GeV , so ~ ~

Possible Origins of the Masses

y =mν

v0.1 eV

174 GeV10−12

–

29

Possible (Weak-Isospin-Conserving) progenitors of Majorana masses:

Majorana neutrino masses must have a different origin than the masses of quarks and charged leptons.

€

HSM HSM νLcνL , HIW =1νL

cνL , mRνRcνR

Not renormalizable This Higgs not in SM

No Higgs

Majorana neutrino masses cannot arise from a coupling of a neutrino field to one power of HSM.

30

Another Theoretical PrejudiceIf some ν has a mass, and we want to avoid its having a Majorana mass, we have to add

to the SM the right-handed field νR.

mRνRcνR

\ Neutrinos probably have Majorana masses.

Obviously, this is not a proof.

Once νR has been introduced, no SM conservation law forbids the Majorana mass .

31

So long as we are willing to add right-handed neutrino fields to the SM, so that the neutrino

field content is like that of the quarks and charged leptons, there is no reason to expect that neutrinos will not have Dirac mass terms coming from the SM Higgs mechanism, just like

the quarks and charged leptons do.

Then the neutrinos will have both Dirac and Majorana mass terms.

So long as there are Majorana mass terms, the neutrino mass eigenstates will be Majorana particles,

whether or not there are also Dirac mass terms.

(What if L comes from an interaction?)

32

The See-Saw Mechanism

n

NVery

heavy neutrino

}Familiar

light neutrino

{

The See-Saw model is the most popular theory of why neutrinos are so light.

Reference: arXiv:hep-ph/0211134

33

The See-Saw — For One Flavor

€

Μν =0 mD

mD mR

$

% &

'

( ) is called the neutrino mass matrix.

€

Lm = −mDνR0νL

0 −mR2

νR0( )

cνR

0 + h.c.

€

= −12

νL0( )

c,νR

0$

% &

'

( )

0 mDmD mR

$

% &

'

( ) νL

0

νR0( )

c

$

%

& &

'

(

) ) + h.c.

We have used .

€

νL0( )

cmD νR

0( )c

= νR0mDνL

0

(“Zero” denotes an underlying “weak eigenstate” fields.)

We add a to the SM, and include bothDirac and Majorana mass terms:

νR0

34

No SM principle prevents mR from being extremely large.

But we expect mD to be of the same order as the masses of the quarks and charged

leptons.

Thus, we assume that mR >> mD.

35

Lm can be recast in terms of mass eigenstatesby the transformation —

ZT MνZ = Dν

With <<1,

€

ρ ≡ mD mR

€

€

Z ≅1 ρ

−ρ 1%

& '

(

) *

i 00 1%

& '

(

) *

and

€

Dν ≅mD

2 mR 00 mR

$

% &

'

( )

Makes eigenvalues positive

36

Define and .

€

νLNL

#

$ %

&

' ( ≡ Z−1 νL

0

νR0( )

c

#

$

% %

&

'

( (

€

ν

N#

$ % &

' ( ≡

νL + νL( )c

NL + NL( )c

#

$ % %

&

' ( (

Then —

€

Lm = −12

mD2

mRν ν −

12

mRN N

Mass of n Mass of N

Majorana neutrinos

(Mass of n) x (Mass of N) = mD2 ~ mquark or lepton

2

The See-Saw Relation

37

Splitting due to mRDirac neutrino

N mN ~ mR–

n mn ~ mD2 / mR–

The Majorana mass term split a Diracneutrino into two Majorana neutrinos.

What Happened?

4

2

2

4 = 2 + 2

38

The straightforward (type-I) See-Saw model adds to the SM 3 RH neutrinos , and—

L new = −12

Mi ν iR0( )

cν iR

0 +i∑ yαi ναL

0 H 0 − ℓαL0 H −#

$%&ν iR

0

α=e,µ,τi =1,2,3

∑ + h. c.

SM Higgs doublet

SM lepton doubletLarge Majorana

masses

Yukawa coupling matrix

The See-Saw — For Three Flavors

Once H0 develops a nonzero vacuum expectation value (vev), we have a Dirac mass term.

ν iR0

The mass eigenstates will be 3 heavy neutrinos Ni, and 3 light neutrinos νi.

39

€

UMνUT = −v2 y MN−1yT( )

Light n mass eigenvalues

Leptonicmixing matrix

Heavy N mass eigenvalues

The Higgs vev

The See-Saw Relation for Three Flavors

Yukawa coupling matrix

40

The Physics of Dirac, and Especially Majorana,

Neutrinos

41

SM Interactions Of A Dirac Neutrino

n

n

n

n

makes !–

makes !+

Conserved L+1

–1

We have 4 mass-degenerate states:

The weak interaction is Left Handed.( (

These states, when Ultra Rel., do not interact.

42

SM Interactions Of A Majorana Neutrino

n

n

We have only 2 mass-degenerate states:

makes !–

makes !+

An incoming left-handed neutral lepton makes !–.

An incoming right-handed neutral lepton makes !+.

The weak interactions violate parity.(They can tell Left from Right.)

43

“n ” =!+ nW+

=!– nW–

“n ”

Spin

Spin

“n ” and “n ” are produced with opposite helicity.Moreover, as long as they are ultra-relativistic,

For example —

For ultra-relativistic neutrinos, helicityis a stand-in for lepton number.

However, for non-relativistic neutrinos, there can be a big difference between the behavior

of Majorana neutrinos and Dirac neutrinos.

Majorana neutrinos behave indistiguishably from Dirac neutrinos

in almost all situations.

44

45

Let’s take a different look at why it is so hard to find out whether

neutrinos are their own antiparticles.

46

We assume neutrino interactions are correctly described by the SM. Then the interactions conserve L (n ® !– ; n ® !+).

An Idea that Does Not Work[and illustrates why most ideas do not work]

Produce a ni via—

p+ni µ+ÞSpin Pion Rest Frame

bp(Lab) > bn(p Rest Frame)

p+

Þ niµ+ Lab. Frame

Give the neutrino a Boost:

47

The SM weak interaction causes—

Þni

µ+

RecoilTarget at rest

—

Þ

Þ

ni— Þ ,

will make µ+ too.

ni = ni means that ni(h) = ni(h).helicity

= ni

our ni

If

48

Minor Technical Difficulties bp(Lab) > bn(p Rest Frame)

Ep(Lab) En(p Rest Frame)mp mn

Þ Ep(Lab) > 105 TeV if mn ~ 0.05 eV

Fraction of all p – decay ni that get helicity flipped

» ( )2 ~ 10–18 if mn ~ 0.05 eV

Since L-violation comes only from Majorana neutrino masses, any attempt to observe it will be at the mercy of the neutrino masses.

(B.K. & Stodolsky)

iÞ >

~ i

ii

mn

En(p Rest Frame)

49

To Determine Whether

Majorana Masses Occur in Nature,

So That n = n

50

The Promising Approach — SeekNeutrinoless Double Beta Decay [0nbb]

e– e–

Nucl Nucl’

Observation at any non-zero level would imply —

ØLepton number L is not conserved (ΔL = 2)ØNeutrinos have Majorana massesØNeutrinos are Majorana particles (self-conjugate)

51

0nbbe– e–

u d d u

Whatever diagrams cause 0nbb, its observation would imply the existence of a Majorana mass term:

(Schechter and Valle)

\ 0nbb ni = ni

(n)R nL

W W

n n

n ® n : A (tiny) Majorana mass term

52

0nbbe– e–

u d d u

What is inside?

53

nini

W– W–

e– e–

Nuclear ProcessNucl Nucl’

Uei Uei

SM vertex

åi

Mixing matrix

We anticipate that 0nbb is dominated by a diagram with light neutrino exchange

and Standard Model vertices:

“The Standard Mechanism”

54

But there could be other contributions to 0nbb, which at the quark level is the process

dd ® uuee.

An example from Supersymmetry:

d

u u

e e

ee~

~~

d L

g

55

If the dominant mechanism is —

Then —

nini

W– W–

e– e–

Nuclear ProcessNucl Nucl’

Uei Uei

SM vertex

åi

Mixing matrix

Mass (ni)

Amp[0nbb] µ ½å miUei2½º mbb

In the phase convention where νi

c = (no phase factor) νi.( (

56

How does one calculate the amplitude for an

L-nonconversing process involving Majorana fermions?

A blackboard exercise

For example, Amp(0νββ)

57

Why the neutrino mass is involved

0νββ has ΔL = 2. This ΔL must come from a Majorana mass, since the SM interactions conserve L.

SM SM

Viewed as a perturbation in a small Majorana mass mM, what happens is —

58

Now, how large is mββ?

59

(Mass)2

n1

n2n3

or

n1

n2n3

Dm221 º m2

2 – m21 = 7.5 x 10–5 eV2, Dm2

32 = 2.4 x 10–3 eV2~ ~

NH IH

The Three – Neutrino (Mass)2 Spectrum

(There may be more mass eigenstates.)

60

cij º cos qijsij º sin qijThe leptonic mixing matrix U is —

mββ = m1 cos2θ12 cos2θ13eiα1 + m2 sin2θ12 cos2θ13e

iα2 + m3 sin2θ13e−2iδ

θ12, the solar mixing angle, ≈ 33°.θ13, the reactor mixing angle, ≈ 8º.

eν1 ν2 ν3

Majorana phases

IH: mββ ≅m2 1− sin2 2θ12 sin2 α2 −α1

2#

$%

&

'(

€

U =

c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13

$

%

& & &

'

(

) ) )

×diag eiα1 2, eiα2 2,1( )

61

(From Steve Elliott)

mββ

(meV

)

Target of the coming round

Present bound: mββ < (61 – 165) meV. KamLAND-Zen

62

Observation of 0nbb at any non-zero level would imply —

ØLepton number L is not conserved (ΔL = 2)ØNeutrinos have Majorana massesØNeutrinos are Majorana particles (self-conjugate)

But

T1/20νββKnowing the mass spectrum is , and establishing a

would not prove that neutrinos are Dirac particles.bound larger than what corresponds to mbb = 15 meV,

63

(Mass)2 or

}Dm2sol

Dm2atm

}Dm2sol

Dm2atm

Loophole 1: A Doubled Spectrum

The small spacings can be too small to have been detected in neutrino oscillation.

Each pair is split by a tiny Majorana mass, and its contributions to S miUei

2 almost cancel.

64

Loophole 2: 1 eV – Scale Sterile NeutrinosThere are then more than 3 neutrino mass eigenstates.

The exchanges of the new mass eigenstatescontribute to 0nbb, and in particular to —

mbb =½å miUei2½

(The mixing matrix U is now bigger than 3 x 3.)

The extra contributions can cancel those from the 3 confirmed neutrino mass eigenstates, leading to —

mbb = 0.(Girardi, Meroni and Petcov)

65

And there are other loopholes.

Nevertheless, if we know the mass spectrum to be , and we establish a bound —

the possibility that neutrinos are Majorana particles is disfavored.

T1/2(0νββ) > T1/2(0νββ; mbb = 15 meV) ,