14. Neutrino Masses, Mixing, and Oscillations

1

14. Neutrino Masses, Mixing, and Oscillations
Revised September 2023 by M.C. Gonzalez-Garcia (YITP, Stony Brook; ICREA, Barcelona; ICC,
U. of Barcelona) and M. Yokoyama (UTokyo; Kavli IPMU (WPI), UTokyo).

14.1
14.2

Neutrinos in the Standard Model: Massless Neutrinos . . . . . . . . . . . . . . . .
Extending the Standard Model to Introduce Massive Neutrinos . . . . . . . . . . .
14.2.1 Dirac Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2.2 The See-saw Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2.3 Light Sterile Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2.4 Neutrino Masses from Generic New Physics . . . . . . . . . . . . . . . . . . .
14.3 Lepton Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.4 Mass-Induced Flavour Oscillations in Vacuum . . . . . . . . . . . . . . . . . . . . .
14.5 Propagation of Massive Neutrinos in Matter . . . . . . . . . . . . . . . . . . . . . .
14.5.1 The Mikheyev-Smirnov-Wolfenstein Effect for Solar Neutrinos . . . . . . . . .
14.6 Experimental Study of Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . .
14.6.1 Solar Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.6.2 Atmospheric Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.6.3 Accelerator Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.6.4 Reactor Antineutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.7 Combined Analysis of Experimental Results: The 3ν Paradigm . . . . . . . . . . .
14.7.1 3ν Oscillation Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.7.2 3ν Oscillation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.7.3 Convention-independent Measures of Leptonic CP Violation in 3ν Mixing . .
14.8 Beyond 3ν: Additional Neutrinos at the eV Scale . . . . . . . . . . . . . . . . . . .
14.9 Laboratory Probes of ν Mass Scale and its Nature . . . . . . . . . . . . . . . . . .
14.9.1 Constraints from Kinematics of Weak Decays . . . . . . . . . . . . . . . . . .
14.9.2 Dirac vs. Majorana: Neutrinoless Double-beta Decay . . . . . . . . . . . . . .
14.9.3 Experimental Search for Neutrinoless Double-beta Decay . . . . . . . . . . .

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3
4
5
5
5
6
8
10
13
15
15
19
22
27
31
32
35
36
37
40
40
42
44

14.1 Neutrinos in the Standard Model: Massless Neutrinos
The gauge symmetry principle is one of the pillars of the great success of modern particle
physics as it establishes an unambiguous connection between local (gauge) symmetries and forces
mediated by spin-1 particles. In the Standard Model (SM) of particle physics the strong, weak, andelectromagnetic interactions are connected to gauge symmetry under SU (3)C × SU (2)L × U (1)Y
where C stands for colour, L for left-handedness, and Y for hypercharge. The SM gauge symmetryis spontaneously broken to SU (3)C × U (1)EM where U (1)EM couples to the electromagnetic charge
QEM = TL3 +Y (TL3 is the weak isospin which is the third generator of SU (2)L ). The model explains
all the interactions of the known fermions once they are assigned to a well defined representation of
the gauge group. The construction and tests of the Standard Model as a gauge theory are coveredin Chapter 9 “Quantum chromodynamics” and Chapter 10 “Electroweak model and constraints
on new physics” of this Review. Here we emphasize that the gauge invariance principle requiresthat all terms in the Lagrangian, including the mass terms, respect the local symmetry. This has
important implications for the neutrino and in particular for the question of the neutrino mass 1 .
1
The physics of massive neutrinos has been the subject of excellent books such as [1–5] and multiple review
articles. The contents of the present review is built upon the structure and the contents of the review articles [6, 7].

S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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14. Neutrino Masses, Mixing, and Oscillations

In the SM, neutrinos are fermions that do not have strong nor electromagnetic interactions.
Consequently, they are! singlets of the subgroup SU (3)C × U (1)EM . They are part of the lepton
ν`doublets LL` =where fL is the left-handed component of the fermion f , fL = PL f ≡`L
1−γ5
2 f . In what follows we will refer as active neutrinos to neutrinos that are part of these lepton
doublets. In the SM there is one active neutrino for each charged leptons, ` = e, µ, τ . SU (2)L gauge

invariance dictates the form of weak charged current (CC) interactions between the neutrinos andtheir corresponding charged leptons and neutral current (NC) among themselves to be:
g X+−LCC = √ν̄L` γ µ `−L Wµ + h.c. ,2 `X
g−LNC =
ν̄L` γ µ νL` Zµ0 .
2 cos θW `(14.1)(14.2)In the above equations, g is the coupling constant associated with SU (2) and θW is the Weinberg
angle. Equations(14.1) and (14.2) describe all the neutrino interactions in the SM. In particular,Eq.(14.2) determines the decay width of the Z boson into light (mν ≤ mZ /2) left-handed neutrinos
states. Thus from the measurement of the total decay width of the Z one can infer the number ofsuch states. At present the measurement implies Nν = 2.984 ± 0.008 (see Particle Listing). As aresult any extension of the SM should contain three, and only three, light active neutrinos.
Sterile neutrinos are defined as having no SM gauge interactions, that is, they are singlets of thecomplete SM gauge group. Thus the SM, as the gauge theory able to describe all known particle
interactions, contains no sterile neutrinos.
The SM with its gauge symmetry and the particle content required for the gauge interactions,that is, in the absence of SM singlets, respects an accidental global symmetry that is not imposed
but appears as a consequence of the gauge symmetry and the representation of the matter fields:
Gglobal= U (1)B × U (1)Le × U (1)Lµ × U (1)Lτ ,SM(14.3)

where U (1)B is the baryon number symmetry, and U (1)Le ,Lµ ,Lτ are the three lepton flavour symmetries. The total lepton number, Le + Lµ + Lτ , is then also an accidental symmetry since it is a
subgroup of Gglobal
SM . This fact has consequences that are relevant to the question of the neutrinomass as we argue next.
In the SM, the masses of the fermions are generated via a Yukawa coupling of the scalar Higgs
doublet φ with a fermion right-handed and left-handed component. The former is an SU (2)L singlet,
the latter is part of a doublet. For leptons, we can build such a term coupling the left-handed lepton
doublets LL with the right-handed charged lepton fields ER :− LYukawa,lep = Yij` L̄Li φERj + h.c. .

(14.4)

After spontaneous symmetry breaking these terms lead to charged lepton massesvm`ij = Yij` √ ,2(14.5)

where v is the vacuum expectation value of the Higgs field. However, since the model does notcontain right-handed neutrinos, no such Yukawa interaction can be built for the neutrinos, which
are consequently massless at the Lagrangian level.
In principle, a neutrino mass term could be generated at loop level. With the particle content of
the SM the only possible neutrino mass term that could be constructed is the bilinear L̄L LcL , where
31st May, 2024

14. Neutrino Masses, Mixing, and Oscillations

3T
LcL is the charge conjugated field, LcL = C L¯L and C is the charge conjugation matrix. However
this term is forbidden in the SM because it violates the total lepton symmetry by two units andtherefore it cannot be induced by loop corrections because it breaks the accidental symmetry of thecmodel. Also, because U (1)B−L is a non-anomalous subgroup of Gglobal
SM , the bilinear L̄L LL , cannot
be induced by nonperturbative corrections either since it breaks B − L.
We conclude that within the SM neutrinos are precisely massless. Consequently one must go
beyond the SM in order to add a mass to the neutrino.

14.2 Extending the Standard Model to Introduce Massive Neutrinos
From the above discussion, we conclude that it is not possible to construct a renormalizable
mass term for the neutrinos with the fermionic content and gauge symmetry of the SM. The obvious
consequence is that in order to introduce a neutrino mass in the theory one must extend the particle
content of the model, depart from gauge invariance and/or renormalizability, or do both.
As a matter of fact, neutrino mass terms can be constructed in different ways. In the following
we shall assume to maintain the gauge symmetry and explore the different possibilities to introduce
a neutrino mass term adding to the SM an arbitrary number of sterile neutrinos νsi (i = 1, . . . m).
In the SM extended with the addition of m number of sterile neutrinos one can construct twogauge invariant renormalizable operators leading to two types of mass terms
1c− LMν = MDij ν̄si νLj + MN ij ν̄si νsj+ h.c. ,2(14.6)

where ν c is the neutrino charge conjugated field (defined in section 14.1). MD is a complex matrixof dimension m × 3 and MN is a symmetric m × m matrix.
The first term is generated after spontaneous electroweak symmetry breaking from Yukawa
interactions,v
Yijν ν̄si φ̃† LLj ⇒ MDij = Yijν √ ,(14.7)
2in a similar way to Eqs.(14.4) and (14.5) for the charged fermion masses. It is correspondingly
called a Dirac mass term. It conserves total lepton number but it can break the lepton flavour
number symmetries.
The second term in Eq.(14.6) is a Majorana mass term and it differs from the Dirac mass terms
in several relevant aspects. First, it is a singlet of the SM gauge group and, as such, it can appear
as a bare mass term in the Lagrangian. Second, since it involves two neutrino fields (right-handedin this case), it breaks lepton number by two units. In general, such a term is not allowed if theneutrinos carry any additive conserved charge.
It is possible to rewrite Eq.(14.6) as:T
10MD− LMν = (~νLc , ~νs )M D MN2!

!

~νL+ h.c. ≡ ~ν c Mν ~ν + h.c. ,~νsc(14.8)

where ~ν = (~νL , ν~sc )T is a (3 + m)-dimensional vector. The matrix Mν is complex and symmetric 2 .
Thus it can be diagonalized by a unitary matrix V ν of dimension(3 + m), so(V ν )T Mν V ν = diag(m1 , m2 , . . . , m3+m ) .

(14.9)

One can express the original weak eigenstates in terms of the resulting 3 + m mass eigenstates~νmass = (V ν )†~ν ,2(14.10)

Notice that Eq.(14.8) corresponds to the tree-level neutrino mass matrix. Corrections are induced at the loop
level, which in particular lead to non-vanishing ν̄Lc νL entry [8].
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414. Neutrino Masses, Mixing, and Oscillations

and in terms of the mass eigenstates, Eq.(14.8) takes the form:−LMν =
=



X
1 3+m
c
c
mk ν̄mass,kνmass,k + ν̄mass,k νmass,k
2 k=1
X
1 3+m
mk ν̄M k νM k ,2 k=1(14.11)

wherec
νM k = νmass,k + νmass,k= (V ν †~ν )k + (V ν †~ν )ck .

(14.12)So these states obey the Majorana conditioncνM = νM,(14.13)and are referred to as Majorana neutrinos. The Majorana condition implies that only one field
describes both neutrino and antineutrino states, unlike the case of a charged fermion for which
particles and antiparticles are described by two different fields. So a Majorana neutrino can be
described by a two-component spinor unlike the charged fermions, which are Dirac particles, andare represented by four-component spinors.
Inverting Eq.(14.12) we can write the weak-doublet components of the neutrino fields as:νLi = PL

3+m
X

Vijν νM j

i = 1, 2, 3 ,(14.14)j=1

where PL is the left projector.
In the following, we will discuss some interesting particular cases of this general framework:
light Dirac neutrinos in Sec.14.2.1, and light Majorana neutrinos and the see-saw mechanism in
Sec.14.2.2. A special case of the second example is the possibility of light-sterile neutrinos discussed
in Sec.14.2.3. In Sec.14.2.4 we shall discuss the effective generation of neutrino masses from nonrenormalizable operators (of which the see-saw mechanism is a particular realization).
14.2.1 Dirac Neutrinos
Imposing MN = 0 is equivalent to imposing lepton number symmetry on the model. In doing
this only the first term in Eq.(14.6), the Dirac mass term, is allowed. If there are only three sterile( m = 3), we can identify them with the right-handed component of a four-spinor neutrino field.
In this case the Dirac mass term can be diagonalized with two 3 × 3 unitary matrices, V ν and VRν
as:VRν † MD V ν = diag(m1 , m2 , m3 ) .
(14.15)The neutrino mass term can be written as:− L Mν =3X

mk ν̄Dk νDk ,(14.16)

νDk = (V ν †~νL )k + (VRν †~νs )k ,(14.17)k=1

where
so the weak-doublet components of the neutrino fields areνLi = PL

3
X

Vijν νDj .

j=1
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i = 1, 2, 3 .

(14.18)

14. Neutrino Masses, Mixing, and Oscillations

5

Let us stress that in this case both the low-energy matter content and the assumed symmetries
are different from those of the SM. Consequently, the SM is not even a good low-energy effective
theory. Furthermore, this scenario does not explain the fact that neutrinos are much lighter thanthe corresponding charged fermions, because all of them acquire their mass via the same mechanism.
14.2.2 The See-saw Mechanism
If the mass eigenvalues of MN are much higher than the scale of electroweak symmetry breakingv, the diagonalization of Mν leads to three light neutrinos, νl , and m heavy neutrinos, N :1
1− LMν = ν̄l M l νl + N̄ M h N ,2
2(14.19)with−1T
M l ' −VlT MD
MN
MD Vl ,

and



Vν '

†
∗ − 1M −1 M
MN
1 − 21 MD
D
N
−1
−MN
MD Vl



M h ' VhT MN Vh ,(14.20)†∗ −1 V
MD
MN
h

Vl




†
∗ −1 VMN
1 − 12 MN −1 MD MDh,(14.21)

where Vl and Vh are 3 × 3 and m × m unitary matrices respectively. From Eq.(14.20) we see that−1,
the masses of the heavier states are proportional to MN while those of the lighter ones to MN
hence the name see-saw mechanism [9–13]. Also, as seen from Eq.(14.21), the heavy states aremostly right-handed while the light ones are mostly left-handed. Both the light and the heavy
neutrinos are Majorana particles. Two well-known examples of extensions of the SM leading to asee-saw mechanism for neutrino masses are SO(10) Grand Unified Theories [10, 11] and left-rightsymmetryc models [13].
In this case, the SM is a good effective low energy theory. Indeed the see-saw mechanism is aparticular example of a full theory whose low energy effective realization is the SM with three light
Majorana neutrinos which we describe in Sec.14.2.4.
14.2.3 Light Sterile Neutrinos
If the scale of some ns ≤ m eigenvalues of MN are not higher than the electroweak scale, the low
energy spectrum contains ns additional light states with a large admixture of sterile component.
As in the case with Dirac Neutrinos, the SM is not a good low energy effective theory: there are
more than three (3+ns ) light neutrinos, and they are admixtures of doublet and singlet fields. As
in the general case, both light and heavy neutrinos are Majorana particles.
14.2.4 Neutrino Masses from Generic New Physics
Under the generic hypothesis that new physics (NP) beyond the SM only manifests itself directly above some scale ΛNP , we can consider that the SM is an effective low energy theory which
is valid to describe the physical world at energies well below ΛNP with the same gauge group,
fermionic spectrum, and the pattern of spontaneous symmetry breaking of the SM. However, thisis an effective theory, holding only till energy below ΛNP , and consequently does not need to berenormalizable. In this case, the low energy Lagrangian can contain non-renormalizable higher
dimensional terms whose effect will be suppressed by powers 1/Λdim−4.
NP
In this approach, the least suppressed NP effects at low energy are expected to come from
dim= 5 operators. With the SM fields and gauge symmetry one can only construct the following
set of dimension-five terms
ν 


Zij
O5 =
L̄Li φ̃ φ̃T LC(14.22)
Lj + h.c. .
ΛNPThis set violates (14.3), which poses no problem since, in general, there is no reason for the NP torespect the accidental symmetries of the SM. In particular, it violates the total lepton number by
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614. Neutrino Masses, Mixing, and Oscillations

two units, and after spontaneous symmetry breaking it generates a bilinear neutrino field term:− L Mν =

ν
Zij
v2
c
ν̄Li νLj+ h.c. .
2 ΛNP(14.23)This is a Majorana mass term (see Eq.(14.8)). It is built with the left-handed neutrino fields andwith mass matrix:2ν v(Mν )ij = Zij.
(14.24)
ΛNP
We conclude that Eq.(14.24) would arise in a generic extension of the SM and that neutrino masses
are very likely to appear if there is NP. Comparing Eq.(14.24) and Eq.(14.5), we also find that thescale of neutrino masses is suppressed by v/ΛNP when compared to the scale of charged fermion
masses, which provides an explanation for their smallness. Furthermore, both total lepton number
and the lepton flavour symmetry U (1)e × U (1)µ × U (1)τ are broken by Eq.(14.24), which means
that, generically, in the absence of additional symmetries on the coefficients Zij , we can expect
lepton flavour mixing and CP violation as we discuss in the next section.
Finally, we notice that, as mentioned in Sec.14.2.2, a theory where the NP is composed of mheavy sterile neutrinos, provides an specific example of a theory which at low energy contains threelight mass eigenstates with an effective dim-5 interaction of the form (14.22) with ΛNP = MN . In
this case, the NP scale is the characteristic mass scale of the heavy sterile neutrinos.

14.3 Lepton Mixing
Let us start by considering n = 3 + m massive neutrino states and denote the neutrinomass eigenstates by (ν1 , ν2 , ν3 , . . . , νn ). The neutrino interaction eigenstates are denoted by ~ν =(νLe , νLµ , νLτ , νs1 , . . . , νsm ). We label the corresponding mass and interaction eigenstates for the
charged leptons as (e, µ, τ ) and (eI , µI , τ I ), respectively. The Lagrangian for the leptonic charged
current interactions in the mass basis takes the form:




ν1
ν 
 2
 
g
µ  ν3 − LCC = √ (ēL , µ̄L , τ̄L )γ U   Wµ+ + h.c. ,2 .. 
 . (14.25)νnwhere U is a 3 × n matrix [14–16]. It satisfies the unitary condition
U U † = I3×3 .

(14.26)

However, in general U † U 6= In×n .
In the interaction basis, the mass terms for the leptons are:




eIR


− LM = [(ēIL , µ̄IL , τ̄LI )M` µIR  + h.c.] − LMν ,τRI(14.27)

with LMν given in Eq.(14.8). M` can be diagonalized with two 3 × 3 unitary matrices V ` and VR`
which satisfy†
V ` M` VR` = diag(me , mµ , mτ ) .
(14.28)Then for the charged leptons we have
− L M` =3Xm`k `¯k `k ,k=1
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(14.29)14. Neutrino Masses, Mixing, and Oscillations

7
with

†

†

`k = (V ` `IL )k + (VR` `IR )k .

(14.30)Inverting the equation above we find that the weak-doublet components of the charged lepton fields
are`ILi = PL

3
X

Vij` `j .

i = 1, 2, 3

(14.31)j=1

From Eqs.(14.14), (14.18) and (14.31) we find that the mixing matrix U can be expressed as:†ν
Uij = P`,ii Vik` Vkj(Pν,jj ).

(14.32)†The matrix V ` V ν contains a number of phases that are not physical. Three of them are eliminatedby the diagonal 3 × 3 phase matrix P` that absorbs them in the charged lepton mass eigenstates. If
neutrinos are Dirac states, further n − 1 are similarly eliminated by absorbing them in the neutrinomass eigenstates with the diagonal n × n phase matrix Pν . For Majorana neutrinos, Pν = In×n
because one cannot rotate by an arbitrary phase a Majorana field without physical effects. If one
rotates a Majorana neutrino by a phase, this phase will appear in its mass term, which will nolonger be real. Consequently, the number of phases that can be absorbed by redefining the mass
eigenstates depends on whether the neutrinos are Dirac or Majorana particles. Altogether for n ≥ 3
Majorana [Dirac] neutrinos, the U matrix contains a total of 6(n − 2) [5n − 11] real parameters, ofwhich 3(n − 2) are angles, and 3(n − 2) [2n − 5] can be interpreted as physical phases.
The possibility of arbitrary mixing between massive neutrino states was first discussed in thecontext of two neutrinos in Ref. [17] (the possibility of two mixed massless flavour neutrino states
had been previously considered in the literature [18], and the possibility of mixing between neutrino
and antineutrino states even earlier, in the seminal paper of Pontecorvo [19]). For the case where
only mixing between two generations is considered with n = 2 distinct neutrino masses, the U
matrix is 2 × 2 and contains one mixing angle if the neutrinos are Dirac and an additional physicalphase if they are Majorana.
If there are only n = 3 Majorana neutrinos, U is a 3 × 3 matrix analogous to the CKM
matrix for the quarks [20, 21], but due to the Majorana nature of the neutrinos it depends on six
independent parameters: three mixing angles and three phases. In this case the mixing matrix canbe conveniently parameterized as:


 

1
0
0
c13

 
0
U = 0 c23 s23  · 
0 −s23 c23−s13 eiδCP

 

 

0 s13 e−iδCPc12 s12 0
eiη1
 
 
1
0
 · −s12 c12 0 ·  0
0
0 1
0
0
c13

0
eiη2
0



0

0 , (14.33)
1

where cij ≡ cos θij and sij ≡ sin θij . The angles θij can be taken without loss of generality to lie
in the first quadrant, θij ∈ [0, π/2] and the phases δCP , ηi ∈ [0, 2π]. This is to be compared to thecase of three Dirac neutrinos. In this case, the Majorana phases, η1 and η2 , can be absorbed in theneutrino states so the number of physical phases is one (similar to the CKM matrix). Thus we canwrite U as:c12 c13U = −s12 c23 − c12 s13 s23 eiδCPs12 s23 − c12 s13 c23 eiδCPs12 c13c12 c23 − s12 s13 s23 eiδCP−c12 s23 − s12 s13 c23 eiδCPs13 e−iδCPc13 s23  .
c13 c23(14.34)This matrix is often called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix.
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814. Neutrino Masses, Mixing, and Oscillations

Notice that when the charged leptons have no other interactions than the SM ones, one canidentify their interaction eigenstates with the corresponding mass eigenstates up to phase redefinition. This implies that, in this case, U is just a 3 × n sub-matrix of the unitary neutrino mass
diagonalizing matrix V ν .
Finally, let us point out that for the case of 3 light Dirac neutrinos, the procedure above leads
to a unitary U matrix for the light states. But for three light Majorana neutrinos, this is not thecase since the full spectrum contains states which are heavy and are not in the low energy spectrum
as seen, for example, in Eq.(14.21). This implies that, strictly speaking, the parametrization in
Eq.(14.33) is not valid to describe the flavour mixing of the three light Majorana neutrinos in thesee-saw mechanism. The violation of unitarity, however, is rather small, of the order O(MD /MN )as seen in Eq.(14.21). It is also severely constrained experimentally [22, 23]. For all these reasons,for all practical purposes, we will consider the U matrix for the 3ν mixing case to be unitary
independently of whether neutrinos are Dirac or Majorana particles.

14.4 Mass-Induced Flavour Oscillations in Vacuum
If neutrinos have masses and lepton flavours are mixed in the weak CC interactions, lepton
flavour is not conserved in neutrino propagation [19,24]. This phenomenon is usually referred to as
neutrino oscillations. In brief, a weak eigenstates, να , which by default is the state produced in the
weak CC interaction of a charged lepton `α , is the linear combination determined by the mixing
matrix Un|να i =

X∗
Uαi
|νi i ,

(14.35)i=1where νi are the mass eigenstates, and here n is the number of light neutrino species (implicit inour definition of the state |νi is its energy-momentum and space-time dependence). After travelinga distance L (L ' ct for relativistic neutrinos), that state evolves as:|να (t)i =

n
X

∗
Uαi
|νi (t)i .

(14.36)i=1This neutrino can then undergo a charged-current (CC) interaction producing a charge lepton `β ,να (t)N 0 → `β N , with a probability
Pαβ = |hνβ |να (t)i|2 = |

n
n X
X

∗
Uαi
Uβj hνj |νi (t)i|2 .

(14.37)i=1 j=1q

Assuming that |νi is a plane wave, |νi (t)i = e−i Ei t |νi (0)i, 3 with Ei = p2i + m2i and mi being,respectively, the energy and the mass of the neutrino mass eigenstate νi . In all practical cases
neutrinos are very relativistic ,so pi ' pj ≡ p ' E. We can then write
m2i,(14.38)2E
and use the orthogonality of the mass eigenstates, hνj |νi i = δij , to arrive to the following form for
Pαβ :
Ei =

q

Pαβ = δαβ − 4

p2i + m2i ' p +

n
X

∗ ∗
Re[Uαi UβiUαj Uβj ] sin2 Xij

i<j

+2

n
X

∗ ∗
Im[Uαi UβiUαj Uβj ] sin 2Xij ,(14.39)i<j3
For a pedagogical discussion of the quantum mechanical description of flavour oscillations in the wave package
approach see for example Ref. [3]. A recent review of the quantum mechanical aspects and subtleties on neutrino
oscillations can be found in in Ref. [25].

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14. Neutrino Masses, Mixing, and Oscillations

9
where

(m2i − m2j )L∆m2ij L/E
= 1.267.
(14.40)4E
eV2 m/MeV
If we had made the same derivation for antineutrino states, we would have ended with a similar
expression but with the exchange U → U ∗ . Consequently, we conclude that the first term in theright-hand-side of Eq.(14.39) is CP conserving since it is the same for neutrinos and antineutrinos,
while the last one is CP violating because it has opposite signs for neutrinos and antineutrinos.
Xij =Table 14.1: Characteristic values of L and E for experiments performed
using various neutrino sources and the corresponding ranges of |∆m2 | towhich they can be most sensitive to flavour oscillations in vacuum. SBL
stands for Short Baseline, VSBL stands for Very Short Baseline , MBL
stands for Medium Baseline, and LBL for Long Baseline.
Experiment
Solar
Atmospheric
Reactor
Accelerator

VSBL–SBL–MBL
LBL
SBL
LBL

L (m)
1010
4
10 − 107
10 − 103
104 − 105102
5
10 − 106

E (MeV)1
102 –1051
103 –104103 − 104|∆m2 | (eV2 )10−10−110 − 10−4
1 − 10−3
10−4 − 10−5
> 0.1
10−2 − 10−3

Equation (14.39) is oscillatory in distance with oscillation lengths
Losc0,ij =

4πE,|∆m2ij |(14.41)and with amplitudes proportional to products of elements in the mixing matrix. Thus, neutrinos
must have different masses (∆m2ij 6= 0) and they must have not vanishing mixing (Uαi Uβi 6= 0)
in order to undergo flavour oscillations. Also, from Eq.(14.39) we see that the Majorana phases
cancel out in the oscillation probability. This is expected because flavour oscillation is a total lepton
number conserving process.
Ideally, a neutrino oscillation experiment would like to measure an oscillation probability overa distance L between the source and the detector, for neutrinos of a definite energy E. In practice,neutrino beams, both from natural or artificial sources, are never monoenergetic but have an energy
spectrum Φ(E). In addition, each detector has a finite energy resolution. Under these circumstances
what is measured is an average probabilityR

hPαβ i =dΦdE dEσ(E)Pαβ (E)(E)

R

dΦ
dE dE
σCC (E)(E)= δαβ − 4nX

∗ ∗Re[Uαi UβiUαj Uβj ]hsin2 Xij i + 2

i<j

n
X

(14.42)∗ ∗Im[Uαi UβiUαj Uβj ]hsin 2Xij i .

i<jσ is the cross-section for the process in which the neutrino flavour is detected, and (E) is thedetection efficiency. The minimal range of the energy integral is determined by the energy resolutionof the experiment.
2
It is clear from the above expression that if (E/L)  |∆m2ij | (L  Losc0,ij ) so sin Xij  1, theoscoscillation phase does not give any appreciable effect. Conversely, if L  L0,ij , many oscillation
31st May, 2024

1014. Neutrino Masses, Mixing, and Oscillations

cycles occur between production and detection, so the oscillating term is averaged to hsin2 Xij i =1/2.
We summarize in Table 14.1 the typical values of L/E for different types of neutrino sourcesand experiments and the corresponding ranges of ∆m2 to which they can be most sensitive.
Historically, the results of neutrino oscillation experiments were interpreted assuming twoneutrino states so there is only one oscillating phase; the mixing matrix depends on a single mixing
angle θ, and no CP violation effect in oscillations is possible. At present, as we will discuss in
Sec.14.7, we need at least the mixing among three-neutrino states to fully describe the bulk ofexperimental results. However, in many cases, the observed results can be understood in terms ofoscillations dominantly driven by one ∆m2 . In this limit Pαβ of Eq.(14.39) takes the form [24]Pαβ = δαβ − (2δαβ − 1) sin2 2θ sin2 X .

(14.43)In this effective 2 − ν limit, changing the sign of the mass difference, ∆m2 → −∆m2 , and changing
the octant of the mixing angle, θ → π2 − θ, is just redefining the mass eigenstates, ν1 ↔ ν2 : Pαβmust be invariant under such transformation. So the physical parameter space can be covered witheither ∆m2 ≥ 0 with 0 ≤ θ ≤ π2 , or, alternatively, 0 ≤ θ ≤ π4 with either sign for ∆m2 .
However, from Eq.(14.43) we see that Pαβ is actually invariant under the change of sign of themass splitting and the change of octant of the mixing angle separately. This implies that there
is a two-fold discrete ambiguity since the two different sets of physical parameters, (∆m2 , θ) and(∆m2 , π2 − θ), give the same transition probability in vacuum. In other words, one could not tellfrom a measurement of, say, Peµ in vacuum whether the larger component of νe resides in theheavier or in the lighter neutrino mass eigenstate. This symmetry is broken when one considers
mixing of three or more neutrinos in the flavour evolution and or when the neutrinos traverse
regions of dense matter as we describe in Sec.14.7.1 and Sec.14.5, respectively.

14.5 Propagation of Massive Neutrinos in Matter
Neutrinos propagating in a dense medium can interact with the particles in the medium. The
probability of an incoherent inelastic scattering is very small. For example the characteristic crosssection for ν-proton scattering is of the orderG2 sE
σ ∼ F ∼ 10−43 cm2
π
MeV


2

,

(14.44)

where GF is the Fermi constant and s is the square of the center of mass energy of the collision.
But when neutrinos propagate in dense matter, they can also interact coherently with theparticles in the medium. By definition, in coherent interactions, the medium remains unchanged so
it is possible to have interference of the forward scattered and the unscattered neutrino waves which
enhances the effect of matter in the neutrino propagation. In this case, the effect of the medium is
not on the intensity of the propagating neutrino beam, which remains unchanged, but on the phasevelocity of the neutrino wave, and for this reason the effect is proportional to GF , instead of the G2F
dependence of the incoherent scattering. Coherence also allows decoupling the evolution equationof the neutrinos from those of the medium. In this limit, the effect of the medium is introduced in
the evolution equation for the neutrinos in the form of an effective potential which depends on thedensity and composition of the matter [26].
As an example, let us consider the evolution of νe in a medium with electrons, protons, andneutrons with corresponding ne , np , and nn number densities. The effective low-energy Hamiltonian
describing the relevant neutrino interactions at point x is given by
1
GF
HW = √ J (+)α (x)Jα(−) (x) + J (N )α (x)Jα(N ) (x) ,
4
2




31st May, 2024

(14.45)

14. Neutrino Masses, Mixing, and Oscillations

11

where the Jα ’s are the standard fermionic currents
Jα(+) (x) = ν̄e (x)γα (1 − γ5 )e(x) ,(14.46)Jα(−) (x) = ē(x)γα (1 − γ5 )νe (x) ,(14.47)Jα(N ) (x) = ν̄e (x)γα (1 − γ5 )νe (x)− ē(x)[γα (1 − γ5 ) − 4 sin2 θW γα ]e(x)(p)+ p̄(x)[γα (1 − gA γ5 ) − 4 sin2 θW γα ]p(x)(14.48)(n)− n̄(x)γα (1 − gA γ5 )n(x) ,(n,p)

and gAare the axial couplings for neutrons and protons, respectively.
Let us focus first on the effect of the charged current interactions. The effective CC Hamiltonian
due to electrons in the medium is
GF(e)
HC = √ZDE

d3 pe f (Ee , T ) × he(s, pe ) | ē(x)γ α (1 − γ5 )νe (x)ν̄e (x)γα (1 − γ5 )e(x) | e(s, pe )i

2
Z
D
E
GF
d3 pe f (Ee , T ) he(s, pe ) | ē(x)γα (1 − γ5 )e(x) | e(s, pe )i .
= √ ν̄e (x)γα (1 − γ5 )νe (x)2(14.49)In the above equation, we denote by s the electron spin, and by pe its momentum, and f (Ee , T ) is
the energy distribution function of the electrons in the medium which is assumed to be homogeneousand isotropic and is normalized asZD

d3 pe f (Ee , T ) = 1 .

(14.50)E

We denote by . . . the averaging over electron spinors and summing over all electrons in themedium. Coherence dictates that s, pe are the same for initial and final electrons. The axial current
reduces to the spin in the non-relativistic limit and therefore averages to zero for a background of
non-relativistic electrons. The spatial components of the vector current cancel because of isotropy.
Therefore, the only non-trivial average isZDE

d3 pe f (Ee , T ) he(s, pe ) | ē(x)γ0 e(x) | e(s, pe )i = ne (x) ,(14.51)

which gives a contribution to the effective Hamiltonian√(e)
HC = 2GF ne ν̄eL (x)γ0 νeL (x) .

(14.52)This can be interpreted as a contribution to the νeL potential energy√
VC = 2GF ne .

(14.53)√
Should we have considered antineutrino states, we would have ended up with VC = − 2GF ne Fora more detailed derivation of the matter potentials see, for example, Ref. [3].
With an equivalent derivation, we find that for νµ and ντ , the potential due to its CC interactions
is zero for most media since neither µ’s nor τ 0 s are present, while the effective potential for any
active neutrino due to the neutral current interactions is found to be√hi2
VN C =
GF −ne (1 − 4 sin2 θw ) + np (1 − 4 sin2 θw ) − nn .
(14.54)2
31st May, 2024

1214. Neutrino Masses, Mixing, and Oscillations

In neutral matter, ne = np and the contribution from electrons and protons cancel each other. So
we are left only with the neutron contribution√
VN C = −1/ 2GF nn .
(14.55)After including these effects, the evolution equation for n ultrarelativistic neutrinos propagatingin matter written in the mass basis is (see for instance Ref. [27–29] for the derivation):id~ν
= H ~ν ,dx

H = Hm + U ν † V U ν .

(14.56)

Here ~ν ≡ (ν1 , ν2 , . . . , νn )T , Hm is the the kinetic Hamiltonian,Hm =1
diag(m21 , m22 , . . . , m2n ),2E(14.57)and V is the effective neutrino potential in the interaction basis. U ν is the n × n submatrix ofthe unitary V ν matrix corresponding to the n ultrarelativistic neutrino states. For the three SM
active neutrinos with purely SM interactions crossing a neutral medium with electrons, protons andneutrons, the evolution equation takes the form (14.56) with U ν ≡ U , and the effective potential: √V = diag ± 2GF ne (x), 0, 0 ≡ diag (Ve , 0, 0) .

(14.58)The sign + (−) in Eq.(14.58) applies to neutrinos (antineutrinos), and ne (x) is the electron number
density in the medium, which in general is not constant along the neutrino trajectory so the potential
is not constant. The characteristic value of the potential at the Earth core is Ve ∼ 10−13 eV, whileat the solar core Ve ∼ 10−12 eV. Since the neutral current potential Eq.(14.55) is flavour diagonal,
it can be eliminated from the evolution equation as it only contributes to an overall unobservable
phase.
The instantaneous mass eigenstates in matter, νim , are the eigenstates of the Hamiltonian H in
(14.56) for a fixed value of x, and they are related to the interaction basis by~ν = Ũ (x)ν~m .

(14.59)The corresponding instantaneous eigenvalues of H are µi (x)2 /(2E) with µi (x) being the instantaneous effective neutrino masses.
Let us take for simplicity a neutrino state which is an admixture of only two neutrino species|να i and |νβ i, so the two instantaneous mass eigenstates in matter ν1m and ν2m have instantaneous
effective neutrino massesµ21,2 (x) =
1
∓
2

q

m21 + m22
+ E[Vα + Vβ ]2

(14.60)[∆m2 cos 2θ − A]2 + [∆m2 sin 2θ]2 ,and Ũ (x) is a 2x2 rotation matrix with the instantaneous mixing angle in matter given bytan 2θm =∆m2 sin 2θ
.
∆m2 cos 2θ − A(14.61)In the Eqs.(14.60) and (14.61) A isA ≡ 2E(Vα − Vβ ) ,31st May, 2024

(14.62)

13

14. Neutrino Masses, Mixing, and Oscillations

and its sign depends on the composition of the medium and on the flavour composition of theneutrino state considered. From the expressions above, we see that for a given sign of A themixing angle in matter is larger(smaller) than in vacuum if this last one is in the first (second)
octant. We see that the symmetry about θ = 45◦ which exists in vacuum oscillations between two
neutrino states is broken by the matter potential in propagation in a medium. The expressions
above show that signficant effects are present when A, is close to ∆m2 cos 2θ. In particular, as seenin Eq.(14.61), the tangent of the mixing angle changes sign if, along its path, the neutrino passes
by some matter density region satisfying, for its energy, the resonance condition
AR = ∆m2 cos 2θ .

(14.63)This implies that if the neutrino is created in a region where the relevant potential satisfies A0 > AR
(A0 here is the value of the relevant potential at the production point), then the effective mixing
angle in matter at the production point is such that sgn(cos 2θm,0 ) = − sgn(cos 2θ). So the flavour
component of the mass eigenstates is inverted as compared to their composition in vacuum. In
particular, if at the production point we have A0 = 2AR , then θm,0 = π2 − θ. Asymptotically, for
A0  AR , θm,0 → π2 . In other words, if in vacuum the lightest (heaviest) mass eigenstate hasa larger projection on the flavour α (β), inside a matter with density and composition such that
A > AR , the opposite holds. So if the neutrino system is traveling across a monotonically varying
matter potential, the dominant flavour component of a given mass eigenstate changes when crossing
the region with A = AR . This phenomenon is known as level crossing.
Taking the derivative of Eq.(14.59) with respect to x and using Eq.(14.56), we find that in the
instantaneous mass basis the evolution equation reads:
"

#



1
dŨ (x) md~ν m=diag µ21 (x), µ22 (x), . . . , µ2n (x) − i Ũ † (x)
~ν .
i
dx
2E
dx

(14.64)The presence of the last term, Eq.(14.64) implies that this is a system of coupled equations. So,
in general, the instantaneous mass eigenstates, νim are not energy eigenstates. For constant or
slowly enough varying matter potential this last term can be neglected and the instantaneous mass
eigenstates, νim , behave approximately as energy eigenstates, and they do not mix in the evolution.
This is the adiabatic transition approximation. On the contrary, when the last term in Eq.(14.64)
cannot be neglected, the instantaneous mass eigenstates mix along the neutrino path. This implies
there can be level-jumping [30–33], and the evolution is non-adiabatic.
For adiabatic evolution in matter, the oscillation probability take a form very similar to thevacuum oscillation expression, Eq.(14.39). For example, neglecting CP violation:
Pαβ =i
Ũαi (0)Ũβi (L) exp −
2E
i

X

Z L
0

!2

µ2i (x0 )dx0.

(14.65)

To compute Pαβ in a varying potential, one can always solve the evolution equation numerically.
Also, several analytic approximations for specific profiles of the matter potential can be found in
the literature [34].
14.5.1 The Mikheyev-Smirnov-Wolfenstein Effect for Solar Neutrinos
The matter effects discussed in the previous section are of special relevance for solar neutrinos.
As the Sun produces νe ’s in its core, we consider the propagation of a νe − νX neutrino system (Xis some superposition of µ and τ , which is arbitrary because νµ and ντ have only and equal neutral
current interactions) in the matter density of the Sun.
31st May, 2024

1414. Neutrino Masses, Mixing, and Oscillations

The density of solar matter is a monotonically decreasing function of the distance R from thecenter of the Sun, and it can be approximated by an exponential for R < 0.9R
ne (R) = ne (0) exp (−R/r0 ) ,(14.66)

with r0 = R /10.54 = 6.6 × 107 m = 3.3 × 1014 eV−1 .
As mentioned above, the nuclear reactions in the Sun produce electron neutrinos. After crossing
the Sun, the composition of the neutrino state exiting the Sun will depend on the relative size of∆m2 cos 2θ versus A0 = 2 E GF ne,0 (where 0 refers to the neutrino production point, which is near
but not exactly at the center of the Sun, R = 0).
If the relevant matter potential at production is well below the resonant value, AR = ∆m2 cos 2θ 
A0 , matter effects are negligible. With the characteristic matter density and energy of the solar
neutrinos, this condition is fulfilled for values of ∆m2 such that ∆m2 /E  LSun−Earth . So the propagation occurs as in vacuum with the oscillating phase averaged to 1/2 and the survival probability
at the exposed surface of the Earth is11
Pee (∆m2 cos 2θ  A0 ) = 1 − sin2 2θ > .
2
2(14.67)

If the relevant matter potential at production is only slightly below the resonant value, AR =∆m2 cos 2θ & A0 , the neutrino does not cross a region with resonant density, but matter effects are
sizable enough to modify the mixing. The oscillating phase is averaged in the propagation between
the Sun and the Earth. This regime is well described by an adiabatic propagation, Eq.(14.65).
Using that Ũ (0) is a 2x2 rotation of angle θm,0 – the mixing angle in matter at the neutrino
production point–, and Ũ (L) is the corresponding rotation with vacuum mixing angle θ, we get
Pee (∆m2 cos 2θ ≥ A0 ) = cos2 θm,0 cos2 θ + sin2 θm,0 sin2 θ =1[1 + cos 2θm,0 cos 2θ] .
2(14.68)This expression reflects that an electron neutrino produced at A0 is an admixture of ν1 withfraction Pe1,0 = cos2 θm,0 and ν2 with fraction Pe2,0 = sin2 θm,0 . On exiting the Sun, ν1 consists ofνe with fraction P1e = cos2 θ, and ν2 consists of νe with fraction P2e = sin2 θ so Pee = Pe1,0 P1e +Pe2,0 P2e = cos2 θm,0 cos2 θ + sin2 θm,0 sin2 θ [35–37], exactly as given in Eq.(14.68). Since A0 < AR
the resonance is not crossed so cos 2θm,0 has the same sign as cos 2θ and still Pee ≥ 1/2.
Finally, in the case that AR = ∆m2 cos 2θ < A0 , the neutrino can cross the resonance on its
way out. In the convention of ∆m2 > 0 this occurs if cos 2θ > 0 (θ < π/4). which means that in
vacuum νe is a combination of ν1 and ν2 with a larger ν1 component, while at the production point
νe is a combination of ν1m and ν2m with larger ν2m component. In particular, if the density at the
production point is much higher than the resonant density, ∆m2 cos 2θ  A0 ,θm,0 =π2⇒

cos 2θm,0 = −1 ,(14.69)and the produced νe is purely ν2m .
In this regime, the evolution of the neutrino ensemble can be adiabatic or non-adiabatic depending on the particular values of ∆m2 and the mixing angle. The oscillation parameters (seeSecs.14.6.1 and 14.7) happen to be such that the transition is adiabatic in all ranges of solar neutrinoenergies. Thus the survival probability at the exposed surface of the Earth is given by Eq.(14.68)
but now with mixing angle (14.69) soPee (∆m2 cos 2θ < A0 ) =1[1 + cos 2θm,0 cos 2θ] = sin2 θ .
231st May, 2024

(14.70)15

14. Neutrino Masses, Mixing, and Oscillations

So, in this case, Pee can be much smaller than 1/2 because cos 2θm,0 and cos 2θ have opposite
signs. This is referred to as the Mihheev-Smirnov-Wolfenstein (MSW) effect [26, 38], which plays afundamental role in the interpretation of the solar neutrino data.
The resulting energy dependence of the survival probability of solar neutrinos is shown in
Fig.14.3 (together with a compilation of data from solar experiments). The plotted curve corresponds to ∆m2 ∼ 7.5 × 10−5 eV2 and sin2 θ ∼ 0.3 (the so-called large mixing angle, LMA, solution).
The figure illustrates the regimes described above. For these values of the oscillation parameters,
neutrinos with E  1 MeV are in the regime with ∆m2 cos 2θ  A0 so the curve represents thevalue of vacuum averaged survival probability, Eq.(14.67), and therefore Pee > 0.5. For E > 10
MeV, on the contrary, ∆m2 cos 2θ  A0 and the survival probability is given by Eq.(14.70), so
Pee = sin2 θ ∼ 0.3. In between, the survival probability is given by Eq.(14.68) with θ0 changing
rapidly from its vacuum value to the asymptotic matter value (14.69), 90◦ .

14.6 Experimental Study of Neutrino Oscillations
Neutrino flavour transitions, or neutrino oscillations, have been experimentally studied using
various neutrino sources and detection techniques. Intense sources and large detectors are mandatory because of a large distance necessary for observable oscillation effects in addition to the tiny
cross-sections. Also, the relevant neutrino flux before oscillations should be known with sufficientprecision for a definitive measurement. Here, the experimental status of neutrino oscillations withthe different neutrino sources: the Sun, Earth’s atmosphere, accelerators, and nuclear reactors arereviewed.
14.6.1 Solar Neutrinos
14.6.1.1 Solar neutrino flux
In the Sun, electron neutrinos are produced in the thermonuclear reactions which generate solar
energy. These reactions occur via two main chains, the pp chain and the CNO cycle. The pp chain
includes reactions p + p → d + e+ + ν (pp), p + e− + p → d + ν (pep), 3 He + p → 4 He + e+ + ν(hep), 7 Be + e− → 7 Li + ν(+γ) (7 Be), and 8 B → 8 Be∗ + e+ + ν (8 B). The CNO cycle involves
13 N → 13 C + e+ + ν (13 N), 15 O → 15 N + e+ + ν (15 O), and 17 F → 17 O + e+ + ν (17 F). Thosereactions result in the overall fusion of protons into 4 He, 4p →4 He + 2e+ + 2νe , where the energy
released in the reaction, Q = 4mp − m4 He − 2me ∼ 26 MeV, is mostly radiated through the photons
and only a small fraction is carried by the neutrinos, hE2νe i = 0.59 MeV. In addition, electron
capture on 13 N, 15 O, and 17 F produces line spectra of neutrinos called ecCNO neutrinos. Dividing
the solar luminosity by the energy released per neutrino production, the total neutrino flux can be
estimated. At Earth, the pp solar neutrino flux is about 6 × 1010 cm−2 s−1 .
The solar neutrino flux has been calculated based on the Standard Solar Model (SSM). The
SSM describes the structure and evolution of the Sun based on a variety of inputs such as themass, luminosity, radius, surface temperature, age, and surface elemental abundances. In addition,
the knowledge of the absolute nuclear reaction cross sections for relevant fusion reactions andradiative opacities are necessary. John Bahcall and his collaborators continuously updated the SSM
calculations over several decades [39, 40]. Figure 14.1 shows the solar neutrino fluxes predicted by
the SSM calculation in [41] and ecCNO neutrinos in [42].
14.6.1.2 Detection of solar neutrinos and the solar neutrino problem
Experiments that observed solar neutrinos are summarized in Table 14.2. A pioneering solar
neutrino experiment was carried out by R. Davis, Jr. and collaborators at Homestake starting in
the late 1960s [44]. The Davis’ experiment utilizes the reaction νe + 37 Cl → e− + 37 Ar. Because thisprocess has an energy threshold of 814 keV, the most relevant fluxes are the 7 Be and 8 B neutrinos.
The detector contained ∼ 615 t of C2 Cl4 . The produced 37 Ar, which has a half-life of 34.8 days,
31st May, 2024

16

14. Neutrino Masses, Mixing, and Oscillations

ec

Figure 14.1: Spectrum of solar neutrino fluxes predicted by SSM calculation in [41]. In addition
to standard fluxes, ecCNO neutrinos have been added based on [42]. Electron capture fluxes are
given in cm−2 s−1 . Taken from [43].

Table 14.2: List of solar neutrino experiments
NameTarget material Energy threshold (MeV)
Mass (ton)Years
HomestakeC2 Cl40.814
615
1970–1994
SAGE
Ga
0.233
50
1989–
GALLEXGaCl3
0.233
100 [30.3 for Ga] 1991–1997
GNO
GaCl3
0.233
100 [30.3 for Ga] 1998–2003
Kamiokande
H2 O
6.5
3,000
1987–1995
Super-Kamiokande
H2 O
3.5
50,000
1996–
SNO
D2 O
3.5
1,000
1999–2006
KamLAND
Liquid scintillator
0.5/5.5
1,000
2001–
Borexino
Liquid scintillator
0.19
300
2007–2021

was chemically extracted and introduced into a low-background proportional chamber every few
months. The Auger electrons from electron capture of 37 Ar were counted to determine the reaction
rate.
From the beginning, the observed number of neutrinos in the Homestake mine experiment wassignificantly smaller than the prediction by SSM — it was almost one-third. After thorough checksof both experimental and theoretical work, the discrepancy remained. This became to be known as
the solar neutrino problem. The final result from the Homestake experiment is 2.56 ± 0.16 ± 0.16
SNU [45], where SNU (solar neutrino unit) is a unit of event rate, 1 SNU = 10−36 captures/(satom). On the other hand, prediction based on SSM is 8.46+0.87−0.88 SNU [46].
31st May, 2024

14. Neutrino Masses, Mixing, and Oscillations

17

The detection of neutrinos from other production processes was recognized as an important
input to investigate the origin of the solar neutrino problem. In particular, the pp neutrino is mostabundant, and its flux prediction has the smallest uncertainty. Using the radiochemical techniquewith gallium, the reaction νe + 71 Ga → e− + 71 Ge has an energy threshold of 233 keV and can beused for the pp neutrino detection. According to the SSM, more than half of the events on 71 Ga are
due to the pp neutrinos, with the second dominant contribution coming from the 7 Be neutrinos.
71 Ge decays via electron capture with a half-life of 11.4 days. The SAGE experiment in Baksan [47]used about 50 t of liquid metallic gallium as a target. The GALLEX experiment in LNGS [48] used
101 t of GaCl3 , containing 30.3 t of gallium. Both experiments used natural gallium, containing
39.9% of 71 Ga isotope. GALLEX was followed by its successor GNO experiment. The measured
capture rate is 69.3 ± 4.1 ± 3.6 SNU for GALLEX+GNO [49] and 65.4+3.1+2.6−3.0−2.8 SNU for SAGE [50].
A SSM prediction is 127.9+8.1
SNU
[46].
−8.2
The radiochemical detectors measure the reaction rate integrated between extractions. The
real-time measurement of solar neutrinos was realized by the Kamiokande experiment [51]. The
Kamiokande detector was a 3,000-t water-Cherenkov detector in the Kamioka mine. SuperKamiokande, the successor of Kamiokande, started operation in 1996. It is a large upright cylindrical water Cherenkov detector containing 50 kt of pure water 4 . An inner detector volume
corresponding to 32 kt water mass is viewed by more than 11,000 inward-facing 50 cm diameterphotomultiplier tubes (PMTs). Kamiokande and Super-Kamiokande can observe solar neutrinos
using ν-e elastic scattering (ES), νx + e− → νx + e− . The ES reaction occurs via both charged andneutral current interactions. Consequently, it is sensitive to all active neutrino flavours, although
the cross-section for νe , which is the only flavour to interact via charged current, is about six times
larger than that for νµ or ντ . Because the energy threshold is 6.5 MeV for Kamiokande and 3.5 MeV
for the present Super-Kamiokande (for the kinetic energy of recoil electron), these experiments are
sensitive to primarily to 8 B neutrinos.
The results from Kamiokande [52] and Super-Kamiokande [53] showed significantly smaller
numbers of observed solar neutrinos compared to the prediction. The latest solar neutrino flux
observed by Super-Kamiokande is (2.345±0.014±0.036)×106 cm−2 s−1 [54], while a prediction basedon the SSM is (5.46±0.66)×106 cm−2 s−1 [55]. In addition, no significant zenith angle variation nor
spectrum distortion were observed in the initial phase of Super-Kamiokande, which placed strong
constraints on the solution of the solar neutrino problem [56].
14.6.1.3 Solution of the solar neutrino problem
The SNO experiment in Canada used 1,000 t of heavy water (D2 O) contained in a sphericalacrylic vessel which was surrounded by an H2 O shield. An array of PMTs installed on a stainlesssteel structure detected Cherenkov radiation produced in both the D2 O and H2 O. The SNO detector
observed 8 B neutrinos via three different reactions. In addition to the ES scattering with an electron,with D2 O target the CC νe + d → e− + p + p and the NC νx + d → νx + p + n interactions are
possible. The CC reaction is sensitive to only νe , while NC reaction is sensitive to all active flavoursof neutrinos with equal cross-sections. Therefore, by comparing the measurements of different
reactions, SNO could provide a model-independent test of the neutrino flavour change [57].
In 2001, SNO reported the initial result of CC measurement [60]. Combined with the highstatistics measurement of ν-e elastic scattering from Super-Kamiokande [61], it provided a direct
evidence for the existence of non-νe component in solar neutrino flux. The result of the NC
measurement in 2002 [62] established it with 5.3σ of statistical significance. Figure 14.2 shows
the fluxes of electron neutrinos (φ(νe )) and muon and tau neutrinos (φ(νµ,τ )) with the 68%, 95%,
and 99% joint probability contours obtained with the SNO data. Finally, together with the reactor
4

Recently, gadolinium (0.01% by weight in 2020, 0.03% in 2022) is loaded in the water.

31st May, 2024

14. Neutrino Masses, Mixing, and Oscillationsφμτ (× 10 6 cm -2 s-1)

18

BS05

φSSM 68% C.L.

6

NC

φμ τ 68%, 95%, 99% C.L.

5
4
3
SNO

2

φCC 68% C.L.

1

φES 68% C.L.

0
0

SNO

φNC 68% C.L.
SNO
SKφES 68% C.L.
0.5

1

1.5

2

2.5

3

3.5

φe (× 106 cm-2 s-1)

Figure 14.2: Fluxes of 8 B solar neutrinos, φ(νe ), and φ(νµ,τ ), deduced from the SNO’s CC, ES,
and NC results [58]. The Super-Kamiokande ES flux is from [59]. The BS05(OP) standard solarmodel prediction [40] is also shown. The bands represent the 1σ error. The contours show the 68%,
95%, and 99% joint probability for φ(νe ) and φ(νµ,τ ). The figure is from [58].

neutrino experiment KamLAND (see Sec.14.6.4), the solution of solar neutrino problem was foundto be the MSW adiabatic flavour transitions in the solar matter, the so-called large mixing angle(LMA) solution. From a combined result of three phases of SNO [63], the total flux of 8 B solar+0.11
neutrino is found to be (5.25 ± 0.16−0.13) cm−2 s−1 , consistent with the SSM prediction. Thisconsistency is one of the major accomplishments of SSM.
In order to understand the SSM as well as to study the MSW effect for the solar neutrino,
measurements of solar neutrinos other than 8 B are important. The Borexino experiment at Gran
Sasso detected solar neutrino via ν-e scattering in real-time with a low energy threshold. The
Borexino detector consisted of 300 t of ultra-pure liquid scintillator, which achieved 0.19 MeV ofenergy threshold and 5% energy resolution at 1 MeV. Borexino reported the first real-time detectionof 7 Be solar neutrinos [65]. They also measured the fluxes of pep [66], pp [67], and CNO [68] neutrino
for the first time. Together with 8 B [69] neutrino measurement, Borexino provides important data
to study the MSW effect. The KamLAND experiment also measured solar neutrinos [70, 71].
Figure 14.3 shows the survival probability of solar νe as a function of neutrino energy. The
data points are from the Borexino results [72,73] and the SNO+SK 8 B data. The theoretical curveshows the prediction of the MSW-LMA solution. All the data shown in this plot are consistent
with the theoretically calculated curve. This indicates that these solar neutrino measurements are
consistent with the MSW-LMA solution of the solar neutrino problem.
The matter effects can also be relevant to the propagation of solar neutrinos through the Earth.
Because solar neutrinos go through the Earth before interaction in the detector during the nighttime, a comparison of measured event rate between daytime and nighttime provides a clean anddirect test of matter effects on neutrino oscillations. Super-Kamiokande reported the indicationof the day/night asymmetry in 8 B solar neutrinos [54]. The measured asymmetry, defined as thedifference of the average day rate and average night rate divided by the average of those two rates,
31st May, 2024

14. Neutrino Masses, Mixing, and Oscillations

19
1
0.9
0.8
0.7

pp
7

Pee

0.6

Be
pep
8

0.5

B

0.4
0.3
0.2
0.1
0 −1
10

1
Neutrino Energy [MeV]10

Figure 14.3: Electron neutrino survival probability as a function of neutrino energy. The points
represent, from left to right, the Borexino pp, 7 Be, pep, and 8 B data (red points) and the SNO+SK
8 B data (black point). The three Borexino 8 B data points correspond, from left to right, to the lowenergy (LE) range, LE+HE range, and the high-energy (HE) range. The electron neutrino survival
probabilities from experimental points are determined using a high metallicity SSM from [55]. The
error bars represent the ±1σ experimental + theoretical uncertainties. The curve corresponds tothe ±1σ prediction of the MSW-LMA solution using the parameter values given in [64]. This figure
is provided by A. Ianni.

is (−3.3 ± 1.0 ± 0.5)%, corresponding to a statistical significance of 2.9σ.
14.6.2 Atmospheric Neutrinos
14.6.2.1 Atmospheric neutrino flux
Atmospheric neutrinos are produced by the decays of pions and kaons generated in the interaction of cosmic rays and nucleons in the Earth’s atmosphere. They have a broad range of energy(∼0.1 GeV to >TeV) and long travel distances before detection (∼10 to ∼ 104 km), and contain all
the flavours of neutrinos and antineutrinos.
Considering their dominant production modes, some generic relations for flux ratios of different
flavours of neutrinos can be derived without detailed calculations. From the decay chain of acharged pion π + → µ+ νµ followed by µ+ → e+ νe ν̄µ (and the charge conjugate for π − ), the ratio(νµ + ν̄µ )/(νe + ν̄e ) is expected to be around 2 at low energies (∼ 1 GeV), where most muons decay
in the atmosphere. For higher energies, some muons reach the Earth before they decay and theratio increases. One can also expect that the zenith angle distributions of atmospheric neutrinos are
symmetric between upward-going and downward-going neutrinos. It is true for the energy above
1 GeV, but at lower energies, the Earth’s geomagnetic field induces up-down asymmetries in the
primary cosmic ray. The zenith angle corresponds to the flight length of atmospheric neutrinos.
Vertically upward-going neutrinos come from the other side of the Earth with flight lengths of∼ 104 km, while downward-going neutrinos produced just above the experimental site travel ∼10 km
before detection.
The atmospheric neutrino fluxes are calculated in detail based on the energy spectrum and31st May, 2024

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14. Neutrino Masses, Mixing, and Oscillations

composition of primary cosmic rays and their hadronic interactions in the atmosphere. The effects
of solar activity and geomagnetic field are also taken into account. Results of calculations by several
groups are available [74–77]. A typical uncertainty of the absolute flux is 10–20%, while the ratioof fluxes between different flavour has much smaller uncertainty (< 5%).
14.6.2.2 Observation of atmospheric neutrino oscillations
The first detection of atmospheric neutrinos was reported in the 1960’s by the undergroundexperiments in the Kolar Gold Field experiment in India [78] and in South Africa [79]. In the
1980’s, experiments searching for nucleon decays started operation. They used large undergrounddetectors which could also observe atmospheric neutrinos that were studied as backgrounds tonucleon decays. Among the early experiments were Kamiokande [80] and IMB [81] using water
Cherenkov detectors, and Frejus [82] and NUSEX [83] using iron tracking calorimeters.
The flavour of an atmospheric neutrino can be identified in charged current interaction with
nuclei, which produces the corresponding charged lepton. Those detectors originally designed
for nucleon decay search had the capability to distinguish muons and electrons. For example, awater Cherenkov detector can utilize the information from Cherenkov ring patterns for particle
identification; e-like particles (e± , γ) produce more diffuse ring than µ-like particles (µ± , π ± )because of electromagnetic cascades and multiple Coulomb scattering effects.
To reduce the uncertainty, in early results the flux ratio νµ /νe ≡ (νµ + ν̄µ )/(νe + ν̄e ) was
measured, and the double ratio between observation and expectation (νµ /νe )obs /(νµ /νe )exp wasreported. The Kamiokande experiment reported an indication of a deficit of (νµ + ν̄µ ) flux [80].
IMB also observed a similar deficit [81], but measurements by Frejus [82] and NUSEX [83] wereconsistent with the expectations. This was called the atmospheric neutrino anomaly. Kamiokandereported studies with an increased data set of the sub-GeV (< 1.33 GeV) [84] as well as the multiGeV (> 1.33 GeV) [85] samples. In the latter, they reported an analysis of zenith angle distributions,which showed an indication that the muon disappearance probability is dependent on the zenith
angle, hence the travel length of neutrinos. However, the statistical significance was not sufficient
to provide a conclusive interpretation.
The solution to the atmospheric neutrino anomaly was brought by Super-Kamiokande, which
reported compelling evidence for neutrino oscillations in atmospheric neutrinos in 1998 [86]. The
zenith angle (θz , with θz = 0 for vertically downward-going) distributions of µ-like events showed aclear deficit of upward-going events, while no significant asymmetry was observed for e-like events.
The asymmetry is defined as A = (U − D)/(U + D), where U is the number of upward-going(−1 < cos θz < −0.2) events and D is the number of downward-going (0.2 < cos θz < 1.0) events.
With multi-GeV (visible energy > 1.33 GeV) µ-like events alone, the measured asymmetry was
A = −0.296 ± 0.048 ± 0.001, deviating from zero by more than 6σ. The sub-GeV (< 1.33 GeV)µ-like, upward through going, and upward stopping µ samples which correspond to different energy
ranges of neutrinos, showed the consistent behaviour which strengthens the credibility of the observation. Super-Kamiokande’s results were confirmed by other atmospheric neutrino observations
MACRO [87] and Soudan2 [88].
Although the energy and zenith-angle-dependent muon neutrino disappearance observed withatmospheric neutrinos could be consistently explained by the neutrino oscillations predominantly
between νµ and ντ , other exotic explanations such as neutrino decay [89] or decoherence [90] were
not initially ruled out. By using a selected sample from Super-Kamiokande’s atmospheric datawith good L/E resolution, the L/E dependence of the survival probability was measured [91]. The
observed dip in the L/E distribution was consistent with the expectation from neutrino oscillation,
while alternative models were strongly disfavored.
As an experimental proof of νµ -ντ oscillation, an appearance signal of ντ was searched for in the
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14. Neutrino Masses, Mixing, and Oscillations

21

2000

Super-Kamiokande I-IV
328 kt y
Prediction

Number of Events

1000

200

1000

νµ → ντ

0-1

Sub-GeV e-like10294 Events
0

1

0
-1

Sub-GeV µ-like10854 Events
0

1

0
-1

400

1000

1000

200

500

500

0
-1

Multi-GeV e-like2847 Events
0

1

0
-1

Multi-GeV µ-like + PC
5932 Events
0

1

0
-1

UpStop µ1456 Events-0.5

0UpThrough µ6266 Events-0.5

0cos zenith
Figure 14.4: The zenith angle distributions of Super-Kamiokande atmospheric neutrino events.
A data set corresponding to 328 kton-years of exposure is used. Fully contained 1-ring e-like andµ-like events with visible energy < 1.33 GeV (sub-GeV) and > 1.33 GeV (multi-GeV), as well as
upward stopping and upward through going µ samples are shown. Partially contained (PC) eventsare combined with multi-GeV µ-like events. The blue histograms show the non-oscillated Monte
Carlo events, and the red histograms show the best-fit expectations for neutrino oscillations. (Thisfigure is provided by the Super-Kamiokande Collaboration)

atmospheric neutrino data. Because of the high energy threshold (> 3.5 GeV) of ντ CC interaction
and the short lifetime of τ lepton (0.3 ps), identifying the appearance of ντ experimentally ischallenging. Super-Kamiokande reported evidence of tau neutrino appearance using atmospheric
neutrino data with 4.6σ significance [92]. The definitive observation of ντ appearance was madeby the long-baseline experiment, OPERA (See Sec.14.6.3.3), and recently IceCube also reportedthe ντ appearance analysis [93] using atmospheric neutrinos.
14.6.2.3 Neutrino oscillation measurements using atmospheric neutrinos
Figure 14.4 shows the zenith angle distributions of atmospheric neutrino data from SuperKamiokande. For a wide range of neutrino energy and path length, the observed distributions
are consistent with the expectation from neutrino oscillation. Atmospheric neutrinos in the energy region of a few to ∼10 GeV provide information for the determination of the neutrino massordering [94].
The neutrino telescopes primarily built for high-energy neutrino astronomy such as ANTARESand IceCube can also measure neutrino oscillations with atmospheric neutrinos. ANTARES consistsof a sparse array of PMTs deployed under the Mediterranean Sea at a depth of about 2.5 km toinstrument a 105 m3 volume. IceCube is a detector deployed in ice in Antarctica at the South Pole,at a depth between 1.45 and 2.45 km. In the bottom center of IceCube there is a region of ∼ 107 m3
volume with denser PMT spacing called DeepCore to extend the observable energies to the lower
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14. Neutrino Masses, Mixing, and Oscillations

energy region. By observing the charged current interaction of up-going νµ , they measure the νµdisappearance. ANTARES reported a measurement of νµ disappearance with 20 GeV threshold [95].
The νµ disappearance measurements from IceCube DeepCore [96] provided a precision comparable
to the measurements by Super-Kamiokande and long-baseline accelerator neutrino experiments.
There are several projects for atmospheric neutrino observations either proposed or under preparation. The atmospheric neutrino observation program is included in the plans for future neutrino
telescopes such as ORCA in the KM3NeT project [97] in the Mediterranean Sea and the IceCube
Upgrade [98]. In India, a 50 kt magnetized iron tracking calorimeter ICAL is planned at theINO [99]. Future large underground detectors, Hyper-Kamiokande in Japan [100] and DUNE in
US [101] can also study the atmospheric neutrinos.
14.6.3 Accelerator Neutrinos
14.6.3.1 Accelerator neutrino beams
A comprehensive description of the accelerator neutrino beams is found in [102]. Conventional
neutrino beams from accelerators are produced by colliding high-energy protons onto a target,
producing π and K, which then decay into neutrinos. Undecayed mesons and muons are stopped
in a beam dump and soil. Because pions are the most abundant product of the high energy
collisions, a conventional neutrino beam contains a dominant amount of muon-type neutrinos (orantineutrinos).
Focusing devices called magnetic horns are used to concentrate the neutrino beam flux towards
the desired direction [103]. A magnetic horn is a pulsed electromagnet with toroidal magnetic fields
to focus charged particles that are parents of neutrinos. One can choose the dominant componentof the beam to be either neutrinos or antineutrinos by selecting the direction of current in themagnetic horns. Even with the focusing with horns, wrong sign neutrinos contaminate in thebeam. Also, there is a small amount of contamination of νe and ν̄e coming primarily from kaon
and muon decays.
In order to maximize the sensitivity of the experiment, the ratio of baseline and neutrino energy(L/E) should be chosen to match the oscillation effects to be studied. In addition to maximizing
the flux of neutrinos with relevant energy, neutrinos with irrelevant energy that result in unwanted
background process should be suppressed. The energy of a neutrino from a pion decay isEν =[1 − (mµ /mπ )2 ]Eπ,
1 + γ 2 θ2

(14.71)

where Eν and Eπ are the energy of neutrino and pion, respectively, θ is the angle between the pionand neutrino direction, and γ = Eπ /mπ . For θ = 0, the energy of neutrino is linearly proportional
to the energy of pion. In this case, a narrow band beam can be made by selecting the momentumof pions. On the other hand, for θ 6= 0, the energy of neutrino is not strongly dependent on theparent energy for a wide range of pion energy, but dependent on the off-axis angle θ. Using thisrelation, a neutrino beam with narrow energy spectrum, around the energy determined by θ, canbe produced. This off-axis beam method was first introduced for BNL E889 proposal [104] andadopted in T2K and NOvA experiments. For a list of neutrino beamlines, see also Chapter 33 of
this Review, “Neutrino Beam Lines at High-Energy Proton Synchrotrons.”
As indicated in Table 14.1, there are two different scales of baselines for accelerator-based
experiments to study different ranges of ∆m2 . The atmospheric mass splitting ∆m2 ∼ 2.5 ×
10−3 eV2 gives rise to the first oscillation maximum at L/E ∼ 500 GeV/km. In order to study
this parameter region with a ∼ 1 GeV accelerator neutrino beam, a long baseline of a few hundred
to a thousand km is necessary. On the other hand, there have been reports of possible neutrino
oscillations at the ∼ 1 eV2 scale, which can be studied at ∼ 1 km baseline with neutrinos fromaccelerators. These experiments are called short-baseline oscillation experiments.
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14. Neutrino Masses, Mixing, and Oscillations

The flux of a neutrino beam is calculated using Monte Carlo simulation based on the configuration of the beamline. An important ingredient of the neutrino flux prediction is the hadronproduction cross-section. Data from dedicated hadron production experiments [105–108] are used
to tune the beam simulation and constrain the uncertainty. The uncertainty of predicted neutrino
flux for the most relevant energy region is 5–10% with the latest hadron production data [109–111].
14.6.3.2 Near detectors and neutrino interaction cross sections
Many long-baseline experiments use two detectors to reduce the systematic uncertainties arising from neutrino flux and neutrino-nucleus interactions. The near detectors either use the same
technology as the far detector or consist of sub-detectors with complementary functions to obtain
detailed information of the neutrino beam and interactions. The near detectors provide information
for the neutrino flux, energy spectrum, and the interaction cross-sections, which is used as input
to make predictions of observables at the far detector. However, even with the two-detector configuration, one should note that the neutrino flux is inevitably different between the near and thefar detectors. In addition to the fact that the neutrino source looks like a line source for the neardetector while it looks like a point source for the far detector, the neutrino oscillations alter theflavour composition of the neutrino beam quite significantly, as the design of a neutrino oscillation
experiment requires.
For the precision measurements of neutrino oscillations with long-baseline experiments, theunderstanding of the neutrino-nucleus interaction becomes crucial. Because heavy nuclei are usedas the interaction target, the nuclear effects complicate the understanding of the neutrino-nucleus
interaction. For more information on the neutrino cross-sections, see also Chapter 52 of this Review,“Neutrino Cross Section Measurements.”
14.6.3.3 Long-baseline experiments
Table 14.3: List of long-baseline neutrino oscillation experiments
Name
Beamline
Far Detector
L (km) Eν (GeV)Year
K2K
KEK-PS
Water Cherenkov
250
1.3
1999–2004
MINOS
NuMI
Iron-scintillator
735
3
2005–2013
MINOS+NuMI
Iron-scintillator
735
7
2013–2016
OPERA
CNGS
Emulsion hybrid730
17
2008–2012
ICARUS
CNGS
Liquid argon TPC
730
17
2010–2012
T2K
J-PARC
Water Cherenkov
295
0.6
2010–
NOvA
NuMI Liquid scint. tracking calorimeter 810
2
2014–
DUNE
LBNF
Liquid argon TPC
1300
2–3
Hyper-Kamiokande J-PARC
Water Cherenkov
295
0.6

Long-baseline neutrino oscillation experiments are summarized in Table 14.3. The first longbaseline experiment was the K2K experiment which used a neutrino beam from the KEK 12 GeV
proton synchrotron directed towards Super-Kamiokande with a baseline of 250 km. The beam
had an average energy of 1.3 GeV. The K2K near detectors, located 300 m downstream of the
production target, consisted of a combination of a 1 kt water Cherenkov detector and a set offine-grained detectors. K2K confirmed the muon neutrino disappearance originally reported bySuper-Kamiokande atmospheric neutrino observation [112].
The MINOS experiment used a beam from Fermilab and a detector in the Soudan mine 735 km
away. The neutrino beam is produced in the NuMI beamline with a 120 GeV proton beam from the
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14. Neutrino Masses, Mixing, and Oscillations

Main Injector. The MINOS detectors were both iron-scintillator tracking calorimeters with toroidal
magnetic fields. The far detector was 5.4 kt, while the near detector had a total mass of 0.98 kt
and was located 1 km downstream of the production target. The energy spectrum of the NuMI
beamline can be varied by changing the relative position of the target and horns. Most of MINOS
data were taken with the “low energy” configuration with a peak energy of around 3 GeV. Since
2013, NuMI was operated with a “medium energy” configuration with the peak neutrino energyof around 7 GeV. The experiment for this period was called MINOS+. MINOS and MINOS+
combined accelerator and atmospheric neutrino data in both disappearance and appearance modes
to measure oscillation parameters [113, 114].
In Europe, the CNGS neutrino beamline provided a beam with mean energy of 17 GeV from
CERN to LNGS for long-baseline experiments with about 730 km of baseline. The beam energy waschosen so that CC interaction of ντ can occur for direct confirmation of ντ appearance. There wasno near detector in CNGS because it was not necessary for the ντ appearance search. The OPERA
experiment used a detector consisting of an emulsion/lead target with about 1.25 kt total mass,complemented by electronic detectors. The excellent spatial resolution of the emulsion enabled
the event-by-event identification of τ leptons. OPERA observed ten ντ CC candidate events with
2.0±0.4 expected background [115] and confirmed νµ → ντ oscillation in appearance mode witha statistical significance of 6.1σ. Another neutrino experiment, ICARUS [116], which used 600 t
liquid argon time projection chambers (TPCs), was operated in Gran Sasso from 2010 to 2012.
The first generation of long-baseline experiments confirmed the existence of neutrino oscillation.
The major initial goal of second-generation experiments was the observation of νµ → νe oscillation.
Using this appearance mode, and by comparison of neutrino and antineutrino oscillation probabilities, search for CP violation in the neutrino mixing and measurement of the mass ordering andthe octant of θ23 become possible.
The T2K experiment started in 2010 using a newly constructed high-intensity proton synchrotron J-PARC and the Super-Kamiokande detector. It is the first long-baseline experiment toemploy the off-axis neutrino beam. The off-axis angle of 2.5◦ was chosen to set the peak of theneutrino energy spectrum at 0.6 GeV, matching the first maximum of oscillation probability at the
295 km baseline for ∆m2 ∼ 2.5 × 10−3 eV2 . T2K employs a set of near detectors at about 280 m
from the production target. In 2011, T2K reported the first indication of νµ → νe oscillation witha statistical significance of 2.5σ [117]. In the framework of 3ν mixing, it corresponds to detecting
non-zero amplitude generated by the mixing angle θ13 (see Eq.14.33). Later νµ → νe oscillation wasestablished by T2K with more than 7σ in 2014 [118]. Figure 14.5 shows the observed kinematic
distributions from T2K, for neutrino and antineutrino beam mode and also for muon and electron
candidates. By a combined analysis of the neutrino and antineutrino data, T2K reported a hint of
CP violation at the 2σ level [119].
The NOvA experiment uses the upgraded NuMI beamline with an off-axis configuration. The
14 kt NOvA far detector is located near Ash River, Minnesota, 810 km away from the source.
At 14.6 mrad off-axis from the central axis of the NuMI beam, the neutrino energy spectrum at
the far detector has a peak of around 2 GeV. The near detector, located around 1 km from thesource, has a functionally identical design to the far detector with a total mass of 290 t. Bothdetectors are tracking calorimeters consisting of planes of polyvinyl chloride cells alternating invertical and horizontal orientation filled with liquid scintillator. The physics run of NOvA wasstarted in 2014. After confirmation of νe appearance from νµ beam [120], NOvA started datataking with antineutrino beam in 2016. Using the antineutrino beam data, NOvA has reported
the observation of ν̄e appearance from ν̄µ beam with 4.4σ significance [121]. Figure 14.6 shows
the reconstructed neutrino energy distributions from NOvA [122]. The NOvA data do not show
a strong asymmetry in the rate of νe versus ν̄e appearance, excluding δCP = 90◦ for the inverted
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Data

10

Best fit

20

0

180

Reconstructed ν-energy [GeV]

150

Data

4

Best fit

120
90

2

60
30
0

Data

5

Best fit

180

Lepton momentum [MeV/c]

150

Data

3

Best fit

120

2

90
60

1

30
100 50

00

0.5

1

1.5

2

2.5

3

0

0
20

10

00

200

1.5

Best fit

1

90
60

0.5

30

Lepton angle [deg]

Data

120

800 1000 1200

180

Lepton momentum [MeV/c]

150

Data

0.3

Best fit

120

0.2

90
60

0.1

0

0

5

00

200

400

600

800 1000 1200

0

Lepton momentum [MeV/c]

Data
Best fit

5

Reconstructed ν-energy [GeV]

150

600

10

Best fit

180

400

Lepton momentum [MeV/c]

0

0

30

Data

10

Lepton angle [deg]10Best fit0

Reconstructed ν-energy [GeV]

0

Data

Lepton angle [deg]20

Lepton angle [deg]Lepton angle [deg]

25

0

180

Lepton momentum [MeV/c]

150

0.6

Data

Best fit

120

0.4

90
60

0.2

30
50

00

0.5

1

1.5

2

2.5

3

Reconstructed ν-energy [GeV]

0

0

5

00

200

400

600

800 1000 1200

0

Lepton momentum [MeV/c]

Figure 14.5: Observed kinematic distributions from T2K compared to the expectations with thebest-fit parameters [119]. Top and bottom panels correspond to data from neutrino and antineutrino
beam mode, respectively. Left, middle, and right panels show distributions for single ring µ-likeevents, single ring e-like events with no associated decay electron, and single ring e-like events with
one associated decay electron (only for neutrino beam mode data), respectively.

mass ordering (m3 < m2 < m1 , see Sec.14.7 for definitions) with > 3σ and disfavouring δCP = 270◦
for the case of normal mass ordering (m1 < m2 < m3 , see Sec.14.7 for definitions) with > 2σ.
Two large-scale long-baseline experiments are under construction. DUNE [101] will be a 1,300 km
long-baseline experiment based in the US. The DUNE far detector will consist of four modules ofat least 10 kt fiducial mass liquid argon time projection chambers, located 1.5 km underground at
the Sanford Underground Research Facility in South Dakota. The beamline for DUNE, 1.2 MW atstart and upgradable to 2.4 MW, as well as the facility for near detectors will be newly constructedat Fermilab. The Hyper-Kamiokande detector [100] in Japan will be the successor of the SuperKamiokande detector. It will be a water Cherenkov detector with 260 kt total water mass. Withupgrades to the existing accelerator and beamline, J-PARC will provide a 1.3 MW neutrino beam
to Hyper-Kamiokande with a baseline of 295 km. Both DUNE and Hyper-Kamiokande will have arich physics program besides the long-baseline experiment, such as searches for nucleon decays andstudy of supernova neutrinos.
14.6.3.4 Short-baseline experiments
The LSND experiment searched for neutrino oscillation using neutrinos from stopped pions at
Los Alamos. A 800 MeV linac was used to produce pions that stopped in the target. Most of
π − s are absorbed by the nuclei inside the target, while π + s and their daughter µ+ s decay andproduce neutrinos. Therefore, the produced neutrinos are mostly νµ , ν̄µ , and νe with minimal
contamination of ν̄e . The detector was a tank filled with 167 t of diluted liquid scintillator, locatedabout 30 m from the neutrino source. LSND searched for ν̄µ → ν̄e appearance using the inverse
beta decay process, ν̄e + p → e+ + n, and found an excess of 87.9 ± 22.4 ± 6.0 events over the
expected background [123].
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26

ν-beamν-beam50

FD Data20

Bkg.

Best-fit Pred.
1-σ syst.
range

Core
Peripheral

25

Low CNNevt High CNNevt
Best-fit
FD Data40
pred.
1-σ syst.
WS bkg.
30
range
Beam
bkg.
20
Cosmic
bkg.

15

10

5

Eventsν-beam15

ν-beam10
8Core
Peripheral

Events / 0.1 GeV

10

10
6
4

5

2

0

1

2

3

4

Reco. νµ / νµ energy (GeV)

5

1

2

3

4

1

2

3

4

Reco. νe / νe energy (GeV)

Figure 14.6: Reconstructed neutrino energy distributions from the NOvA far detector [122]. Topplots are for neutrino beam mode, and bottom plots are for antineutrino beam mode. Left: muontype candidates. Right: electron-type candidates, split into a low and high purity sample as wellas the event counts in the peripheral sample which occurred near the edge of the detector.

The KARMEN experiment was performed at the neutron spallation facility ISIS of the Rutherford Appleton Laboratory. The KARMEN 2 detector was a segmented liquid scintillation calorimeter with a total volume of 65 m3 located at a mean distance of 17.7 m from the ISIS target.
KARMEN found a number of events consistent with the total background expectation, showing no
signal for ν̄µ → ν̄e oscillations [124]. The resulting limits exclude large regions of the parameterarea favored by LSND.
The MiniBooNE experiment at Fermilab used a conventional neutrino beam to search for νeand ν̄e appearance in the same parameter region as LSND. The booster neutrino beamline (BNB)
with a single magnetic horn uses an 8 GeV proton beam from the Fermilab booster to produce aneutrino (antineutrino) beam with an energy spectrum peak of 600 (400) MeV. The MiniBooNE
detector consists of a 12.2 m diameter sphere filled with 818 t of mineral and oil located 541 m fromthe target. MiniBooNE reported νe and ν̄e event excess in both neutrino and antineutrino runningmodes. In total, 638.0 ± 52.1 ± 122.2 excess events are observed over the expected backgrounds,
corresponding to 4.8σ significance [125].
Both LSND and MiniBooNE are single detector experiments. The short-baseline neutrino(SBN) program at Fermilab BNB [126] is further investigating the reported excess with a multidetector configuration. The SBN program comprises three liquid argon time projection chambers at
different baselines in the same neutrino beamline. The 85 t MicroBooNE detector was operated at
470 m from the target. The 112 t Short-Baseline Near Detector is located at 110 m from the target.
The ICARUS detector was transported from Europe after refurbishment at CERN and is located31st May, 2024

27

14. Neutrino Masses, Mixing, and Oscillations

at a baseline of 600 m as the SBN Far Detector. As of the writing of this review, MicroBooNE isthe only experiment that has presented results from SBN, observing no excess of νe events [127]and no evidence of light sterile neutrino oscillations [128].
JSNS2 experiment at J-PARC has started the search for neutrino oscillations with ∆m2 ∼1 eV2 [129]. 1MW proton beam from the 3 GeV Rapid Cycling Synchrotron of J-PARC produces
neutrinos from muon decay at rest. With a detector filled with a gadolinium-loaded liquid scintillator of 17 t fiducial mass at 24 m from the target, JSNS2 is aiming to provide a direct test of the
LSND anomaly. The second phase with an additional second detector at 48 m is under preparation.
14.6.4 Reactor Antineutrinos
14.6.4.1 Reactor antineutrino flux
Nuclear reactors are very intense sources of ν̄e ’s in the MeV energy region, which are generated
in nuclear fission of heavy isotopes (mainly 235 U, 238 U, 239 Pu, and 241 Pu). The ν̄e flux from areactor can be estimated based on the thermal power output and fuel composition as a function oftime. On average, about six ν̄e ’s are emitted, and about 200 MeV of energy is released per fission.
Therefore, a 1 GWth (thermal power) reactor produces about 2 × 1020 ν̄e ’s per second.
The detailed estimate of ν̄e flux and energy spectrum can be obtained by either summing up the
spectra of beta decays involved using available nuclear data information of each fission fragment andits decays, or by using measurements of cumulative electron spectra associated with the beta decaysof fission fragments. Because the fission of four main fuel isotopes involves thousands of beta-decaybranches, a complete ab initio calculation is challenging. The cumulative electron spectra for 235 U,
239 Pu, and 241 Pu were measured at the Institut Laue-Langevin (ILL) reactor in Grenoble, France
in the 1980s [130–132]. For the prediction of ν̄e flux from 238 U, a summation calculation in [133]was often used together with the ILL results.
A recent calculation of the reactor ν̄e flux [134] uses an improved ab initio approach for 238 U
and combined information from nuclear databases and electron spectra measured at ILL for 235 U,
239 Pu, and 241 Pu. Another calculation [135] is provided for 235 U, 239 Pu, and 241 Pu based on the
ILL measurement of electron spectra, taking into account higher-order corrections and minimizing
the use of nuclear databases. Both calculations predict a few percent higher normalization for theenergy-averaged antineutrino fluxes of 235 U, 239 Pu, and 241 Pu compared to the original analyses
of ILL data. However, the reactor antineutrino flux measurement at Daya Bay [136] is consistent
with the old flux predictions and the flux measurement results. Also, an excess of ν̄e flux around
5 MeV, compared to the prediction, has been observed by recent reactor experiments [137–139].
Table 14.4: List of reactor antineutrino oscillation experiments
Name
Reactor power (GWth ) Baseline (km) Detector mass (t)Year
KamLAND
various180 (ave.)
1,000
2001–
Double Chooz
4.25×2
1.05
8.3
2011–2018
Daya Bay
2.9×6
1.65
20×4
2011–2020
RENO
2.8×6
1.38
16
2011–
JUNO
26.6 (total)
53
20,00014.6.4.2 Reactor antineutrino oscillation experiments
Charged current interaction cannot happen if a reactor ν̄e changes its flavour to ν̄µ or ν̄τ ,because its energy is not sufficient to produce heavier charged leptons. Thus, ν̄e disappearance isthe only channel to study neutrino flavour change with reactor experiments. The inverse beta decay
31st May, 2024

28

14. Neutrino Masses, Mixing, and Oscillations

ν̄e + p → e+ + n provides a way to detect ν̄e in the relevant energy region. The energy of prompt
signal from e+ , Ep , is related to the energy of ν̄e , Eν̄ ∼ Ep + 0.8 MeV. The delayed coincidence withthe signal from γ ray emitted by neutron capture on nucleus after thermalization very efficiently
suppresses the backgrounds. A liquid scintillator is often used to realize large detectors containinghydrogen as the target of inverse beta decay. In order to increase the neutron detection efficiency,
a liquid scintillator is sometimes loaded with gadolinium because of a large neutron capture crosssection and higher energy of emitted γ rays, the total energy of about 8 MeV, by gadolinium, incontrast to 2.2 MeV for the capture by hydrogen.
Table 14.4 shows a list of reactor antineutrino experiments measuring neutrino oscillations.
As was also shown in Table 14.1, experiments are designed with different baselines because of thedifferent scale of mass splittings found by solar and atmospheric neutrino experiments. Experimentswith O(100) km baseline are sensitive to ∆m2 of 10−4 –10−5 eV2 , while ∼ 1 km of baseline results
in a sensitivity in the range of 10−2 –10−3 eV2 .
The KamLAND detector consists of 1,000 t of ultra-pure liquid scintillator contained in a 13m diameter spherical balloon [140]. The detector is located in the original Kamiokande cavern,
where the ν̄e flux was dominated by a few reactors at an average distance of ∼180 km until 2011.
KamLAND reported the first results in 2002 showing that the ratio of the observed number ofν̄e events and expectation without disappearance is 0.611 ± 0.085 ± 0.041, evidence for reactor ν̄edisappearance at the 99.95% confidence level [140]. It confirmed a large value of the mixing angle
corresponding to the LMA solution, which was reported by solar neutrino experiments. KamLAND
also showed the evidence of ν̄e spectrum distortion consistent with the expectation from neutrino
oscillations [141]. Figure 14.7 shows the ratio of observed ν̄e spectrum to the expectation for nooscillation as a function of L0 /E (L0 = 180 km) for the KamLAND data. A clear oscillatory
signature can be seen.
Following the establishment of neutrino oscillations with atmospheric, solar, accelerator, andreactor experiments, the measurement of the remaining mixing angle θ13 was recognized as thenext major milestone. A reactor neutrino experiment with a baseline of ∼1 km can make an almost
pure measurement of sin2 2θ13 from the disappearance of ν̄e . To be sensitive to a small value of θ13 ,experiments with two detectors were proposed. Among several proposals, three experiments havebeen realized: Double Chooz in France, Daya Bay in China, and RENO in Korea.
These three experiments employ a similar detector design optimized for the precise measurementof a reactor antineutrino. An antineutrino detector consists of a cylindrical stainless steel vessel thathouses two nested acrylic cylindrical vessels. The innermost vessel is filled with gadolinium-doped
liquid scintillator as the primary antineutrino target. It is surrounded by a liquid scintillator layer
to contain γ rays from the target volume. A buffer layer of mineral oil is placed outside to shield
inner volumes from radioactivity of PMTs and surrounding rock. The light from liquid scintillator
is detected by an array of PMTs mounted on the stainless steel vessel. Optically separated by
the stainless steel vessel, the outside region is instrumented as a veto detector with either liquid
scintillator (Double Chooz) or water Cherenkov (Daya Bay and RENO) detector.
The Double Chooz detector has a gadolinium-doped liquid scintillator with a mass of 8.3 t.
The far detector at a baseline of ∼1050 m from the two 4.25 GWth reactors started physics datataking in 2011. The near detector, located at ∼400 m from the reactors, was completed in the endof 2014. Double Chooz finished data taking in early 2018. Daya Bay has two near (flux-weighted
baseline 470 m and 576 m), and one far (1648 m) underground experimental halls near six reactors
with 2.9 GWth each. Daya Bay has eight antineutrino detectors in total; two detectors in each of
the near detector halls, and four detectors in the far detector hall. Each detector contains 20 t
of gadolinium-loaded liquid scintillator. RENO has two identical detectors located at 294 m and1383 m from the center of an array of six 2.8 GWth reactors. The mass of the gadolinium-loaded
31st May, 2024

14. Neutrino Masses, Mixing, and Oscillations

29

Survival Probability

1
0.8
0.6
0.4
0.2
0
20

3-ν best-fit oscillation

30

40

50

Data - BG - Geo νe6070

L0/Eν (km/MeV)

80

90

100

e

Figure 14.7: Ratio of the observed ν̄e spectrum to the expectation for no-oscillation versus L0 /E
for the KamLAND data. L0 = 180 km is the flux-weighted average reactor baseline. The 3-νhistogram is the best-fit survival probability curve from the three-flavour unbinned maximumlikelihood analysis using only the KamLAND data. This figure is taken from [142].

liquid scintillator is 16 t per detector. RENO started data taking with both near and far detectors
in August 2011.
All three reactor neutrino experiments published their first results in 2012. First, Double Chooz
reported an indication of reactor electron antineutrino disappearance with the ratio of observed to
expected events of R = 0.944 ± 0.016 ± 0.04, ruling out the no-oscillation hypothesis at the 94.6%
CL [143]. Daya Bay observed R = 0.940 ± 0.011 ± 0.004, corresponding to 5.2σ significance of
non-zero value of θ13 [144]. RENO also reported R = 0.920 ± 0.009 ± 0.014, indicating a non-zerovalue of θ13 with a significance of 4.9σ [145]. These results established a non-zero value of θ13 .
Both Daya Bay [146] and RENO [137] report results constraining mass-squared difference aswell as the mixing angle by using both relative ν̄e rate and energy spectra information. Double
Chooz has reported the analysis based on both far and near detectors [138] for the mixing angle,
using neutron capture on any elements (primarily gadolinium and hydrogen) to increase the effective
target mass. Figure 14.8 shows the energy spectra of the prompt signals observed in the far detectorof three experiments.
In all three experiments, an excess of ν̄e events over the expected energy spectrum has been
observed around 5 MeV, as mentioned earlier. This excess is observed in both near and far detectors
and scales with the reactor power. Thanks to the cancellation between the near and far detectors,
the neutrino oscillation measurements are not affected in a multi-detector setup.
With a baseline of ∼50 km and an excellent energy measurement, reactor antineutrino experiments have significant sensitivity to the mass ordering. The JUNO detector, which is underconstruction, will consist of a 20 kt liquid scintillator and be located at 53 km from two nuclear
power plants in China [147]. The JUNO experiment aims to determine the mass ordering with
31st May, 2024

14. Neutrino Masses, Mixing, and Oscillations

30
EH3

×103

105

250

103

200

102
4

6

8

10 12

100
50

pred

Nobs/Nno-osc.

6000

Events / 0.2 MeV

2

2

4

6

8

10

12

1
0.9

2

4

6

8

Prompt energy [MeV]1012

FD Data
No-oscillatted MC
Accidentals

4000

9

2000

Far Data
Prediction (best fit)
Prediction (no oscillation)

1.11

2

3

4

5

6

7

8

2

3

4
5
E p (MeV)

67

8

Events / 0.25 MeV

150

Data / Prediction

Entries [MeV-1]

104

3

10

Li

Fast Neutrons

102

1
0.9
0.81

10

Double Chooz IV: Far (818 live-days)
5

10
15
Visible Energy (MeV)

20

Figure 14.8: Energy spectra for prompt events at the far detectors for Daya Bay [146],
RENO [137], and Double Chooz [138].
Table 14.5: List of reactor antineutrino experiments for O(eV2 ) oscillations
Reactor power Baseline Detector Detector
σE /E
S/B
(MWth )
(m)mass (t) technology @1 MeV(%)
NEOS
2,800
24
1
Gd-LS
5
22
DANSS
3,100
10–13
0.9
Gd-PS
34
∼30
STEREO
57
9–11
1.7
Gd-LS
10
0.9
6 Li-LS
PROSPECT
85
7–9
4
4.5
1.3
NEUTRINO-4
100
6–12
1.5
Gd-LS
16
0.5
6
SoLid
80
6–9
1.6
Li-PS
14
Name

this technique as its primary goal. It can also provide precision measurements of neutrino mixing
parameters as well as a broad non-oscillation science program.
14.6.4.3 Reactor experiments sensitive to O(1) eV2 oscillations
Possible hints of neutrino oscillation at a scale of ∆m2 ∼1 eV2 (see Sec.14.8) have motivatedreactor experiments at a distance of ∼10 m from the core. Recent experiments searching for ∼1 eV2oscillation at reactors are summarized in Table 14.5.
As the antineutrino source, some use commercial reactors, which can provide a large flux leading
to high statistical precision. On the other hand, though the flux is orders of magnitude smaller, aresearch reactor could have favorable conditions such as relatively easier access to a short baseline,
simpler fuel composition, and compact core size.
The detectors are based on organic scintillators, either liquid scintillator (LS) or solid plastic
scintillator (PS), which contain hydrogen as the target for inverse beta decay (ν̄e + p → e+ + n).
To identify the signal, neutron capture on either gadolinium (Gd) or 6 Li is detected with delayedcoincidence. When a neutron is captured by Gd, γ rays with a total energy of 8 MeV are emitted.
After neutron capture, 6 Li decays into triton and α. The effect of neutrino oscillation appears as a
distortion of energy spectrum. To be independent from the reactor neutrino spectrum uncertainties,some experiments compare the spectra at different baselines by using a segmented detector or
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31

14. Neutrino Masses, Mixing, and Oscillations

moving the detector.
NEOS [139, 148] uses about 1 t of gadolinium-loaded liquid scintillator in an unsegmented
detector. It is located at 23.7 m from the center of a commercial reactor and covered by an
overburden of about 20 meters of water equivalent.
DANSS [149] is another experiment using a commercial reactor. The detector is highly segmented, consisting of 2,500 plastic scintillator strips, each with the size of 1 × 4 × 100 cm3 andcoated with a thin gadolinium-loaded reflective layer. The detector is placed on a movable platform
below the reactor core, which varies the distance to the core from 10.7 to 12.7 m.
The STEREO detector [150,151] has six identical target cells of 37 cm length, ∼ 2m3 of volume
in total, filled with gadolinium-loaded liquid scintillator. They are placed from 9.4 to 11.1 m fromthe compact (80 cm high, 40 cm diameter) core of the ILL research reactor.
The PROSPECT detector [152] consists of a segmented 4 t 6 Li-doped liquid scintillator detector.
It covers a baseline range of 7–9 m from the highly 235 U enriched reactor core of the High Flux
Isotope Reactor (HFIR) at Oak Ridge National Laboratory. Thin reflecting panels divide theLS volume into an 11 × 14 two-dimensional array of 154 optically isolated rectangular segments(14.5 × 14.5 × 117.6 cm3 ).
NEUTRINO-4 [153,154] uses a gadolinium-loaded liquid scintillator detector segmented in 10×5
sections with a total volume of 1.8 m3 . The detector is installed on a movable platform and moved
to various positions with baselines of 6–12 m from the SM-3 reactor core in Russia.
The SoLid detector [155] is a finely segmented detector made of 5 × 5 × 5 cm3 plastic scintillator
cubes and 6 LiF:ZnS sheets. Using the difference of time constant of scintillation between ZnS andthe plastic scintillator a n-γ separation capability is achieved. A detector with 1.6 t of active volume
is installed at a distance of 6–9 m from the research reactor core in the BR2 reactor, Belgium.

14.7 Combined Analysis of Experimental Results: The 3ν Paradigm
From the experimental situation described in Sec.14.6 we conclude that• Atmospheric νµ and ν̄µ disappear most likely converting to ντ and ν̄τ . The results show anenergy and distance dependence perfectly described by mass-induced oscillations.
• Accelerator νµ and ν̄µ disappear over distances of ∼ 200 to 800 km. The energy spectrum of
the results show a clear oscillatory behavior also in accordance with mass-induced oscillations
with a wavelength in agreement with the effect observed in atmospheric neutrinos.
• Accelerator νµ and ν̄µ appear as νe and ν̄e at distances ∼ 200 to 800 km.
• Solar νe convert to νµ and/or ντ . The observed energy dependence of the effect is welldescribed by massive neutrino conversion in the Sun matter according to the MSW effect• Reactor ν̄e disappear over distances of ∼ 200 km and ∼ 1.5 km with different probabilities.
The observed energy spectra show two different mass-induced oscillation wavelengths: at
short distances in agreement with the one observed in accelerator νµ disappearance, and along distance compatible with the required parameters for MSW conversion in the Sun.
The minimum scenario to describe these results requires the mixing between the three flavour
neutrinos of the standard model in three distinct mass eigenstates. In this case U in Eq. (14.32) is
a 3 × 3 matrix analogous to the CKM matrix for the quarks [21], but due to the possible Majorana
nature of the neutrinos it can depend on six independent parameters: three mixing angles andthree phases. There are several possible conventions for the ranges of the angles and ordering of
the states. The community finally agreed to a parametrization of the leptonic mixing matrix asin Eq. (14.33). The angles θij can be taken without loss of generality to lie in the first quadrant,
θij ∈ [0, π/2], and the phase δCP ∈ [0, 2π]. Values of δCP different from 0 and π imply CP violationin neutrino oscillations in vacuum [156–158]. The Majorana phases η1 and η2 play no role in neutrino
31st May, 2024

14. Neutrino Masses, Mixing, and Oscillations

32

oscillations [157, 159]. Hence for the study of neutrino oscillations in the 3ν mixing scenario one
can use the parametrization in Eq. (14.34) irrespective of whether neutrinos are Dirac or Majorana
particles. Indeed, Majorana phases are very hard to measure since they are only physical if neutrinomass is non-zero, and therefore, the amplitude of any process involving them is suppressed by afactor mν /E to some power where E is the energy involved in the process, which is typically muchlarger than the neutrino mass. The most sensitive experimental probe of Majorana phases is therate of neutrinoless ββ decay discussed in Secs. 14.9.3 and 14.9.2.
In this convention there are two non-equivalent orderings for the spectrum of neutrino masses:– Spectrum with Normal Ordering (NO) with m1 < m2 < m3
– Spectrum Inverted ordering (IO) with m3 < m1 < m2 .
Furthermore, the data show a hierarchy between the mass splittings, ∆m221  |∆m231 | ' |∆m232 |with ∆m2ij ≡ m2i − m2j .
In this section, we follow the convention used in the listing section of the PDG and discuss theresults for both, NO and IO, using ∆m221 , which is always the smallest mass splitting, and ∆m232 ,which up to a sign, is the largest mass splitting for IO, while for NO the largest mass splitting is∆m231 = ∆m232 + ∆m221 .
With what we know of the mass differences (see table 14.7) and the neutrino mass scale (seeSec. 14.9), depending on the value of the lightest neutrino mass, the neutrino mass spectrum canbe further classified in:– Normal HierarchicalSpectrum (NH): m1 q
qm2 < m3 ,⇒ m2 '∆m221 ∼ 8.6 × 10−3 eV, m3 '∆m232 + ∆m221 ∼ 0.05eV,– Inverted HierarchicalSpectrum (IH): m3  m1 <qm2 ,q⇒ m1 '|∆m232 + ∆m221 | ∼ 0.0492eV, m2 '|∆m232 | ∼ 0.05eV,– Quasidegenerate Spectrum (QD): m1 ' m2 ' m3 q|∆m232 |.

Sometimes in the literature the determination of the neutrino mass spectrum is referred to as
determination of the neutrino hierarchy. However, as described above, with what we know so farof the neutrino mass scale, the neutrino spectrum may or may not be hierarchical. Therefore
determination of neutrino mass ordering is a more precise expression, and it is the one used in thisreview.
In total, the 3-ν oscillation analysis of the existing data involves six parameters: 2 mass differences (one of which can be positive or negative), 3 mixing angles, and the CP phase. The different
experiments described in Sec.14.6 provide information on different subsets of these parameters.
The precise statistical analysis of the data requires the numerical evaluation of the corresponding oscillation probabilities by solving the evolution equation of the neutrino ensemble from theirsource to the experiment. Nevertheless, the dominant effects in the different experiments can bequalitatively understood in terms of approximate expressions for the oscillation probabilities which,for convenience, we briefly summarize here.
14.7.1 3ν Oscillation Probabilities
The relevant survival probabilities for solar and KamLAND experiments in the framework of
three neutrino oscillations can be written as:
3ν
2ν
Pee
= sin4 θ13 + cos4 θ13 Pee(∆m221 , θ12 ) ,31st May, 2024

(14.72)

14. Neutrino Masses, Mixing, and Oscillations

33

2
where we have used the fact that Losc
0,32 = 4πEν /∆m32 is much shorter than the distance traveled byboth solar and KamLAND neutrinos, so that the oscillations related to Losc
0,32 are averaged. In the2ν2presence of matter effects Pee (∆m21 , θ12 ) should be calculated, taking into account the evolution
2−5 . ∆m2 /eV2 . 10−4 , P 2ν (∆m2 , θ )in an effective matter density neffe = ne cos θ13 . For 10
ee
21 12
presents the following asymptotic behaviors [160]:1
sin2 (2θ12 )22ν,sunPee' sin2 (θ12 )2ν,sunPee'1−2ν,kamPee= 1 − sin2 (2θ12 ) sin

for Eν . few × 100 keV ,(14.73)for Eν & few × 1 MeV ,(14.74)2
2 ∆m21 L

4Eν

.

(14.75)

At present most of the precision of the solar analysis is provided by SNO and SK, for which therelevant MSW survival probability provides a direct measurement of sin2 θ12 , as seen in Eq. (14.74).
In the MSW regime, the determination of ∆m221 in solar experiments comes dominantly fromthe ratio between the solar potential and the ∆m221 term required to simultaneously describe the
CC/NC data at SNO and the undistorted spectra of 8 B neutrinos as measured in both SK and
SNO. Conversely, KamLAND ν̄e survival probability proceeds dominantly as vacuum oscillations
and provides a most precise determination of ∆m221 via the strong effect of the oscillating phasein the distortion of the reactor energy spectrum. On the contrary, it yields a weaker constrainton θ12 as the vacuum oscillation probability depends on the double-valued and “flatter” function
sin2 (2θ12 ).
In what respects the interpretation of νµ disappearance data, at LBL experiments, the νµsurvival probability can be expanded in the small parameters sin θ13 and α ≡ ∆m221 /∆m231 to good
accuracy as [161, 162]Pνµ →νµ ≈ 1 − sin2 2θµµ sin2with∆m2µµ L∆m232 L≈ 1 − cos2 θ13 sin2 (2θ23 ) sin2
+ O(α, s213 ) ,4Eν
4Eν

(14.76)sin2 θµµ = cos2 θ13 sin2 θ23 ,= sin2 θ12 ∆m231 + cos2 θ12 ∆m232∆m2µµ+ cos δCP sin θ13 sin 2θ12 tan θ23 ∆m221 .

At present νµ disappearance results at LBL provide the best determination of |∆m232 | and θ23 ,but as seen above, the probability is symmetric with respect to the octant of θµµ , which implies
symmetry around s223 = 0.5/c213 .
The relevant oscillation probability for νe appearance at LBL experiments can be expanded at
the second order in the small parameters sin θ13 and α, and assuming a constant matter density it
takes the form [163–165]:Pνµ →νe ,(ν̄µ →ν̄e ) ≈ 4 sin2 θ13 sin2 θ23

sin2 ∆(1 − A)sin2 A∆
2
2
2
+
α
sin
2θ
cos
θ
12
23
(1 − A)2
A2
sin ∆A sin ∆(1 − A)max
+ 8 α JCP
cos(∆ ± δCP ), (14.77)
A
1−A

with
max
JCP
= cos θ12 sin θ12 cos θ23 sin θ23 cos2 θ13 sin θ13 ,and∆≡∆m231 L,
4EνA≡

31st May, 2024

2Eν V
,
∆m231

(14.78)(14.79)14. Neutrino Masses, Mixing, and Oscillations

34

where V is the effective matter potential in the Earth’s crust. Results on νe appearance at LBL
provide us with the dominant information on leptonic CP violation. Furthermore, α, ∆, and A are
sensitive to the sign of ∆m232 (i.e., the type of the neutrino mass ordering). The plus (minus) signin Eq. (14.77) applies for neutrinos (antineutrinos), and for antineutrinos V → −V , which implies
A → −A. Numerically one finds for a typical Earth crust matter density of 3 g/cm3 that at T2K
with E ∼ 0.7 GeV, matter effects cause ∼ ±10% differences in the rates, whereas in NOvA withE ∼ 2 GeV, we can have |A| ∼ 0.2. Also, α2 ≈ 10−3 , which implies that the second term in thefirst line of Eq. (14.77) gives a very small contribution compared to the other terms. Also, the firstterm in Eq. (14.77) (which dominates for large θ13 ) depends on sin2 θ23 and therefore is sensitive
to the octant.
The νe survival probability relevant for reactor experiments with medium baseline (MBL),
L ∼ 1 km, can be approximated as [162, 166]:Pνe →νe = 1 − sin2 2θ13 sin2∆m2ee L+ O(α2 ) ,4Eν

(14.80)where∆m2ee = cos2 θ12 ∆m231 + sin2 θ12 ∆m232 .

(14.81)

These MBL reactor experiments provide the most precise determination of θ13 . Furthermore there
is an additional effect sensitive to the mass ordering when comparing the disappearance of νµ at
LBL experiments – which is symmetric with respect to the sign of ∆m2µµ given in Eq.(14.7.1) – withthat of νe disappearance at MBL reactors which is symmetric with respect to the slightly different
effective mass-squared difference ∆m2ee given in Eq. (14.81).
Table 14.6: Experiments contributing to the present determination of the oscillation parameters.
Experiment
Dominant
ImportantSolar Experiments
θ12∆m221 , θ13
2Reactor LBL (KamLAND)∆m21
θ12 , θ132
θ13 , |∆m31,32 |Reactor MBL (Daya-Bay, Reno, D-Chooz)
Atmospheric Experiments (SK, IC-DC)
θ23 ,|∆m231,32 |, θ13 ,δCP
Accel LBL νµ ,ν̄µ , Disapp (K2K, MINOS, T2K, NOνA) |∆m231,32 |, θ23
Accel LBL νe ,ν̄e App (MINOS, T2K, NOνA)
δCP
θ13 , θ23
Finally, for atmospheric neutrinos, the fluxes contain νe , νµ , ν̄e , and ν̄µ , and for a good fractionof the events, neutrinos travel through the Earth’s matter. In the context of 3ν mixing, the dominant oscillation channel of atmospheric neutrinos is νµ → ντ driven by |∆m232 | with an amplitudecontrolled by θ23 with subleading oscillation modes, triggered by ∆m221 and/or θ13 , which dependon the octant of θ23 , on the mass ordering, and on δCP . In that respect, an interesting observable
is the deviation of e-like events relative to the no-oscillation prediction Ne0 , since in the two-flavour
limit one expects Ne = Ne0 . Such deviation can be written in the following way (see, e.g., [167]):Ne− 1 ≈ (r sin2 θ23 − 1)P2ν (∆m232 , θ13 ) + (r cos2 θ23 − 1)P2ν (∆m221 , θ12 )Ne0− sin θ13 sin 2θ23 r <(A∗ee Aµe ) .

(14.82)

Here r ≡ Φµ /Φe is the flux ratio with r ≈ 2 in the sub-GeV range and r ≈ 2.6 → 4.5 in the multiGeV range. P2ν (∆m2 , θ) is an effective two-flavour oscillation probability, and Aee , Aµe are elements
31st May, 2024

35

14. Neutrino Masses, Mixing, and Oscillationsof a transition amplitude matrix. The three terms appearing in Eq. (14.82) have a well-defined
physical interpretation. The first term is important in the multi-GeV range and is controlled by themixing angle θ13 in P2ν (∆m232 , θ13 ). This probability can be strongly affected by resonant matter
effects [168–173]. Depending on the mass ordering, the resonance will occur either for neutrinos
or antineutrinos. The second term is important for sub-GeV events, and it takes into account theeffect of oscillations due to ∆m221 and θ12 [174–177]. Via the pre-factor containing the flux ratio
r both, the first and second terms in Eq. (14.82) depend on the octant of θ23 , though in opposite
directions: the multi-GeV (sub-GeV) excess is suppressed (enhanced) for θ23 < 45◦ . Finally, thelast term in Eq. (14.82) is an interference term between θ13 and ∆m221 amplitudes and this termshows also dependence on the CP phase δCP [167, 177].
Subdominant three neutrino effects can also affect µ-like events. For example, for multi-GeV
muon events one can write the excess in µ-like events as [178, 179]Nµ1
1
− 1 ≈ sin2 θ23− sin2 θ23 P2ν (∆m232 , θ13 ) − sin2 2θ23 [1 − <(A33 )] .
Nµ0
r
2




(14.83)

The first term is controlled by θ13 and is subject to resonant matter effects, similar to the first termin Eq. (14.82), though with a different dependence on θ23 and the flux ratio. In the second term,
A33 is a probability amplitude satisfying P2ν (∆m232 , θ13 ) = 1 − |A33 |2 . In the limit θ13 = 0 we have<(A33 ) = cos(∆m232 L/2E), such that the second term in Eq. (14.83) just describes two-flavourνµ → νµ vacuum oscillations.
14.7.2 3ν Oscillation Analysis
We summarize in Table 14.6 the different experiments which dominantly contribute to thepresent determination of the different parameters in the chosen convention.
The table illustrates that the determination of the leptonic parameters requires global analysesof the data from the different experiments. Over the years, these analyses have been in the handsof a few phenomenological groups. We show in Table 14.7 the results from the latest analyses in
Refs. [180–183]. For the sake of comparison, all results are presented in the convention of thelisting section as described above.
The table illustrates the dependence of the present determination of the parameters on variationsof the statistical analysis performed by the different groups and on the data samples included. Inthat last respect, the main difference resides in the results from Super-Kamiokande atmospheric
data [94] which, at present, can only be included in this analysis by directly adding the χ2 tabulated
χ2 map provided by the experiment.
Altogether the different analyses find consistent results, in particular on the better-known parameters, θ12 , θ13 and ∆m221 and |∆m232 |. The issues which still require clarification are the massordering discrimination, the determination of θ23 and the leptonic CP phase δCP :• In all analyses the best fit is for the normal mass ordering. Inverted ordering is disfavoured
with a ∆χ2 , which ranges from slightly above 1.5σ – driven by the interplay of long-baseline
accelerator and short-baseline reactor data – to 2.5σ when adding the atmospheric χ2 tablefrom Ref. [184].
• There is no statistically significant preference for the octact of θ23 in any of the analysis.
• The best fit for the complex phase in NO is at δCP ∼ 200◦ . CP conservation (for δCP ∼ 180◦ )is allowed at a confidence level (CL) of 1-2σ.
31st May, 2024

14. Neutrino Masses, Mixing, and Oscillations

36

Table 14.7: 3ν oscillation parameters obtained from different global analyses of neutrino data. In all cases, the numbers labeled as NO (IO) are
obtained assuming NO (IO), i.e., relative to the respective local minimum. SK-ATM makes reference to the tabulated χ2 map from the SuperKamiokande analysis of their data in Ref. [184].
Ref. [181] w/o SK-ATM
Ref. [181] w SK-ATM
Ref. [182] w SK-ATM
Ref. [183] w SK-ATM
NOBest Fit Ordering
Best Fit Ordering
Best Fit Ordering
Best Fit OrderingParam
bfp ±1σ
3σ rangebfp ±1σ
3σ rangebfp ±1σ
3σ rangebfp ±1σ
3σ rangesin2 θ12
+0.12
+0.12
+0.13
+0.16
3.03−0.11
2.70 → 3.41
3.03−0.12
2.70 → 3.41
3.03−0.13
2.63 → 3.45
3.18−0.16
2.71 → 3.69
10−1
31.31 → 35.74
31.31 → 35.74
30.85 → 35.97
31.4 → 37.4
33.41+0.75
33.41+0.75
33.40+0.80
34.3+1.0
θ12 /◦
−0.72
−0.72
−0.82
−1.0
sin2 θ23
+0.18
+0.19
+0.18
+0.14
5.72
4.06
→
6.20
4.51
4.08
→
6.03
4.55
4.16
→
5.99
5.74
4.34 → 6.10
−0.23
−0.16
−0.15
−0.14
10−1
+1.1
+1.0
+0.8
49.1+1.0
39.6
→
51.9
42.2
39.7
→
51.0
42.4
40.2
→
50.7
49.3
41.2 → 51.3
θ23 /◦
−1.3
−0.9
−0.9
−0.8
sin2 θ13+0.056+0.056+0.07+0.0692.203−0.059 2.029 → 2.391 2.225−0.059 2.052 → 2.398
2.23−0.06
2.04 → 2.44
2.200−0.062 2.00 → 2.405
10−2
8.19 → 8.89
8.23 → 8.91
8.21 → 8.99
8.13 → 8.92
θ13 /◦
8.54+0.11
8.58+0.11
8.59+0.13
8.53+0.13
−0.12
−0.11
−0.12
−0.12
+36
+32
δCP /◦
197+42
108
→
404
232
144
→
350
223
139
→
355
194+24128 → 359
−25
−26
−23
−22
∆m221
+0.21
+0.21
+0.16
+0.22
7.41−0.20
6.82 → 8.03
7.41−0.20
6.82 → 8.03
7.36−0.15
6.93 → 7.93
7.50−0.20
6.94 → 8.14
10−5 2eV2∆m32+0.028+0.026+0.023+0.02
2.437−0.027 2.354 → 2.523 2.433−0.027 2.353 → 2.516 2.448−0.031 2.367 → 2.521
2.47−0.03
2.40 → 2.46
10−3 eV2IO∆χ2 = 2.3
∆χ2 = 6.4
∆χ2 = 6.5
∆χ2 = 6.4
sin2 θ12
3.03+0.12
3.03+0.12
3.03+0.13
3.18+0.16
2.70 → 3.41
2.70 → 3.41
2.63 → 3.45
2.71 → 3.69
−0.11
−0.11
−0.13
−0.16
10−1
+0.75
+0.75
+0.80
◦
33.41−0.72
31.31 → 35.74
33.41−0.72
31.31 → 35.74
33.40−0.82
30.85 → 35.97
34.3+1.0
31.4 → 37.4
θ12 /
−1.0
sin2 θ23
+0.16
+0.16
+0.13
−0.10
5.78
4.12
→
6.23
5.69
4.12
→
6.13
5.69
4.17
→
6.06
5.78
4.33 → 6.08
+0.17
−0.21
−0.21
−0.21
10−1
+1.0
+0.7
+0.6
49.5+0.9
39.9
→
52.1
49.0
39.9
→
51.5
49.0
40.2
→
51.1
49.5
41.2
→ 51.2
θ23 /◦
−1.2
−1.2
−1.4
−1.0
sin2 θ13+0.060+0.058+0.06+0.0642.219−0.057 2.047 → 2.396 2.223−0.058 2.048 → 2.416
2.23−0.06
2.225−0.070 2.02 → 2.42
2.03 → 2.45
10−2
8.57+0.12
8.23 → 8.90
8.57+0.11
8.23 → 8.94
8.59+0.13
8.19 → 9.00
8.58+0.12
8.17 → 8.96
θ13 /◦
−0.11
−0.11
−0.12
−0.14
+22
+25
δCP /◦
286+27
192
→
360
276
194
→
344
274
193
→
342
284+26200 → 353
−32
−29
−27
−28
∆m221
+0.21
+0.21
+0.16
+0.22
7.41−0.20
7.41−0.20
7.36−0.15
7.50−0.20
6.82 → 8.03
6.82 → 8.03
6.93 → 7.93
6.94 → 8.14
10−5 2eV2∆m32+0.032+0.028+0.025+0.03−2.498−0.025 −2.581 → −2.408−2.486−0.025 −2.570 → −2.406−2.492−0.030 −2.578 → −2.413−2.52±−0.02 −2.60 → −2.44
10−3 eV2

14.7.3 Convention-independent Measures of Leptonic CP Violation in 3ν Mixing
In the framework of 3ν mixing leptonic CP violation can also be quantified in terms of the
leptonic Jarlskog invariant [185], defined by:∗∗= Uαi Uαj
Uβi
Uβj ≡





X

X

max
JCP αβγ ijk ≡ JCP
sin δCP .

γ=e,µ,τ k=1,2,3(14.84)max is the combination of mixing angles in Eq. (14.78). For
With the convention in Eq. (14.33) JCP
example, from the analysis in Ref. [180, 181]max
JCP
= 0.0330 ± 0.0006 (±0.0019) ,(14.85)at 1σ (3σ) for both orderings, and the preference of the present data for non-zero δCP implies abest = −0.009.
small non-zero best fit value JCP
The status of the determination of leptonic CP violation can also be graphically displayed byprojecting the results of the global analysis in terms of leptonic unitarity triangles [186–188]. Since
in the analysis U is unitary by construction, any given pair of rows or columns can be used todefine a triangle in the complex plane. There are a total of six possible triangles corresponding to
the unitary conditions
X
i=1,2,3∗Uαi Uβi= 0 with α 6= β ,X
α=e,µ,τ

31st May, 2024∗
Uαi Uαj= 0 with i 6= j .

(14.86)

14. Neutrino Masses, Mixing, and Oscillations

37

As an illustration we show in Fig. 14.9 the recasting of the allowed regions of the analysis inRef. [180, 181] in terms of one leptonic unitarity triangle. We show the triangle corresponding
to the unitarity conditions on the first and third columns (after the shown rescaling), which isthe equivalent to the one usually shown for the quark sector. In this figure, the absence of CP
violation would imply a flat triangle, i.e., =(z) = 0. So the CL at which leptonic CP violation is
being observed would be given by the CL at which the region crosses the horizontal axis. Notice
however, that this representation is made under the assumption of a unitary U matrix and therefore
does not provide any test of unitarity in the leptonic sector.

Im(z)

0.5

Ue1 U∗e3z = − ⎯⎯⎯⎯Uμ1 U∗μ3

0

U e1

∗
Uμ1 Uμ3∗
U e3∗
U τ1 U τ3

-0.5
-1

-0.5

0

Re(z)

0.5

1

Figure 14.9: Leptonic unitarity triangle for the first and third columns of the mixing matrix.
After scaling and rotating the triangle so that two of its vertices always coincide with (0, 0) and(1, 0) the figure shows the 1σ, 90%, 2σ, 99%, 3σ CL (2 dof) allowed regions of the third vertex for
the NO from the analysis in Ref. [180, 181].

14.8 Beyond 3ν: Additional Neutrinos at the eV Scale
Besides the huge success of three-flavour oscillations described in Sec.14.7, as mentioned inSecs.14.6.3 and 14.6.4, there are some anomalies which cannot be explained within the 3ν framework
and which have been interpreted as hints for the existence of additional neutrino states with masses
at the eV scale. In brief:• the LSND experiment [123] reported evidence for ν̄µ → ν̄e transitions with E/L ∼ 1 eV2 ,where E and L are the neutrino energy and the distance between source and detector, respectively (see Short Baseline Experiments subsection of Sec.14.6.3).
• this effect was further explored by the MiniBooNE experiment [189], which reported a yet
unexplained event excess in the low-energy region of the electron neutrino and antineutrino
event spectra. No significant excess is found at higher neutrino energies. Interpreting the datain terms of oscillations, parameter values consistent with the ones from LSND were obtained,thought the test was not definitive;
31st May, 2024

38

14. Neutrino Masses, Mixing, and Oscillations• radioactive source experiments at the Gallium solar neutrino experiments both in SAGE
and GALLEX/GNO obtained an event rate that is somewhat lower than expected. This
result was recently confirmed by the BEST experiment [190]. If not due to uncertainties in
the interaction cross-section, this effect by itself can be explained by the hypothesis of νedisappearance due to oscillations with ∆m2 & 1 eV2 (“Gallium anomaly”) [191–193];• the calculations of the neutrino flux emitted by nuclear reactors in Refs. [134,135] predicted aneutrino rate that is a few percent higher than observed in short-baseline (L . 100 m) reactor
experiments. This “reactor anomaly” could be explained by assuming ν̄e disappearance due tooscillations with ∆m2 ∼ 1 eV2 [194]. The flux predictions from [134,135] were not supportedby the reactor antineutrino flux measurement at Daya Bay [195, 196] which was consistent
with the older (lower) flux predictions and the flux measurement results in the short-baseline
reactor neutrino oscillation experiments. In the last years the reactor flux calculations havebeen revisited [197–199] and in most cases lower fluxes are predicted, in agreement with theflux measurement at Daya Bay. These new fluxes, correspondingly reduce the significance of
the anomaly [200].

These anomalies are under study by the experimental community with a set of follow-up measurements performed at SBL both at reactors and accelerators (see the corresponding subsections in
Sec.14.6.4 and Sec.14.6.3).
As mentioned in Sec.14.1, whatever the extension of the SM we want to consider, it must contain
only three light active neutrinos. Therefore if we need more than three light massive neutrinos, wemust add sterile neutrinos to the particle content of the model.
The most immediate question as these anomalies were reported was whether they could all beconsistently described in combination with the rest of the neutrino data – in particular with thenegative results on the disappearance of νµ at short distances – if one adds those additional sterile
states. Quantitatively one can start by adding a fourth massive neutrino state to the spectrum andperform a global data analysis to answer this question. Although the answer is always the same,the physical reason behind it depends on ordering assumed for the states. In brief, there are six
possible four-neutrino schemes that can, in principle, accommodate the results of solar+KamLAND
and atmospheric+LBL neutrino experiments as well as the SBL result. They can be divided into twoclasses: (2+2) and (3+1). In the (3+1) schemes, there is a group of three close-by neutrino masses(as on the 3ν schemes described in the previous section) that is separated from the fourth one by agap of the order of 1 eV, which is responsible for the SBL oscillations. In (2+2) schemes, there are
two pairs of close masses (one pair responsible for solar results and the other for atmospheric [201])
separated by the O( eV) gap. The main difference between these two classes is the following: if
a (2+2)-spectrum is realized in nature, the transition into the sterile neutrino is a solution ofeither the solar or the atmospheric neutrino problem, or the sterile neutrino takes part in both.
Consequently, a (2+2)-spectrum is easier to test because the required mixing of sterile neutrinos ineither solar and/or atmospheric oscillations would modify their effective matter potential in the Sun
and in the Earth, giving distinctive effects in the solar and/or atmospheric neutrino observables.
Those distinctive effects were not observed so oscillations into sterile neutrinos did not describe
well either solar or atmospheric data. Consequently, as soon as the early 2000’s 2+2 spectra could
be ruled out already beyond 3-4 σ as seen in the left panel in Fig.14.10 taken from Ref. [202].
On the contrary, for a (3+1)-spectrum (and more generally for a 3 + N -spectrum with anarbitrary N number of sterile states), the sterile neutrino(s) could be only slightly mixed withthe active ones and mainly provide a description of the SBL results. In this case, the oscillation
probabilities for experiments working at E/L ∼ 1 eV2 take a simple form:
Pαα = 1 − sin2 2θαα sin2 ∆ ,31st May, 2024

Pµe = sin2 2θµe sin2 ∆ ,(14.87)

14. Neutrino Masses, Mixing, and Oscillations

39
40

atm+SBL
solar
global30

99.73% CL
2 dof

101

2

∆χ

2

Δm2 [eV2]

χPG

20
2

χPC

Appearance

100(

w/o DiF)

10
Disappearance

0

0

0.2

0.4

0.6

0.8

1

10-1
10-4

Free Fluxes
Fixed Fluxes

ηs

10-3

10-2
sin2 2θμe

Figure 14.10: Left: Status of the 2+2 oscillation scenarios from Ref. [202] (ηS =

10-1

X

|Uis |2 whereii runs over the two massive states mostly relevant for solar neutrino oscillations). In the figure
also shown are the values of χ2PC and χ2PG relevant for parameter consistency test and parameter
goodness of fit, respectively. Right: Present status of 3+1 oscillation scenarios from Ref. [203].
where ∆ ≡ ∆m241 L/4E and one can define effective mixing angles
sin2 2θαα ≡ 4|Uα4 |2 (1 − |Uα4 |2 ) ,sin2 2θµe ≡ 4|Uµ4 |2 |Ue4 |2 .

(14.88)In here α = e, µ and Uα4 are the elements of the lepton mixing matrix describing the mixing of the
4th neutrino mass state with the electron and muon flavour. In this scenario, there is no sensitivity
to CP violation in the the ∆ driven oscillations, so the relations above are valid for both neutrinos
and antineutrinos. At linear order in the mixing elements one can derive a relation between theamplitudes of appearance and disappearance probabilities:
4 sin2 2θµe ≈ sin2 2θee sin2 2θµµ .

(14.89)This relation implies a constraint between the possible results in disappearance and appearance
experiments. Consequently, it is not trivial to find a consistent description to all the SBL anomalies.
Over the years, different groups have performed a variety of such global analyses leading to different
quantitative conclusions on the statistical quality of the global fit (see for example [203–208], seealso Refs. [209,210] for recent reviews on the subject). Generically, the results of the global analysis
show that there is significant tension between groups of different data sets – in particular between
appearance and disappearance results – and Eq.(14.89) makes it difficult to obtain a good global
fit as illustrated in the right panel in Fig.14.10 taken from Ref. [203] which concluded that 3+1
scenario is excluded at 4.7σ level.
A straightforward question to ask is whether the situation improves if more neutrino states atthe eV scale are introduced. The simplest extension is the introduction of 2 states with eV scale mass
splittings, ν4 and ν5 . The ordering of the states can be such that ∆m241 and ∆m251 are both positive(“3+2”), or one of them is negative (“1+3+1”). From the point of view of the description of thedata, the most important new qualitative feature is that now, non-zero CP violation at E/L ∼ eV2
is possibly observable [206, 211–213]. This allows some additional freedom in fitting neutrino
31st May, 2024

40

14. Neutrino Masses, Mixing, and Oscillations

versus antineutrino data from LSND and MiniBooNE together. However, it still holds that a nonzero νµ → νe appearance at SBL necessarily predicts SBL disappearance for both νe and νµ . So,
generically, the tension between appearance and disappearance results remains, though differencesin the methodology of statistical quantification of the degree of agreement/disagreement in thesescenarios can lead to different conclusions on whether they can provide a successful description of
all the data [203, 209, 210]. Cosmological observations can provide complementary information on
the number of relativistic neutrino states in thermal equilibrium in the early Universe and on thesum of their masses, which sets further constraints on light sterile neutrino scenarios (see Chapter
26 of this Review, “Neutrinos in cosmology”).

14.9 Laboratory Probes of ν Mass Scale and its Nature
As described in Secs.14.4 and 14.5, neutrino flavour oscillations in vacuum and flavour transitions in matter only depend on the differences between the neutrino masses-squared, ∆m2ij , and on
the mixing matrix elements, Uij . However, they are insensitive to the absolute mass scale for theneutrinos, mi . They also give us no information on whether they are Dirac or Majorana particles.
Clearly, the observation ofq flavour oscillations implies a lower bound on the mass of the heavier
2
neutrino in ∆mij , |mi | ≥ ∆m2ij for ∆m2ij > 0. However, there is no upper bound on mi .
In particular, oscillation results allow theq neutrino spectrum to be approximately degenerate at amass scale that is much higher than the ∆m2ij that they determine. Information of the mass scaleof the neutrino is provided by other types of experiments. Here we briefly summarize the mostsensitive laboratory probes of the neutrino mass scale and on whether they are Dirac or Majorana
particles. Cosmological observations provide, albeit indirectly, complementary information on theneutrino mass scale as it is reviewed in Chapter 26 of this Review, “Neutrinos in cosmology”.
14.9.1 Constraints from Kinematics of Weak Decays
The only model-independent information on the neutrino masses, rather than mass differences,can be extracted from energy-momentum conservation relation in reactions in which a neutrino oran antineutrino is involved.
Historically these bounds were labeled as limits on the mass of the flavour neutrino statescorresponding to the charged flavour involved in the decay. Fermi proposed in 1933 such a kinematic
search for the νe neutrino mass (which we will label here as meffνe ) in the end part of the beta spectra333−
in H beta decay H → He + e + ν̄e .
Because 3 H beta decay is a superallowed transition, the nuclear matrix elements are energy
independent, so the electron spectrum is determined exclusively by the phase spaceqqdN
2 F (E) ≡ R(E) (E − E)2 − (meff )2 .
= C p E (Q − T ) (Q − T )2 − (meff
)
0
νe
νe
dE

(14.90)E0 is the mass difference between the inital and final nucleus, E = T + me is the total electronenergy, p its momentum, Q ≡ E0 − me is the maximum kinetic energy of the electron and Finalstate Coulomb interactions are contained in the Fermi function F (E). R(E) in the second equality
contains all the mν -independent factors.
q1
eff
The Kurie function is defined as K(T ) ≡ dN
dE pEF (E) . From Eq.(14.90), we see that if mνe =0
K(T ) would depend linearly on T . A non-vanishing neutrino mass then provokes a distortion fromeff
the straight-line T -dependence at the endpoint, So for meffνe = 0, Tmax = Q, while for mνe 6= 0,
3
Tmax = Q − meffνe . In H beta decay Q = 18.6 KeV is very small and therefore, this decay is more
effsensitive to this mνe -induced distortion.
The most recent result on the kinematic search for neutrino mass in tritium decay is fromKATRIN [214], an experiment which has found so far no indication of mνe 6= 0 and sets an upper
31st May, 2024

14. Neutrino Masses, Mixing, and OscillationsNuFIT 4.1 (2019)

41

0

-1

-1

10

ε

mν (eV)

10

10

IO
NO
mee (eV)

0

10

-2

10

-3

-2

10

10

-4

0.001

0.01

0.1
mlight (eV)1100.0001

0.001

0.01
0.1
mlight (eV)1

Figure 14.11: Allowed 95% CL ranges (1 dof) for the neutrino mass observable determined in 3 H
beta decay (left panel) and in 0νββ (right panel) in the framework of 3ν mixing as a function of
the lightest neutrino mass. The ranges are obtained by projecting the results of the global analysis
of oscillation data (w/o SK-atm) in Ref. [180, 181]. The region for each ordering is defined withrespect to its local minimum.
limit
meff
νe < 0.8 eV ,(14.91)at 90% CL improving over the previous bound from the Mainz [215] and Troitsk [216] experiments
which constrained meffνe < 2.2 eV at 95% CL. KATRIN continues running with an estimated sensitivity limit of meff∼
0.2 eV. Project 8 is exploring a new technique for β-spectrometry based onνecyclotron radiation [217].
An alternative isotope to Tritium is 163 Ho [218] which presents the advantage of a smallerQ = 2.8 KeV. It decays via electron capture to 163 Dy. Currently, there are three experiments
exploring this decay to probe the neutrino mass: ECHo [219], HOLMES [220], and NuMECS [221].
These experiments are complementary to tritium-based searches from a technical point of view.
Also, the decay of 163 Ho determines the effective electron neutrino mass as opposed to antineutrino
in Tritium.
For the other flavours the present limits compiled in the listing section of the PDG readmeffνµ < 190 keV (90% CL)

from

meff
ντ < 18.2 MeV (95% CL)

from

π − → µ− + ν̄µ ,τ

−→ nπ + ντ .

(14.92)(14.93)In the presence of mixing and for neutrinos with small mass differences, the distortion of thebeta spectrum is given by the sum of the individual spectra generated incoherently by each neutrino
massive state weighted with the relevant mixing matrix element squared [222]:
q
X
dN= R(E)|Uei |2 (E0 − E)2 − m2i Θ(E0 − E − mi ) .
dE
i(14.94)The step function Θ(E0 − E − mi ) arises because a neutrino with a given mass mi can only be
produced if the available energy is larger than its mass. Equation (14.94) shows the two main
effects of the neutrino masses and mixings on the electron energy spectrum: First, kinks appear(i)at the electron energies Ee = E ∼ E0 − mi with sizes that are determined by |Uei |2 . Second, the31st May, 2024

14. Neutrino Masses, Mixing, and Oscillations

42

endpoint shifts to Eep = E0 − m0 , where m0 is the lightest neutrino mass. Corrections are induced
once the energy resolution of the experiment is considered. [223, 224]
In the 3-ν mixing scenario, the distortion of the spectrum can still be effectively described by asingle parameter – which we will still denote as mνe – if for all neutrino states E0 −E = Q−T  mi .
In this case, one can expand Eq.(14.94) as:q
XdN
2' R(E)|Uei |2 (E0 − E)2 − (meffνe ) ,
dE
i

with
2
(meff
νe ) =(14.95)P 2
2
Xi mi |Uei |P=
m2i |Uei |2 ,2i |Uei |(14.96)iwhere unitarity is assumed in the second equality. In this approximation, the distortion of theendpoint of the spectrum is described by a single parameter, and with the present results fromKATRIN, it is bounded to be
1.1 eV ≥ meff
νe =

 q

 m20 + ∆m221 (1 − c213 c212 ) + ∆m232 s213
2
2
mi |Uei | =q m2 + ∆m2 c2 c2 − ∆m2 c2i
0
21 13 1232 13

sX

in NO ,
(14.97)in IO ,

where m0 = m1 (m3 ) is the lightest neutrino mass in the NO (IO) spectrum. CorrespondinglyP
the bounds in Eqs.(14.92) and (14.93) apply to the combinations i m2i |Uαi |2 for α = µ and τ
respectively. So with the values known of the mixing matrix elements, the strongest constraint on
the absolute value of the neutrino mass comes from Tritium beta decay.
From Eq.(14.97) we see that, given the present knowledge of the neutrino mass differences andtheir mixing from oscillation experiments, it is possible to translate the experimental informationof mνe on a corresponding range for the lightest neutrino mass and that such relation depends on
the ordering of the states. We plot in Fig.14.11 the recasting of the allowed regions of the analysis
in Ref. [180, 181] in terms of the allowed range mνe as a function of mlight ≡ m0 . In particular, one
finds that the results of oscillation experiments imply a lower bound on mνe > 0.048 (0.0085) eV
for IO (NO) at 95% CL.

p
n
e−

W
ν
W

e−

n
p

Figure 14.12: Feynman diagram for neutrinoless double-beta decay.
14.9.2 Dirac vs. Majorana: Neutrinoless Double-beta Decay
The most sensitive probe to whether neutrinos are Dirac or Majorana states is the neutrinolessdouble beta decay (0νββ):(A, Z) → (A, Z + 2) + e− + e− .
(14.98)31st May, 2024

14. Neutrino Masses, Mixing, and Oscillations

43

In the presence of neutrino masses and mixing, the process in Eq.(14.98) can be generated at lower
order in perturbation theory by the term represented in Fig.14.12. The corresponding amplitude
is proportional to the product of the two leptonic currents
Mαβ ∝ [ēγα (1 − γ5 )νe ] [ēγβ (1 − γ5 )νe ] ∝X(Uei )2 [ēγα (1 − γ5 )νi ] [ēγβ (1 − γ5 )νi ] .

(14.99)i

The neutrino propagator in Fig.14.12 can only arise from the contraction h0 | νi (x)νi (y)T | 0i. However, if the neutrino is a Dirac particle νi field annihilates a neutrino state and creates an antineutrino state, and neutrino and antineutrino states are different, so the contraction h0 | νi (x)νi (y)T | 0i =0 and Mαβ = 0. On the other hand, if νi is a Majorana particle, neutrino and antineutrino aredescribed by the same field and h0 | νi (x)νi (y)T | 0i =6 0.
The conclusion is that in order to induce the 0νββ decay, neutrinos must be Majorana particles.
This is consistent with the fact that the process (14.98) violates the total lepton number by twounits. Conversely, if 0νββ decay is observed, massive neutrinos cannot be Dirac states [225].
It is important to stress that neutrinoless double-beta decay could be dominantly induced byother new physics effects beyond that of Majorana neutrino masses. Consequently, the connection
between the observation or limitation of the neutrinoless double beta decay and the neutrino masscan only be made under some assumption about the source of total lepton number violation in themodel.
The observable determined by the experiments is the half-life of the decay. Under the assumption that the Majorana neutrino mass is the only source of lepton number violation at low energies,
the decay half-life is given by:
0ν −1
(T1/2
) = G0ν

M

0ν

2



mee
me

2

,

(14.100)

where G0ν is the phase space integral taking into account the final atomic state, |M 0ν | is thenuclear matrix element of the transition, and mee is the effective Majorana mass of νe ,
mee =

X

2
mi Ueiiqq m0 c212 c213 + ∆m221 + m20 s212 c213 e2i(η2 −η1 ) + ∆m232 + ∆m221 + m20 s213 e−2i(δCP +η1 )
=
q
q


 m0 s213 + m20 − ∆m232 s212 c213 e2i(η2 +δCP ) + m20 − ∆m232 − ∆m221 c212 c213 e2i(η1 +δCP )in NO ,
in IO ,

which, in addition to the masses and mixing parameters that affect the tritium beta decay spectrum,
also depends on the leptonic CP -violating phases. We plot in Fig.14.11 the recasting of the allowed
regions of the analysis in Ref. [180,181] in terms of the allowed range mee as a function of mlight ≡ m0
for the two orderings. As a consequence of the dependence on the unknown Majorana phases, theallowed range of mee for a given value of mlight and ordering is substantially broader than thatof mνe . Nevertheless, the results of oscillation experiments imply a lower bound on the effective
Majorana mass for the IO, which at 95%CL reads mee > 0.016 eV.
From Eq.(14.100) we see that nuclear structure details enter relation between the decay rate (orlifetime) and the effective Majorana mass. As a consequence, uncertainties in the nuclear structure0ν by a factor of 2–3 [226].
calculations result in a spread of mee values for a given T1/2
We present in Sec.14.9.3 a brief description of the experimental searches for neutrinoless doublebeta decay. At the time of writing of this review, the strongest bound on 0νββ decay lifetimes
came for Xenon and Germanium from the KamLAND-Zen [227] and GERDA [228] experiments,
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14. Neutrino Masses, Mixing, and Oscillations

0ν > 2.3 × 1026 yr, andrespectively. They have set a bound on the corresponding half-lives of T1/2
0ν > 1.8 × 1026 yr, at 90% CL. Using a variety of nuclear matrix element calculations, theT1/2
corresponding upper bounds on the effective Majorana mass aremee < 36 − 156 meV(14.101)

mee < 79 − 180 meV .

(14.102)14.9.3 Experimental Search for Neutrinoless Double-beta Decay
There are 35 candidate nuclei for double-beta decay. Currently, experiments using 136 Xe and
76 Ge have reported the most sensitive results of 0νββ search. Because of the uncertainties related
to the nuclear matrix element, the complementarity of technologies, different backgrounds, and theinvestigation of the mechanism behind the 0νββ in case of a positive signal, it is important topursue the searches with as many isotopes as possible.
The signature of 0νββ is that the sum of the energy of two electrons is equal to the Q-value ofthe nuclear transition. The energy from electrons is measured with either ionization, scintillation,or through phonons. In some experiments, a combination of two techniques is used to enhance
the sensitivity. In the case of background-free measurement, the sensitivity to the half-life is
proportional to the product of the detection efficiency of the signal
q ε, the source mass M , and theMtmeasurement time t. If background exists, it is proportional to ε b∆E, where b is the backgroundrate, and ∆E is the energy resolution.
Among the experiments using ionization detection, ultra-high-purity germanium detector provides the best sensitivity thanks to high energy resolution and low background. GERDA achieved−4 counts/(keV·kg·year) [228], which enabled a background-freea background level of (5.2+1.6−1.3 ) × 10
search. The final result from GERDA for the half-life limit for 0νββ decay of 76 Ge is T1/2 > 1.8×1026
years at 90% CL. The Majorana Demonstrator achieved energy resolution of 2.5 keV FWHM (fullwidth at half maximum) at the Q-value (2.039 MeV) [229] and set a half-life limit of T1/2 > 8.3×1025
years at 90% CL. GERDA and Majorana collaborations have united and formed, together with newgroups, the LEGEND collaboration. LEGEND has adopted a phased approach toward the deployment of a tonne of enriched Ge detectors [230]. The first phase of LEGEND based on approximately
200 kg of enriched Ge, using the GERDA infrastructure at LNGS, has started and first data arebeing taken. The second phase of LEGEND has been approved as a one ton experiment in a new
cryostat reaching a discovery sensitivity of half-lifes beyond 1028 years. In China, CDEX-300ν using
225 kg of enriched Ge detector is under preparation [231] in the existing CEDX-DM infrastructure
at the China Jinping Underground Laboratory.
Liquid scintillator detectors have a simple structure and can utilize existing large detectors
with low background environments. By adding an inner balloon to contain xenon-loaded liquid
scintillator to the KamLAND detector, KamLAND-Zen uses 745 kg of xenon with 90.9% enrichment
in 136 Xe. KamLAND-Zen reported the half-life limit of 2.3 × 1026 years at 90% CL [227]. The SNO
detector has also been upgraded to be filled with a liquid scintillator in the SNO+ experiment [232].
The SNO+ detector will be loaded with 0.5% natural tellurium, corresponding to approximately
1.3 t of 130 Te, to search for 0νββ.
With a time projection chamber, one can utilize both ionization and scintillation. EXO-200
used a liquid xenon time projection chamber with enriched Xe and reported a half-life limit ofT1/2 > 3.5 × 1025 years at 90% CL [233] for 136 Xe. Based on the technology validated by EXO200, nEXO plans to use about 5,000 kg of enriched xenon [234]. NEXT has been developing ahigh-pressure xenon gas time projection chamber with electroluminescent amplification and opticalreadouts. An energy resolution of 1% FWHM at the Q-value of 136 Xe 0νββ is demonstrated with
NEXT-White detector [235].
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14. Neutrino Masses, Mixing, and Oscillations

CUORE uses a cryogenic bolometer to measure the energy in a calorimetric way. The detector
composed of 988 TeO2 bolometers for a total mass of 742 kg, corresponding to 206 kg of 130 Te, wasoperated at 10 mK. CUORE set a half-life limit of T1/2 > 2.2×1025 years at 90% CL for 130 Te [236].
For further reduction of background towards future search based on the CUORE technology, CUPID
proposes to simultaneously measure the calorimetric signal and the scintillation light with a tonnescale bolometric detector. Using the prototype the technology is demonstrated, and 0νββ is also
searched for with 82 Se [237] and 100 Mo [238].
AMoRE also uses the simultaneous detection of heat and scintillation. Six 100 Mo-enrichedand 48 Ca-depleted CaMoO4 crystals with a total mass of 1.9 kg (AMoRE-Pilot) were operatedin Yangyang underground laboratory located in South Korea, searching for 0νββ of 100 Mo [239].
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information after discovery. The NEMO-3 experiment used 7 isotopes, with the largest mass
comprised of 100 Mo (7 kg) [241].
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