2.3 Neutrino Oscillations
2.3.1 Neutrino Masses
All of the physics of neutrino mixing (oscillations) is defined by the neutrino mass matrix, just as down-quark mixing is defined by the quark mass matrix in Section 2.2.3. Neutrino mass is not part of the GWS electroweak theory and so neutrino mass terms need to be added to the Lagrangian.
There are numerous theories which can introduce the neutrino mass terms, such as the various seesaw mechanisms, but they all must eventually produce terms with the same form. Two classes of mass terms that can be written because neutrinos are neutral particles: Dirac and Majorana. The two classes differ in the relationship between the neutrino and antineutrino.
2.3.1.1 Dirac Mass
In a Dirac mass term, neutrinos are treated as a 4-component spinor like all other fermions, with left- and right-handed particles and antiparticles which are distinct from one another:
νL, νR, ν¯L, ν¯R. (2.69)
The mass term in the Lagrangian has the form:
LDmass=−X
l0,l
¯
νl0LMlD0lνlR+ h.c. (2.70)
where MD is a 3×3 complex matrix and l0, l run over {e, µ, τ}, the neutrinos that couple to the weak force (the ‘flavor basis’). This mass term preserves the invariance of the Lagrangian under
9Electron scattering is sensitive to all flavors of neutrino viaZ0 exchange, but νe’s can also interact via W− exchange.
22 Physics of Neutrinos and Antineutrinos global lepton number transformations
νlL→eiΛνlL, νlR→eiΛνlR, l→eiΛl, (2.71)
¯
νlL→e−iΛν¯lL, ν¯lR→e−iΛν¯lR, ¯l→e−iΛ¯l, (2.72) and consequently conserves total lepton number. In order to determine the physical (i.e. real, single- valued) masses, the matrixMDmust be diagonalized:
MD=U†m V (2.73)
where mis a real, positive diagonal matrix, mij =miδij,with mi>0. The neutrino fields can be rewritten in this ‘mass basis,’
νlL=
3
X
i=1
UliνiL (2.74)
νlR=
3
X
i=1
VliνiR (2.75)
where, again, l ∈ {e, µ, τ} and three neutrino mass states have been presumed. When the mass LagrangianLDmass is rewritten in the mass basis, it takes on the form of a standard mass term,
LDmass=−
3
X
i=1
miν¯iνi. (2.76)
The unitary matrix, U, is called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [15, 56].
2.3.1.2 Majorana Mass
Unlike a Dirac fermion, a Majorana fermion only has two components: the charge-conjugate of the left-handed particle is the right-handed particle. Put another way, the particle is its own antiparticle.
νL, ν¯R≡(νL)c =Cν¯LT (2.77)
whereC is the charge-conjugation operator. Neutrinos are the only Standard Model fermions that can be Majorana particles since all other fermions have electric charges which distinguish the particles from the antiparticles.
By definition, a Lagrangian mass term is a Lorentz-invariant product of the left- and right-handed components of a field. Thus, it must be demonstrated that (νL)c is, in fact, a right-handed field.
The handedness of a field is defined by its behavior under multiplication byγ5:
γ5ψL=−ψL, γ5ψR=ψR. (2.78)
Let us, therefore, examine the behavior of the field in question, (νL)c,
γ5(νL)c=γ5Cν¯LT (2.79)
= ¯νLCTγ5T
T
(2.80) UsingCT =−C, Cγ5TC−1=γ5, the expression becomes
γ5(νL)c=− ν¯LCγT5C−1CT (2.81)
=−(¯νLγ5C)T (2.82)
Finally, using the relation ¯νLγ5=νL,
γ5(νL)c=−(νLC)T (2.83)
=CνLT (2.84)
= (νL)c (2.85)
showing that (νL)cbehaves like the right-handed component of a field and can be used to construct a mass term. That term has the form:
LMmass=−1 2
X
l0,l
¯
νl0LMlM0l(νlL)c+ h.c. (2.86)
or, in matrix form,
LMmass=−1
2ν¯LMM(νL)c+ h.c., (2.87) where MM is a complex matrix and νLT = (νeL νµL ντ L). Unlike the Dirac mass term, this term is not invariant under global gauge transformations. While the left- and right-handed Dirac fields transformed in the same way (Equation 2.71), the relationship between the left- and right-handed Majorana neutrinos requires that
νlL→eiΛνlL, νlR= (νlL)c→e−iΛ(νlL)c, (2.88) and Equation 2.86 is not invariant under this transformation. Consequently, a theory with Majorana neutrinos does not conserve total lepton number.
24 Physics of Neutrinos and Antineutrinos The relationship between the left- and right-handed fields in this mass term provide a constraint on the form ofMM:
¯
νLMM(νL)c= ¯νLMMCν¯TL (2.89)
=−¯νL MMTCTνLT (2.90)
= ¯νL MMTCνLT (2.91)
using the anticommuntation properties of fermion fields. This implies that
MM = MMT (2.92)
or that MM is symmetric. Thus, when the matrix is diagonalized to find the physical neutrino masses, only one unitary matrix,U, is needed:
MM =U m UT (2.93)
where, again,U is a unitary matrix andmij =miδij,withmi>0. Substituting into the Lagrangian, LMmass=−1
2ν¯LU m UTCνLT+ h.c. (2.94)
=−1
2U†νLm U†νLc−1
2(U†νL)cm U†νL (2.95)
=−1
2ν¯Mm νM (2.96)
where
νM =U†νL+ U†νLc=
ν1
ν2
ν3
, (2.97)
once again producing the canonical mass term by transforming the fields into the mass basis (compare to Equation 2.76). It is clear from Equation 2.97 that
νMc=νM, (2.98)
and thus that
νic=νi. (2.99)
This relation is known as the Majorana condition, and it states that in the mass basis the neutrino and antineutrino are the same particle.
As in the Dirac case (Equation 2.74), the mass and flavor bases are related to one another by
W−W−n n
p p
e− e−
¯ νe
¯ νe
νe/¯WWνe−−n n
p p
e− e−
Figure 2.5: Feynman diagrams for 2ν(left) and 0ν(right) double beta decay. Note, the 0νββis only possible if the same particle can act asνe and ¯νe.
the mixing matrixU
νL=UνLM or νlL=
3
X
i=1
UliνiL. (2.100)
It is significant to note that whether neutrinos are Dirac or Majorana, in the end there are still three neutrinos of definite mass m1, m2, m3 which are related to the three neutrinos with definite SU(2)L⊗U(1)Y transformation properties (the flavor neutrinos, νe, νµ, ντ) by the PMNS mixing matrixU. In fact, in either case only left-handed neutrinos and right-handed antineutrinos are ever observed since only those states couple to the electroweak force.
The only known way, experimentally, to distinguish between Dirac and Majorana neutrinos is to search for the rare neutrinoless double beta decay process (0νββ). In a typical double beta decay, two neutrons transition to two protons by emitting two antineutrinos and two electrons (via two W−’s),
2n→2p++ 2e−+ 2¯νe. (2.101)
However, if the neutrino and antineutrino are the same particle, then instead of releasing two ¯νe’s, one virtual ¯νe could be emitted and then absorbed as aνe all within the decay (since the emission of a ¯νeand absorption of aνe are equivalent processes), leaving
2n→2p++ 2e−. (2.102)
Note that in this process, total lepton number is violated by 2, which is forbidden if neutrinos are Dirac particles but is allowed if neutrinos are Majorana particles. Consequently, the observation of neutrinoless double beta decay is a sensitive test of lepton number conservation and hence the nature of the relationship between the neutrino and antineutrino.