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Neutrinoless Double Beta Decay and

Neutrino Mass — A Point of Principle

Neutrinoless Double Beta Decay and

Neutrino Mass — A Point of Principle

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Neutrinoless Double Beta Decay [0]

e–

e–

Nucl Nucl’

This is the promising approach to testing whether neutrinos are their

own antiparticles.

It will be sought with greatly increased sensitivity.

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ii

W– W–

e– e–

Nuclear Process

Nucl Nucl’

Uei

Uei

SM vertex

i

Mixing matrix

we assume that 0 is dominated by

a diagram with Standard Model vertices:

To estimate the sensitivity required —

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the i is emitted [RH + O{mi/E}LH].

Thus, Amp [i contribution] mi

Amp[0] miUei2 m i

ii

W– W–

e– e–

Nuclear Process

Nucl Nucl’

In

Uei

Uei

SM vertex

i

Mixing matrix

Mass (i)

— Down the Garden Path —

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The Trap

This makes it look as if 0 needs mass only because of a helicity

mismatch.

Suppose we had a parity-conserving world,

that treats right-handed and left-handed particles in the same way.

Couldn’t we then have 0 without any mass?No! No!

mass is still mass is still required.required.

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Why?e–

e–

Nucl Nucl’

— manifestly does not conserve L.

But the Standard Model (SM) weak interactions do conserve L. Absent any non-SM L-violating

interactions, the L = 2 of 0 can only come from Majorana neutrino masses, such as —

mL

XL()

RmL( Lc L + LLc)

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W– W–

e– e–

Nuclear Process

Nucl Nucl’

mL

XL()R

Assuming Standard Model vertices, 0 is —

The Majorana neutrino mass term plays two roles:1)Violate L

2)Flip handedness

It will be needed for (1) even when not needed for (2).

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A Parity-Conserving Toy-Model Illustration

Suppose the W couples to the electron and neutrino via a parity-conserving vector current:

€

−LW =g

2Wλ

−e γ λ ν + h.c. =g

2Wλ

− eLγ λ ν L + eRγ λ ν R( ) + h.c.

Suppose the neutrino mass term is also L R invariant:

€

−LM =1

2mM ν L

cν L + ν Rc ν R( ) + mDν Rν L + h.c.

We take mM,D real, mM > mD, and neglect the electron mass.

Suppose there is only one generation.

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The neutrino mass eigenstates are the two Majorana particles —

€

2 =1

2ν L + ν R( ) + ν L + ν R( )

c[ ]

€

1 =1

2ν L −ν R( ) + ν L −ν R( )

c[ ]

, with mass

€

m2 = mM + mD

€

m1 = mM − mD

,

and

, with mass

.

In terms of them —

€

−LW =g

2Wλ

− eLγ λ 1

2ν 2L + ν 1L( ) + eRγ λ 1

2ν 2R −ν 1R( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥+ h.c.

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Consider now the particle-physics part of 0 —

e–e–

W–W–

ii = 1,2

If the electrons both emerge Left-Handed (LH), only the LH leptonic currents contribute.

For this final state, one finds that the amplitude, ALL, obeys —

€

ALL ∝m2

q2 − m22

+m1

q2 − m12≅

m2 + m1

q2=

2mM

q2

The amplitude is proportional to the Majorana mass mM.

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Similarly, if both electrons emerge right-handed, the amplitude, ARR, obeys —

€

ARR ∝2mM

q2

If the electrons emerge with opposite handedness, both the LH and RH currents are involved, and one finds that the amplitude, ALR, obeys —

€

ALR ∝ iq1

q2 − m22

−1

q2 − m12

⎛

⎝ ⎜

⎞

⎠ ⎟≅ iq

m22 − m1

2

q2( )

2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟=

iq

q2( )

24mM mD

The amplitude is proportional to the Majorana mass mM.

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Why ALR is also proportional to the Dirac mass mD: BK, Petcov, and Rosen;

Enqvist, Maalampi, and Mursula

W–

eL–

mM

XL()R

XR

mD W–

eR–

LH current

RH current

The process is —

Two flips of handedness are needed.

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SummarySummary

Absent L-nonconserving interactions from beyond the

Standard Model, 0 requires Majorana neutrino mass.

This mass is needed to introduce L-nonconservation, even if it is not needed for any other reason.