Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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1
6 1 Neutrinos: Past, Present an
d
Futur
e
c
onstrained the
fl
ux o
f
muon neutrinos emitted
f
rom the
g
amma ray bursts to
be lower than
2
.
7
×
1
0
−
3
er
g
cm
−
2
in the ener
g
yran
g
e3Te
V
<E
ν
<
2
.
8
Pe
V
(
Abbas
i
e
ta
l
., 2010
)
.Ak
m
3
-
sca
l
eun
d
erwater neutrino te
l
escope,
k
nown a
s
the KM3NeT
(
Carr
et al.
, 2007
)
, is now under consideration. The discovery o
f
u
ltrahigh-energy cosmic neutrinos will shed light on the origin of ultrahigh-
energy cosmic rays and examine the intrinsic properties of neutrinos them
-
s
e
l
ves. In particu
l
ar, we
h
ope t
h
at u
l
tra
h
i
gh
-ener
g
y cosmic neutrinos cou
ld
finally become a useful messenger between human beings and the distant o
r
even invisi
bl
e
U
niverse
.
Question 6: how to detect cosmic back
g
round neutrinos
?
A
ccordin
g
to the
s
tandard Big Bang model of cosmology, today’s abundance of relic neutrino
s
a
nd that of relic photons are of the same order
(
Weinberg, 2008
)
. The pre
-
c
ision measurement o
f
the
C
MB has remarkably improved our knowled
g
eo
n
the matter content and structure formation of the Universe. Unfortunately,
i
tisabi
g
challen
g
e to directly detect the
C
ν
B
because the ener
g
ies o
f
reli
c
n
eutrinos are too low
(
∼
1
0
−
4
e
V
)
and their interactions with matter are
too weak. Current proposals for detecting relic neutrinos fall into three cat
-
egories
(
Ringwald, 2009
)
:
(
a
)
using the mechanical force — relic neutrinos
s
catterin
g
off the nuclei of a tar
g
et will cause a mechanical force which can
be measured by the Cavendish-type torsion balance;
(
b
)
using the captur
e
on ra
d
ioactive nuc
l
ei — re
l
ic neutrinos wi
ll b
ea
b
sor
b
e
dby
ra
d
ioactive nu-
c
lei via an inverse beta decay
(
e.g.,
ν
e
+
3
H
→
3
H
e
+
e
−
)
;and
(
c
)
using
the cutoff of an ultrahigh-energy neutrino spectrum — the ultrahigh-energy
c
osmic neutrinos wit
h
ener
g
ie
s
E
ν
>
10
21
e
V will be absorbed or cut o
ff
b
y
the C
ν
B
via the resonant reactio
n
ν
α
+
ν
α
→
Z
0
→
a
nything
(
Weiler, 1982,
1
999; Fargio
n
et al.
,
1999
)
. However, the sensitivities of such methods ar
e
l
imited by the present technolo
g
y and still several orders o
f
ma
g
nitude lower
than that demanded b
y
a successful detection of the
C
ν
B.
Q
uestion 7: what role is played by neutrinos in the evolution o
f
the Uni
-
verse?
T
he existence o
f
neutrinos in the earl
y
Universe has onl
y
indirectl
y
been confirmed by the success of the standard Big Bang cosmology. Cur
-
rent observational data on the abundances o
f
primordial li
g
ht elements, the
power spectra o
fC
MB anisotropies and the lar
g
e-scale structure
f
ormatio
n
a
ll suggest that neutrinos be playing a crucial role in the evolution of the
U
niverse
(
Hannestad, 2006
)
. We expect that this intriguing picture can be
f
urther verified b
y
the future observations. If the observed bar
y
on numbe
r
a
symmetry of the Universe has something to do with the occurrence of heav
y
Majorana neutrinos in the early Universe
(
Fukugita and Yanagida, 1986
),
then the
y
must have affected the evolution of the Universe. How to test suc
h
b
aryogenesis-via-
l
eptogenesis mec
h
anisms is anot
h
er
b
urning question in to
-
d
ay’s particle physics and cosmology
(
Xing, 2008
)
.
S
o far we have onl
y
listed some experimental or observational questions
ab
out neutrinos. T
h
ere are certain
l
ymanyt
h
eoretica
l
questions on t
h
emar
-
ket. For example, we wonder what the ori
g
in o
f
neutrino masses is, why th
e
R
e
f
erences 1
7
pattern o
f
lepton
fl
avor mixin
g
is so di
ff
erent
f
rom that o
f
quark
fl
avor mixin
g,
what the ori
g
in of CP violation is, how the cosmolo
g
ical matter-antimatter
a
symmetry is related to CP violation at low energies, what the role of neu
-
trinos is in
d
ar
k
matter, w
h
et
h
er neutrinos
h
ave somet
h
in
g
to
d
owit
hd
ar
k
energy, whether there exist sterile neutrinos in the Universe, and so forth
.
Motivated by so many interesting and fundamental questions, we are tryin
g
to discover a new ph
y
sics world with the help o
f
neutrinos in the era o
f
the
L
arge Hadron Collider and in the era of precision astronomy and cosmology.
Last but not least, will neutrinos spring many more surprises on us? Th
e
h
istory o
f
neutrino physics is
f
ull o
f
surprises.
S
o it would be surprisin
g
i
f
f
urther sur
p
rises were not in store. In this sense one must be well
p
re
p
are
d
to encounter somet
h
in
g
t
h
at is comp
l
ete
l
y unexpecte
d
,a
g
ain an
d
a
g
ain, o
n
the way of neutrino exploration. The road ahead must be bri
g
ht and excitin
g!
R
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2
N
eutrinos within the
S
tandard Model
N
eutrinos are assumed to be massless in the standard model
(
SM
)
of par
-
tic
l
ep
h
ysics. T
h
ey can on
l
y
h
ave wea
k
interactions wit
h
matter. In t
h
is
c
hapter we shall
fi
rst introduce the
S
Mo
f
electroweak interactions and
then describe neutrino-electron scatterin
g
, neutrino-neutrino scatterin
g
and
n
eutrino-nucleon interactions. The coherent forward scattering of neutrino
s
with matter, which is ver
y
important
f
or neutrino oscillations in matter, wil
l
a
lso be discussed in some detail.
2
.1 Fundamentals o
f
the
S
tandard Mode
l
There exist three generations of quarks and leptons which are the fundamen
-
tal building blocks of matter. The interactions of these elementary particles
c
an be described in the lan
g
ua
g
eo
fg
au
g
e
fi
eld theories. For instance, the elec
-
tromagnetic force between two charged particles is mediated by the photon, a
gauge boson in quantum electrodynamics
(
QED
)
based on the Abelian grou
p
U
(
1
)
. The perfect agreement between the QED prediction and the experimen
-
tal measurement of the muon anomalous magnetic momen
t
a
μ
≡
(
g
−
2
)
/
2
proves the use
f
ulness and exactness o
f
quantum
fi
eld theories. We shall outlin
e
the structure of the SM, which unifies electroma
g
netic and weak interactions
of elementary particles, in this section. Three indispensable ingredients of th
e
S
M—
g
au
g
e principles, spontaneous symmetry breakin
g
and renormalizabil-
i
t
y
, will be explained in order.
2.1.1
G
auge
S
ymmetries
L
et us illustrate the gauge principle by examining QED, which exhibits the
U
(
1
)
gauge invariance. Consider the Lagrangian of a free electron describe
d
b
y
the fermion fiel
d
Ψ
(
x
)
,
L
0
=
Ψ
(
x
)
i
/
∂
/
/
−
m
Ψ
(
x
)
,
(
2.1
)
Z.-Z. Xing et al., Neutrinos in Particle Physics, Astronomy and Cosmology
© Zhejiang University Press, Hangzhou and Springer-Verlag Berlin Heidelberg 2011
22 2 Neutrinos within the
S
tandard Model
whe
r
e
/
∂
//
≡
γ
μ
∂
μ
i
sdefi
n
ed a
n
d
m
d
enotes t
h
ee
l
ectron mass. It is eas
y
to ver
-
if
ythat
L
0
i
s
in
va
ri
a
n
tu
n
de
r
t
h
et
r
a
n
sfo
rm
at
i
o
n
Ψ
(
x
)
→
Ψ
(
x
)
=
e
−
i
α
Ψ
(
x
)
w
i
th
α
bein
g
a spacetime-independent real constant. I
f
one promotes th
e
transformation int
o
Ψ
(
x
)
→
Ψ
(
x
)
=
e
−
i
α
(
x
)
Ψ
(
x
)
wit
h
α
(
x
)
being an arbi
-
trar
y
real
f
unction o
f
x
,h
owever,
L
0
w
i
ll
n
ot be
in
va
ri
a
n
tu
n
de
r
th
i
s
n
ew
transformation. In order to preserve the local
g
au
g
e invariance of
L
0
,
we
i
n
-
tro
d
uce t
h
ecovariant
d
erivative
D
μ
≡
∂
μ
−
i
eA
μ
(
x
)
,wher
e
e
is t
h
ee
l
ectric
c
har
g
eo
f
the proton an
d
A
μ
(
x
)
is a four-vector gauge field. The requirement
tha
t
D
μ
Ψ
(
x
)
transforms in the same way as
Ψ
(
x
)
allows us to determine ho
w
A
μ
(
x
)
transforms. More explicitly, we have
D
μ
Ψ
(
x
)
→
D
μ
Ψ
(
x
)
=
∂
μ
−
i
e
A
μ
(
x
)
e
−
i
α
(
x
)
Ψ
(
x
)
=
e
−
i
α
(
x
)
D
μ
Ψ
(
x
)
,
(
2.2
)
w
h
ic
hg
ives rise t
o
A
μ
(
x
)
→
A
μ
(
x
)
=
A
μ
(
x
)
−
1
e
∂
μ
α
(
x
)
.
(
2.3
)
It is the local gauge invariance that necessitates the introduction of the gaug
e
fie
l
d
A
μ
(
x
)
and uniquely fixes the interaction between the charged particle
s
a
nd
A
μ
(
x
)
. The kinetic term of
A
μ
(
x
)
can be constructed from the strengt
h
t
enso
r
F
μν
F
≡
∂
μ
A
ν
(
x
)
−
∂
ν
A
μ
(
x
)
, which is evidently gauge invariant as indi
-
c
ated by Eq.
(
2.3
)
. Therefore, the total gauge-invariant Lagrangian of QED
c
an be written as
L
Q
ED
=
−
1
4
F
μ
ν
F
μν
F
+
Ψ
(
x
)
i
/
D
//
−
m
Ψ
(
x
)
,
(
2.4
)
w
h
ere
/
D
//
≡
γ
μ
D
μ
i
s defined. From the above equation, one may easily deriv
e
the electron-
p
hoton verte
x
e
Ψ
(
x
)
γ
μ
Ψ
(
x
)
A
μ
(
x
)
.
I
n 1954,
C
hen Nin
g
Yan
g
and Robert Mills proposed that the local isospi
n
s
ymmetry could be maintained in strong interactions
(
Yang and Mills, 1954
)
.
In this seminal
p
a
p
er the
p
roton and neutron were combined to form a doublet
u
nder the non-
A
belian
g
roup
SU
(
2
)
. It is straightforward to generalize their
i
dea to any non-
A
belian
g
roup SU
(
N
)
. The generators of the group SU
(
N
)
s
atisfy the commutation relatio
n
t
i
,t
j
=i
c
ij
k
t
k
,
(
2.5
)
whe
r
e
i
,
j
,
k
ru
n
ove
r
1
,
2
,
···
,
N
2
−
1a
n
d
c
ij
k
i
sthest
r
uctu
r
eco
n
sta
n
twh
i
ch
i
s full
y
antis
y
mmetric with respect to the superscripts. The collective fields
Ψ
(
x
)
=
[
ψ
1
(
x
)
,
ψ
2
(
x
)
,
·
·
·
,ψ
N
(
x
)]
T
t
ransform in the following way:
Ψ
(
x
)
→
Ψ
(
x
)
=
e
−
i
θ
k
(
x
)
T
k
Ψ
(
x
)
,
(
2.6
)
whe
r
e
T
k
is the matrix representation o
f
the
g
enerator
t
k
,an
d
θ
k
(
x
)(
for
k
=
1
,
2
,
···
,N
)
are the spacetime-dependent real functions. Similar to th
e
A
belian gauge theory, the covariant derivative is now defined a
s
2
.1 Fundamentals o
f
the
S
tandard Model 2
3
D
μ
=
∂
μ
−
i
gT
k
A
k
μ
(
x
)
,
(
2.7
)
whe
r
e
A
k
μ
(
x
)(
for
k
=
1
,
2
,
·
·
·
,
N
)
stand for the gauge fields. If
D
μ
Ψ
(
x
)
i
s required to trans
f
orm as
D
μ
Ψ
(
x
)
→
D
μ
Ψ
(
x
)
=e
−
i
θ
k
(
x
)
T
k
D
μ
Ψ
(
x
)
,then
A
k
μ
(
x
)
must transform as follows:
T
k
A
k
μ
→
T
k
A
k
μ
=e
−
i
θ
k
T
k
T
k
A
k
μ
e
i
θ
k
T
k
−
i
g
∂
μ
e
−
i
θ
k
T
k
e
i
θ
k
T
k
.
(
2.8
)
F
or the in
fi
nitesimal trans
f
ormation, we hav
e
A
k
μ
(
x
)
→
A
k
μ
(
x
)=
A
k
μ
(
x
)
−
1
g
∂
μ
θ
k
(
x
)
−
c
k
l
m
A
l
μ
(
x
)
θ
m
(
x
)
.
(
2.9
)
The
fi
eld stren
g
th tensor reads
F
i
μν
F
=
∂
μ
A
i
ν
(
x
)
−
∂
ν
A
i
μ
(
x
)+
gc
i
jk
A
j
μ
(
x
)
A
k
ν
(
x
)
,
(
2.10
)
which constitutes the kinetic term of the gauge fields. It is obvious that th
e
g
au
g
e-invariant La
g
ran
g
ian takes the
f
orm
L
SU
(
N
)
=
−
1
4
F
i
μν
F
F
i
μ
ν
+
Ψ
(
x
)i
/
DΨ
//
(
x
)
.
(
2.11
)
One of the most successful examples of such a gauge theory is quantu
m
c
hromodynamics
(
QCD
)
, which describes the strong interactions of quarks
with gluons
(
Fritzsc
h
et al.
,
1973
)
. One of the experimentally-verified QC
D
predictions is the existence of three-gluon and four-gluon interaction vertice
s
(
Elli
s
et al.
,
1976; Bran
d
e
l
i
k
et al
., 1979
)
, but glueballs
(
the bound states of
gluons
)
have not been identified in any high-energy physics experiments.
2.1.2 Spontaneous Symmetry Breakin
g
A
lthou
g
h the preliminary idea of
g
au
g
e theories appeared as early as in the
1
920’s
(
Weyl, 1929
)
and the non-Abelia
n
SU
(
2
)
gauge symmetry was ex-
p
l
ore
db
y Yang an
d
Mi
ll
sint
h
e 1950
’
s, t
h
eir importance was not wi
d
e
l
y
reco
g
nized until the mechanism o
f
spontaneous symmetry breakin
g
was in
-
vented in the 1960’s.
A
sim
p
le reason is that the weak interactions must b
e
m
e
d
iate
db
y massive gauge
b
osons,
b
ut a naive mass term
l
i
ke
m
A
i
μ
A
iμ
e
x-
plicitly violates the local
g
au
g
e invariance. Inspired by the Bardeen-
C
ooper
-
Schrieffer theor
y
of superconductivit
y
, Yoichiro Nambu applied the concept
of spontaneous symmetry breaking to particle physics
(
Nambu, 1960; Namb
u
a
nd Jona-Lasinio, 1961a, 1961b; Goldstone, 1961
)
and paved the way for the
H
iggs mechanism and to a true gauge theory of weak interactions
.
T
o explain the mechanism o
f
spontaneous symmetry breakin
g
, let us con
-
s
ider a scalar theory with the
g
loba
l
U
(
1
)
symmetry:
24 2 Neutrinos within the
S
tandard Model
L
φ
=
∂
μ
φ
†
∂
μ
φ
−
μ
2
φ
†
φ
−
λ
φ
†
φ
2
,
(
2.12
)
where
μ
2
a
nd λ are real parameters. One can easil
y
verif
y
tha
t
L
φ
i
s
i
nvar
i
an
t
u
n
de
r
t
h
et
r
a
n
sfo
rm
at
i
o
n
φ
→
φ
=e
−
i
α
φ
, or equivalentl
y
δφ
≡
φ
−
φ
=
−
i
α
φ
f
or the infinitely small constant
α
.
We simi
l
ar
l
y
h
av
e
φ
†
→
φ
†
=
e
i
α
φ
†
or
δφ
†
=
i
α
φ
†
f
or the Hermitian-conju
g
ate
fi
el
d
φ
†
.Th
eva
ri
at
i
o
n
of
L
φ
w
i
th
res
p
ec
tt
o δ
φ
c
an be cast into
δ
L
φ
=
∂
L
φ
∂φ
δ
φ
+
∂
L
φ
∂
(
∂
μ
φ
)
∂
μ
δφ
=
∂
L
φ
∂φ
δ
φ
+
∂
μ
∂
L
φ
∂
(
∂
μ
φ
)
δφ
−
∂
μ
∂
L
φ
∂
(
∂
μ
φ
)
δφ
=
∂
L
φ
∂
φ
−
∂
μ
∂
L
φ
∂
(
∂
μ
φ
)
δ
φ
+
∂
μ
∂
L
φ
∂
(
∂
μ
φ
)
δφ
.
(
2.13
)
A
s the equation of motion must be satisfied, the term inside the squar
e
brackets is vanishin
g
.Requirin
g
the theory to be invariant, one may de
fi
ne
a
symmetry current by identifying the remaining term in Eq.
(
2.13
)
as it
s
d
iver
g
ence. T
h
is current turns out to
be
J
μ
J
=i
φ
†
∂
μ
∂
φ
−
φ
∂
μ
φ
†
,
(
2.14
)
a
nd the correspondin
g
symmetry char
g
ei
s
Q
≡
d
3
x
J
0
JJ
(
x
)
=
i
d
3
x
φ
†
˙
φ
−
φ
˙
φ
†
,
(
2.15
)
whe
r
ethe
irr
eleva
n
tco
n
sta
n
t
α
has bee
n
d
i
sca
r
ded
.Th
eco
n
se
r
vat
i
o
n
of t
h
e
c
urren
t
∂
μ
J
μ
=
0 implies that the char
g
e is independent of time, namely
d
Q
d
t
=
0. The above discussion is actuall
y
a concise version o
f
the
f
amous
N
oet
h
er t
h
eorem, w
h
ic
h
states t
h
at t
h
ere a
l
ways exists a conserve
d
curren
t
i
n the theory with a continuous symmetry
(
Noether, 1918
)
. The commutators
between the charge
Q
and the field operators
(
φ
an
d
φ
†
)
actually generate
t
h
et
r
a
n
s
f
o
rm
at
i
o
n
s
[
Q
,
φ
]
=
−
φ,
Q,
φ
†
=+
φ
†
,
(
2.16
)
where the commutative relations between
fi
eld operators and their conju
g
at
e
m
omenta at the equal time have been used. Takin
g
φ
and
φ
†
a
s inde
p
enden
t
variables, one can immediately verif
y
φ
(
t
,
x
)
,
˙
φ
†
(
t
,
y
)
=
φ
†
(
t
,
x
)
,
˙
φ
(
t
,
y
)
=i
δ
3
(
x
−
y
)
.
(
2.17
)
N
ote that the char
g
eei
g
envalues o
f
φ
a
n
d
φ
†
a
r
e
−
1 and +1, respectivel
y
,
a
s indicated by Eq.
(
2.16
)
. Note also that one or more massless bosons, th
e
2
.1 Fundamentals o
f
the
S
tandard Model 2
5
s
o-called
G
oldstone bosons or Nambu-
G
oldstone bosons, will appear in th
e
theory if the ori
g
inal La
g
ran
g
ian has a continuous symmetry but the vacuum
s
tate has a reduced symmetry
(
Goldstone, 1961; Goldston
e
et a
l
., 1962
)
.
T
o see the occurrence o
f
a
G
oldstone boson in this scalar theor
y
,wede
fi
n
e
the com
p
lex field
φ
=(
η
+i
χ
)
/
√
2
and rewrite the La
g
ran
g
ia
n
L
φ
as
L
φ
=
1
2
∂
μ
η∂
μ
η
+
1
2
∂
μ
χ∂
μ
∂
χ
−
1
2
μ
2
(
η
2
+
χ
2
)
−
1
4
λ
η
2
+
χ
2
2
.
(
2.18
)
O
ne can observe that
L
φ
is
in
va
ri
a
n
tu
n
de
r
the
r
otat
i
o
n
η
→
η
=
+
η
cos
α
+
χ
s
i
n
α
,
χ
→
χ
=
−
η
sin
α
+
χ
cos
α
;
(
2.19
)
or equivalentl
y,
δη
=
+
αχ ,
δχ
=
−
α
η.
(
2.20
)
This transformation is just the rotation in the
(
x,
y
)
plane, and the corre-
s
pon
d
in
g
symmetry
g
roup is
SO
(
2
)
which is locally equivalent to th
e
U
(
1
)
group as it should be. If the vacuum expectation value
(
vev
)
o
f
η
i
s nonzero
,
η
=
u
w
i
th
u
b
eing rea
l
an
d
positive, t
h
en t
h
e vacuum state is no
l
onge
r
i
nvariant under the rotation as the pre
f
erred direction is alon
g
the
x
-
a
xi
s
.
To
find out the ph
y
sical modes in the theor
y
,onema
y
disturb the field variable
s
a
r
ou
n
dt
h
e
ir
c
l
ass
i
ca
l
va
l
ues a
n
d
r
ede
fin
e
η
=
u
+˜
η
.
Therefore, the minimiza-
tion of the scalar potential in Eq.
(
2.18
)
determines the vev
u
=
−
μ
2
/λ
.
G
ive
n
λ>
0,
μ
2
m
ust be negative to produce a meaningful vev. Now we hav
e
L
φ
=
1
2
∂
μ
η∂
˜
μ
˜
η
+
1
2
∂
μ
χ∂
μ
χ
−
1
2
−
2
μ
2
(˜
η
)
2
+
···
,
(
2.21
)
where the ellipsis denotes those non-quadratic terms of ˜
η
a
n
d
χ
.I
t beco
m
es
evident that the masses of two scalar particles are
g
iven b
y
m
2
˜
η
=
−
2
μ
2
a
nd
m
2
χ
= 0. The massless boso
n
χ
,
known as the Goldstone boson
,
occurs after
the spontaneous breaking of the global
U
(
1
)
symmetry.
I
n 1964, Peter Hi
gg
s and some other theorists applied the idea o
f
spon
-
taneous symmetry breaking to the local gauge theories and found that th
e
m
assless gauge bosons could absorb those unwanted Goldstone bosons an
d
then become massive
(
Higgs, 1964a, 1964b, 1966; Englert and Brout, 1964;
Guralnik et al., 1964
)
. This is just the famous Higgs mechanism. Based on th
e
SU
(
2
)
L
×
U
(
1
)
Y
g
auge symmetry
(
Glashow, 1961
)
and the Higgs mechanism
,
Steven Weinber
g
and
A
bdus Salam successfully proposed a coherent unified
theory of electromagnetic and weak interactions — the SM of electroweak
i
nteractions
(
Weinberg, 1967; Salam, 1968
).
2.1.3 Renormalizabilit
y
In quantum
fi
eld theories one
f
requentl
y
encounters various in
fi
nities when
going beyond the tree-level calculations. Richard Feynman, Julian Schwinger