lecture III
82
Outline of lectures
•
•
•
•
introduction to what we know about neutrinos
➡ mass ✓
➡ mixing ✓
★ lepton flavor violation ✓
yesterday
more
➡ oscillation ✓
introductory
➡ matter effects ✓
critical open experimental questions ✓
the neutrino portal
today
neutrinos in cosmology and astrophysics
more
➡ effects of sterile neutrinos
advanced
➡ dark energy coincidence and MaVaNs
}
}
83
Anomalies in ν physics via the ν portal?
84
Review from yesterday
Review of last part of lecture II
•
Unknown territory for new discoveries is not just the high energy
frontier
➡ Are there new light particles, longer range forces?
★ they must be weakly coupled to us!
- in some cases constraints are very strong
- in others not so much
➡ vast landscape of possibilities to consider
☺ avoids “limited palette” of small number of renormalizable
interactions consistent with sacred principles
☹ also avoids being predictive
85
Review from yesterday
Review of last part of lecture II
continued
•
★
★
How to organize our thinking about possible new sectors
containing relatively light particles and long range forces?
➡ Portals!
low dimension gauge invariant operators of Standard Model
☺ these don’t have to be Lorentz scalars
interaction between visible and hidden sector of form:
ξϑ SMϑ hidden
whose effects are visible at relatively low energy
86
Review from yesterday
Dimensional Analysis of Portal
ξϑ SMϑ hidden
•
assume 3+1=4 spacetime dimensions for SM particles
➡ ϑSM has dimension dSM
★
e.g. ϑSM=ℓh has dSM=5/2
➡ ϑhidden has dimension dhidden
e.g. ϑhidden=ψ has dhidden=3/2
➡ dimension of ξ is 4-dSM-dhidden
★ e.g. coefficient of ℓhψ is dimensionless
★
★
ℓhψ is renormalizable operator
87
Condensed Matter terminology
•
•
•
Relevant Operator (in QFT we say superrenormalizable)
➡ operator with scaling dimension d<4
★ becomes more important at low energy/long distance
★ If the operator can create massless particles it is always
important at long distance
Marginal Operator (in QFT we say renormalizable)
➡ operator with scaling dimension d=4
★ equally important at all distance/energy scales
Irrelevant operator (in QFT we say nonrenormalizable)
➡ operator with scaling dimension d>4
★ important at high energy/short distance
★ difficult to detect the effects at low energy
88
Review from yesterday
Dimensional Analysis continued
ξϑ SMϑ hidden
•
In standard model the scaling dimension of some operators
changes below the weak scale
➡ e.g. ϑSM=ℓh has dSM=5/2 above the weak scale
★
ϑSM=ℓ<h > has effective dSM=3/2 below the weak scale
➡ ℓ<h>ψ operator becomes more important at low energy
★
(below the weak scale: the operator is relevant with d=3 )
☺ no matter how small the coefficient, the operator becomes
more important at low energy/long distance
☺ light particles like neutrinos allow us to see the effects of
relevant operators with tiny coefficients
89
relevant irrelevant operators
•
•
•
•
•
•
An irrelevant operator at some scale can be relevant at low
energy
Weinberg operator ϑWeinberg=( h2/M) ℓℓ has d=5, is irrelevant
Below weak scale h→<h>
Below weak scale ϑWeinberg→ < h>2/M) ℓℓhas d=3, is relevant
We understand why ϑWeinberg is unimportant at weak scale:
➡ M >> <h>
Far below weak scale ϑWeinberg becomes important again
➡ without it neutrinos would have no mass
90
Popular Portals
• Scalar: h†h φ, h†h φ2
• Higgs mixing with exotic scalars
• Higgs decay to exotics, Higgs production
• new force if φ is light
• Vector: Bμν Xμν, JμXμ
• exotic vector coupling to electromagnetic charge, B, L
• Z decay to exotics
• new force
• Spin 1/2: hlψ
• neutrino mixing with exotic fermions
• new force, strongest for neutrinos, if ψ experiences dark force
91
Where are all the anomalies in the
neutrino sector?
•
•
Are there no exotic light fermions?
Z boson decays: “there are only 3 neutrinos lighter than mZ/2”
➡ possible to have light “sterile” neutrinos that don’t couple to
the Z
➡ these could be related to the known quarks and leptons in
some unified scheme
➡ or simply part of a hidden sector
➡ they could be massless or light composite fermions
➡ they could live in extra dimensions….
92
LSND: the Effect that
Refuses to Die
_
_
νe appearance
_ at ~30 m
in ~30 MeV νμ beam
3.8 σ excess
93
LSND took data from 1993-98
- 49,000 Coulombs of protons
- L = 30m and 20 < E!< 53 MeV
Saw an excess of !e :
87.9 ± 22.4 ± 6.0 events.
Oscillations?
Signal:
p # e+ n
n p # d $(2.2MeV)
With an oscillation probability of
(0.264 ± 0.067 ± 0.045)%.
3.8 " evidence for oscillation.
HARP recently announced measurements that confirm LSND ve background estimate
from R. Van de Water talk, 2010
94
ν oscillation interpretation
⎛ (m 2 − m22 )L ⎞
Pµ →e = sin 2 (2θ )sin 2 ⎜ 1
⎟⎠
4E
⎝
⎛
⎛ (m12 − m22 ) ⎞ ⎛ L ⎞ ⎞
⎜⎝ eV 2 ⎟⎠ ⎜⎝ meters ⎟⎠ ⎟
⎜
⎟
= sin 2 (2θ )sin 2 ⎜ 1.27
⎛ E ⎞
⎜
⎟
⎜⎝
⎟⎠
⎜
⎟
MeV
⎝
⎠
• LSND studied antineutrino oscillations
• L~ 30 m
• 20 MeV < E < 53 MeV
_
• sensitive to Δm ~eV
2
2
95
(In)compatibility of LSND with Other
Oscillation Signals
three oscillation
signals have 3
different Δm2
⇒ need ≥ 4 ν’s,
3 active + n sterile
3.8σ evidence for
physics beyond the ν
standard model
96
MiniBooNE
⎛ (m 2 − m22 )L ⎞
Pµ →e = sin 2 (2θ )sin 2 ⎜ 1
⎟⎠
4E
⎝
⎛
⎛ (m12 − m22 ) ⎞ ⎛ L ⎞ ⎞
⎜⎝ eV 2 ⎟⎠ ⎜⎝ meters ⎟⎠ ⎟
⎜
2
2
⎜
⎟
= sin (2θ )sin 1.27
⎛ E ⎞
⎜
⎟
⎜⎝
⎟⎠
⎜
⎟
MeV
⎝
⎠
• initial run with muon neutrino beam
• L=541m
• 200 MeV < E < 3 GeV
• similar L/E to LSND, sensitive to Δm ~eV
2
2
97
from R.Van de Water Neutrino 2010
98
from R.Van de Water Neutrino 2010
99
from R.Van de Water Neutrino 2010
100
Data plotted vs L/E
New MiniBooNE result
0.020
New!
P�Ν Μ�Ν e�
0.015
5.66E20 POT
Measured �stat error�
Background �sys error�
MiniBooNE
(!
mode )
0.010
0.005
0.000
0.020
�0.005
0.5
1.0
P�Ν Μ�Ν e�
0.015
•!MiniBooNE L/E bins match
the standard MB energy bins,
just recast in L/E
1.5
2.0
L
m
LSND
�
�
EΝ MeV
2.5
(! )
0.010
0.005
0.000
0.020
�0.005
P�ΝΜ�Νe�
•MB 2007: neutrino data
0.015
does not confirm LSND
(but low energy νe excess)
1.0
0.5
1.0
1.5
2.0
2.5
2.0
2.5
(! mode )
0.010
0.005
0.000
�0.005
P (! " # ) $
L�Eerror�
MiniBooNE
Ν �meters�MeV�
Measured �stat
Background �sys error�
0.5
1.5
L
m
�
�
EΝ MeV
observed event excess
number expected for full transmutation of % µ or % µ
101
LSND and MiniBooNE:
_
same L/E conversion prob for ν?
Μ to e Conversion Probability
0.020
_
0.015
�
0.010
�
�
0.005
�
�
�� �
��
� �
� � �� � � �
� �0.5
�
��
�
�
1.0
�
�
�
�
�
1.5
2.0
2.5
LSND ν
_
MBν
MBν
E776
�L�E���MeV�m�
�0.005
102
Are LSND/MB
discovering CP
Violation from new
ν?
103
νeffects on neutrino oscillations
new ultralight fermion: new mass eigenstates in oscillations
104
How many neutrinos for CPV?
•
•
•
•
•
3, with 2 mass squared differences
with 3+1=4 ν we have more than enough?
At short baseline we can neglect solar, atmospheric mass squared
differences
with 3+1 we only have 1 nonnegligible mass squared difference
➡ NO CPV observable at LSND/MB with 3+1
for CPV at LSND/MB we need 3+2=5 ν
105
3+2, ultralightν
0.004
0.003
0.002
0.001
0
1
2
3
4
5
6
(L/E)/(m/MeV)
Other mass squared differences negligible at this L/E
106
3+2 ultralight
0.004
0.003
0.002
0.001
L/E
0
1
2
3
4
107
5
6
Above Δm2/eV2=100
Oscillations rapid, can
Average over them
(Same results from
decoherence)
Some constraints remain
but mostly on appearance.
Disappearance constraints go
away (CDHS) or weaken
(Bugey, CHOOZ)
108
Effects of light sterile neutrinos
light: new mass eigenstates in oscillations
(either masses large enough that wavepackets do separate
spatially, or oscillations so rapid they must be averaged
over, but small enough so mass has no affect on phase
space factor ) 1012 eV2>>Δm2 > 1000 eV2
large CPV requires at least one xij of order one
109
3+1, ultralight + 1 light
0.004
Note CPV shift of location of oscillation maximum
0.003
0.002
Note
probability
never zero
0.001
0
1
2
3
4
5
6
L/E
Other mass squared differences negligible at this L/E
110
Effects of nonlight ν’s on oscillations
nonlight: new mass eigenstates, too heavy to be produced
with same phase space as light states 1012 eV2<Δm2
PSF=Phase Space factor, <1, Can be 0 (for neutral fermion
which is too heavy to produce)
111
3+1 ultralight neutrinos + 1
nonlight neutrino
0.004
0.003
0.002
0.001
0
1
2
3
4
112
5
6
note
oscillation
prob can
reach 0, but
not at L/E=0
with CPV
Note any 1 laboratory
experiment
typically has wiggles (sensitive
to L/E dependence of
oscillations) only over
a decade or so, likely sensitive
to at most 1 Δm2 .
113
General formula for vacuum (anti)
neutrino oscillation appearance
• Assume 1 x is of order 1
• neglect all x <<1
• For x >>1, neglect interference
• allow for nonunitarity from mixing with nonlight fermion
ij
ij
ij
only 2 new
parameters
CPV from
interference with
short distance
114
constant term
from short distance
Can this formula
fit short
baseline data?
115
Minimal fit to short baseline
ν oscillations
• 3 active + 1 light sterile neutrinos
• CP violation in mixing
• requires nonunitary mixing matrix
• 3 active +1sterile +1 heavy neutrino
Effects of CP violation from Neutral Heavy Fermions on Neutrino Oscillations,
and the LSND/MiniBooNE Anomalies
A. E. N., arXiv:1010.3970v1 [hep-ph]
116
Fit Example: Δm241=0.6 eV2, sin2(2θ)=0.003, α=0, β=-0.3,
χ2=26.4/24 d.o.f
�
0.010
Appearance Probability
�
LSND
0.008
MB
MB
�
NOMAD
KARMEN
0.006
�
�
�
�
�
0.002
� �
� �
�
�0.002
�
�
� �
0.5
�
�
ν
�
��
�
�
_
ν
�
�
0.004
�
�
�
�
1.0
�
1.5
2.0
2.5
3.0
(L/E)/(m/MeV)
�0.004
117
What about disappearance?
102
102
CCFR
Karmen
10
10
LSND
from
A. Strumia
CDHS
2
#m LSND
Dm 2
1
-1
10
-2
10
Bugey
10-3
1
10! 1
CHOOZ
10-4
10-4
10-3
10-2
sin2 2q
10-1
10! 2
10! 4
1
allowed
10! 3
excluded
10! 2
sin 2"LSND
10! 1
1
2
Figure 2: 90% CL regions from Karmen, CDHS, CCFR, Figure 3: The LSND region at 90% and 99% CL, compared
Bugey, Chooz and LSND (shaded). The mixing angle θ with the 90% (dashed line) and 99% CL (continuous line)
on the horizontal axis is different for the different experi- combined exclusion bounds from data in fig. 2 and SK.
ments.
recognized by fitting separately the two incompatible data.
This is what is done in fig. 3.
• Solar data give a 5.4σ evidence for pure active solar
oscillations versus pure sterile oscillations: combining all solar data in a global fit we obtain [19] !
Ignoring the poor quality of the fit, the best combined
fit region for the LSND parameters is shown in fig. 4a. It
agrees reasonably well with the corresponding fig. in [20],
taking into account that we show values of
χ2sun (best sterile) − χ2sun (best active) = 30
and ηssun = 0 ± 0.18. In particular, SNO/SK find a
118 5.1σ direct indication for νµ,τ appearance.
χ2 (θLSND , ∆m2LSND ) = min χ2 (p, θLSND , ∆m2LSND )
p
• Atmospheric data data give a 7σ indication for
pure active atmospheric oscillations versus pure sterile oscillations. In fact, a global fit of atmospheric
data gives [1, 21]∗∗
(where p are all other parameters in which we are not
interested), so that we convert values of χ2 − χ2best into
confidence levels using the gaussian values appropriate for
2 d.o.f. (the 2 LSND parameters), while a statistically less
efficient procedure with more d.o.f. is employed in [20].
χ2atm (best sterile) − χ2atm (best active) ≈ 50
and ηsatm = 0±0.16. This strong evidence is obtained
combining independent sets of data. SK claims [1]
that pure sterile is disfavoured by the up/down ratio
in a NC-enriched sample (3.4 standard deviations)
and by matter effects in partially contained events
2+2 neutrinos
In the jargon 2+2 indicates 2 couples of neutrinos (one
generates the solar anomaly, and the other one the atmospheric anomaly), separated by the large LSND mass gap.
Within this scheme, the sterile neutrino is employed to generate the solar or atmospheric anomaly, or one combination of the two. The fraction of sterile neutrino involved in
solar oscillations, ηssun , plus the fraction of sterile neutrino
involved in atmospheric oscillations, ηsatm , is predicted to
sum to unity [17]
! Some words of caution. Arbitrary choices become more relevant
when fitting disfavoured data (for example: the error is evaluated at
the experimental point or at the theoretical point?). Furthermore,
our bound on the sterile fraction allowed by solar data is obtained
assuming the BP00 [23] prediction for the Boron νe solar flux. It is
proportional to the 7 Be p → 8 B γ cross section: some authors think
that systematic uncertainties in its measurement could be underestimated.
∗∗ A large amount of these atmospheric data is not included in
theoretical reanalyses (because not yet accessible outside the SK collaboration in a form that allows to recompute them) that therefore
obtain a much smaller ∆χ2 ≈ 15 [20, 24] in place of 50 [1, 21]. This
underestimation of the SK bound means that at the moment only
SK can perform a sensible analysis of mixed sterile and active atmospheric oscillations and explains why the authors of [20] do not
recognize that 2+2 oscillations are extremely disfavoured. One mixing angle is set to zero in the SK analysis; relaxing this unjustified
simplification should not significantly weaken the bounds.
ηstot ≡ ηssun + ηsatm = 1.
Experiments now tell that both the solar and atmospheric
anomalies are mostly generated by active neutrinos, and
only a small sterile contribution is allowed. Consequently
2+2 oscillations give a global fit worse than 3+1 oscillations [22, 20]. Let us summarize the present experimental
status of this issue.
3
andvery
get oscillation
the 5th neutrino is heavywhere
x51 varies
rapidly probability
and should
be averaged
over. In this limit it makes no difference to
B. Additional
weak doublets
�|2 [(1the
�
2
2
the oscillation formula whether the neutrino becomes heavy
packets
as .interference
Pν →ν enough
= 2|Ue4 so
|2 |Uthat
− ∗r)wave
+ 2r sin
β + 2rseparate,
sin2 (x41 − β)]
µ4 U U
e5 µ5
III. CP VIOLATING ELECTRON
NEUTRINO
ANTINEUTRINO
IN A 3+2
For anti
neutrinos theOR
matrix
elements
are complexAPPEARANCE
conjugated. This probability
can be written as [? ? ]
2
2
2
φ ≡ arg
effects become unobservable when the
rapid
oscillations
averaged
over.
The∗ 100
appearance
probability
then
becomes (4)
MODEL
WITH
ONE ∆m are
IN THE
RANGE
BETWEEN
EV AND 1000
MEV
µ
We can simplify
e
2
(8)
2
2
2
For anti neutrinos we replace β 4|U
→e4−β.
violation
remains observable in the limit of heavy m . Note
UCP
e4e5U
|2 |Uµ4 |Note
sin2 xthat
|2µ4
|U
(3)
41 + 4|U
µ5 | sin x51 + 8|Ue5 ||Ue4 ||Uµ4 ||Uµ5 | sin x51 sin x41 cos(x51 − x41 ± φ)5
also
that
the
amplitude
of
oscillations
associated
with
m
may
be
enhanced by mixing with a heavy 5th neutrino.
4
2
2 Assuming
2
2
2
where
are
small
so
the||U
mass
components
of the
wavefunction
doexperiments
not
|Ue4is| a|Uphysically
xmass
2|Uwith
|Uµ5
| enough
+ experiments,
4|Uand
||U
| sin
x sin(x
∓
φ)
(5)where
µ4 | sinthe
41 +differences
e5 |violating
e5that
e4
µ4 ||U
µ5
41 disappearance
In
constrast
appearance
indifference
the
absence
of
matter
effects,
much less
observable
CP
the
+(-)
is
used
for41neutrinos(antineutrinos).
In the limitare
� for the probability
separate spatially,
and neglecting
the mass phase
differences
among
the sign
3 light
eigenstates,
the�formula
∗
U
U
sensitive
to
mixing
with
heavy
neutral
fermions.
Averaging
over
the
short
oscillation
length
associated
withto
∆m251 ,
e5
µ5
theof 5th
neutrino
is appearance
heavy x51 in
varies
very
rapidly
and
over.
In
this
limit
it
makes
no
difference
electron
neutrino
a muon
neutrino
beam
is should be averaged
φ ≡ arg
(4)
∗
Ue4probability
Uµ4 This probability
and
neglecting
the
among
three
light
the
vacuum
electron
neutrino
this expression
byformula
defining
themass
CP
oddneutrinos
quantity
βthe
For differences
anti
the matrix
elements
areneutrinos,
complex
canas
beinterference
written
as [? ? ]or
the oscillation
whether
the
neutrino
becomes
heavy
enough
so
that conjugated.
the
wave
packetsfor
separate,
5
electron anti neutrino disappearance
is
�
�
Disappearance vs appearance,
large m limit
effects become unobservable when
oscillations
are
averaged
The8|U
appearance
probability thenInbecomes
� � the
� over.
is a ∗rapid
physically
is used for neutrinos(antineutrinos).
the limit where
2 +(-)
−2ix
−2ix
51
4|U
|241
|U observable
|2 sin2∗xCP
+violating
4|U
|2phase
|U ∗|2and
sinthe
x ∗+�2sign
||U ||U ||U | sin x sin x cos(x
− x ± φ)
µ4
µ5
51µ5
e5
e4
µ4
µ5
51
41
41
Uµ4
e e4
+U
Uµ5 e41
−e5very
Ue4 Urapidly
→ν = Ue4
e5 U
µ4 2− Uand
sin
φ|U
||U
| varies
1 Pν−1
the
5th
neutrino
is e5
heavy
x51
should be
averaged
over.
In this (1)
limit
it makes
no 51
difference
to
µ5
4|Ue4 |2formula
(1e5−
|Uwhether
|2 −the
|Ue5
|2 ) sinbecomes
x41 +heavy
2|Ue5
|2 (1 −
|U
|2 ) wave packets separate,
(9)
e4 2
e5the
2
2
2
2
P
=
(6)
β ≡ |Utan
the
oscillation
neutrino
enough
so
that
as
interference
e→e
/
|
|U
|
sin
x
+
2|U
|
|U
|
+
4|U
||U
||U
||U
|
sin
x
sin(x
∓
φ)
(5)
where
e4
µ4
41
e5
µ5
e5
e4
µ4
µ5
41
41
2
|Ue4effects
||Uµ4
| + unobservable
cos φ|Ue5when
||Uthe
| oscillations are �
become
averaged over. The appearance probability then becomes
µ5rapid
�
µ
where
and the mixing ratio r
e
(3)
and the probability for muon neutrino 2or muon
antineutrino disappearance is
∗ obtained by replacing e → µ in the
2
2
2)
Ue5 Uµ5 µ4 ||Uµ5 | sin x41 sin(x41 ∓ φ)
(mibounds
−
|Um
|Uµ4
|2quantity
sin
x41 + 2|U
|2 +obtained
(5)
L/E
e4 |jodd
We can simplify
this
expression
by defining
the
CP
β e5 |2 |U
preceding
formula.
Typically,
on
disappearance
are
by cancelling(2)
systematic errors using
φ µ5
≡
arg4|Ue5 ||Ue4∗||U
(4)
.
xijstringent
≡ 1.27
Define
2
Ue4 Uµ4
eV
km/GeV
the L/E dependence. ThusWe
Ue5
and
U
are
only
weakly
constrained,
and
in
the
limit
where
the
oscillation
length
�
�
can simplifyµ5this expression by defining the CP odd quantity β
φ|Ue5 ||Uµ5
|� dependence, disappearance
1
�
−1 give any sin
associated with ∆m251is aisphysically
too
to
measurable
L/E
experiments strongly
∗short
∗ CP violating
tan
βU≡
observable
phase
and
signe5 ||U
is used
for neutrinos(antineutrinos). (6)
In the limit where
sin φ|U
1
µ5 |
−1 the +(-)
|U
+
U
U
|
tan
(6)
β
≡
e4
bound only Ue4 and Uthe
2
|U
||U
|
+
cos
φ|U
||U
|
µ5
µ4
µ4 .e5
e4
µ4
e5
µ5
5th neutrino is heavy x51 varies very 2rapidly and
should
beφ|U
averaged
|U ||U
| + cos
||U | over. In this limit it makes no difference to
r ≡ electron antineutrino
(7)electron and
The tension between
appearance
at LSND
MiniBooNE
andthe
short
∗ | whether
the oscillation
formula
the neutrino
becomesand
heavy
enough so that
wavebaseline
packets separate,
as interference
|U
U
e4
µ4
and
the
mixing
ratio
r
and the mixing
r disappearance
muonratio
neutrino
experiments
maywhen
be reduced
allowing are
r toaveraged
be greater
Only electron
neutrino
effects become
unobservable
the rapidbyoscillations
over.than
The 1.
appearance
probability
then becomes
∗
∗ from the E776 experiment [? ] . The
appearance experiments will constrain r. The
strongest
such
constraint
comes
|U
U
+
U
U
|
e5
e4
µ5
µ4
∗|2 sin2 x +∗2|Ur ≡|2 |U |2 + 4|U ||U ||U ||U | sin x sin(x ∓ φ)
(7)
|Ue4
|UU
(5)
|U|2e5
+ Ue4
| e5 µ5|Ue4 Uµ4
probability
µ4µ5
41Uµ4implies
∗ | e5
absence of veryLarge
short baseline
electron
neutrino
appearance
m5 gives
constant
term
which
is not
welle4 µ4 µ5 41 41
e4
and get oscillation
2
Pνµ →νe
r≡
µ4
e5
(7)
∗
|Ue4 U
|2
Weand
canget
simplify
this
byµ4defining
2CP odd quantity β
2expression
oscillation
probability
2 + 4r the
|U
|2fairly
((1
r)large
β) <.0.0008 .
can
be
e4 | 2
= 2|Ue4 | constrained
|Uµ4 |2 [(1 − r)2 + r2r|U
sin
βµ4+
2r−sin
(x41sin
−� β)]
2
µ5
�
| 2 (x41 − β)] .
Pνµ →νe = 2|Ue41|2 |Uµ4−1
|2 [(1 − r)2 +sin
2r φ|U
sin2 e5
β ||U
+ 2r
µ5sin
(8)
(10)
(8)
and get oscillation
probability
≡ tanit is possible to obtain much larger electron neutrino
Therefore
the existence of the heavy fifth neutrinoβ makes
2
|Ue4 ||Uµ4 | + cos φ|Ue5 ||Uµ5 |
For
anti neutrinos
we replace
→ −β. be
Note
that in
CP violation
remains
the limit
of
m50,
. Note
For anti neutrinos we replace β →appearance
−β. Note
that CPat
violation
remains
observable
the
ofobservable
heavy
m
probabilities
non
zero L/E
than β2would
in 2a limit
3+1
model.
For ininstance
forheavy
β =
and
5 . Note
2possible
also
that
amplitude
with m4 may
be enhanced
with a heavy 5th neutrino.
=the
2|U
|2 |U
|r2 [(1 of−oscillations
r)
+ of
2rassociated
β+
− β)]
. byδ mixing
(8)
−4
νµ10
→ν
e4the
µ4allowed
41
e , the
and
mixing
ratio
|U |2 |U |2 = 2P×
maximum
value
rsin
is 3.
For2rasin
3+1(xmodel
with
= 0 and r = 1, the maximum
e4 associated
µ4
also that the amplitude of oscillations
with mIn4constrast
may with
be appearance
enhanced
by mixing
with
a−4heavy
5th neutrino.
experiments,
in the absence
of matter
effects, disappearance
experiments are much less
probability of electron neutrino
a function
of L/EAveraging
is 8 × ∗10
. In
for
r =associated
3, the probability
sensitiveappearance
to mixing withas
heavy
neutral fermions.
over the
short
length
with ∆m251 ,
∗ contrast,
|U−3
+ Ue4inU
| oscillation
In constrast with appearance
in
the
absence
of
matter
effects,
disappearance
experiments
are
much
less
For anti experiments,
neutrinos
we
replace
β
→
−β.
Note
that
CP
violation
remains
observable
the
limit
of
heavy
m5 . even
Note
e5 Uµ5
µ4
and neglecting
the massat
differences
the 4
three
light
neutrinos,
the
probability
for
vacuum
electron
neutrino
or
of electron neutrino appearance
maximizes
a muchamong
larger
×
10
.
In
the
next
section
we
will
find
larger
r≡
∗ |
2
119 isbe enhanced by|Umixing
U
also
that
the
amplitude
of
oscillations
associated
with
m
may
with
a
heavy
5th
neutrino.
electron
anti
neutrino
disappearance
e4
4
µ4
sensitive to mixing with heavy neutral
fermions.
Averaging
overmix
thewith
short
oscillation
associated
with ∆m51 ,
enhancements
possible
when neutrinos
fermions
which arelength
too heavy
to be produced.
In constrastamong
with appearance
experiments,
in the absence
ofe4 |matter
are much
2
2disappearance
2 experiments
2
(1 − |Ue4effects,
|2 − |U
sin2 x41 + 2|Uelectron
)
and neglecting the mass differences
the three
neutrinos,
the 4|U
probability
for
or less (9)
e5 | )vacuum
e5 | (1 − |Ue5 | neutrino
andlight
get oscillation
probability
sensitive to mixing with heavy neutral fermions. Averaging over the short oscillation length associated with ∆m251 ,
electron anti neutrino disappearance
is
the probability for muon neutrino or muon
antineutrino
obtained
e → µ in the
2
2
2 disappearance
Pν OSCILLATIONS
= 2|Ue4 |2 |U
[(1 −
+ 2r
sin2vacuum
β +is2rWITH
sinelectron
(x41by
−replacing
β)]neutrino
.
CP VIOLATION
IN NEUTRINO
IN
Ar)3+2
MODEL
ONE
FERMION
and neglecting IV.
the mass
differences and
among
the three light
the
for
or
→νneutrinos,
µ4 | probability
µ
e
preceding formula. Typically,
stringent
bounds on disappearance are obtained by cancelling systematic errors using
(6)
(7)
(8)
THAN
mµ constrained, and in the limit where the oscillation length
electron anti neutrino
disappearancethe2isL/E dependence.
ThusHEAVIER
Ue5 2and Uµ5 are
only2weakly
For anti
−β.give
Note
that) CP violation
remains observable
in experiments
the(9)
limit ofstrongly
heavy m5 . Note
2 2|U β |→(1
4|Ue4 |2 (1 − |Ue4 |2 − |U
| )neutrinos
sin2with
x41we
+replace
|Uany
∆m
is too
L/E dependence,
disappearance
e5associated
e5short to−
e5 |measurable
also that
of oscillations
associated with
m4 may 2be enhanced by mixing with a heavy 5th neutrino.
2
2 the amplitude
2
2
2
51
bound only Ue4 and Uµ4 .
(1
−in|Udetail
−a|Umodel
| ) sin
x41
2|U
(1sterile
−
|Uofe5neutrinos,
| and
) MiniBooNE
(9) and
In this section we4|U
consider
with
2 +
additional
onedisappearance
of which isexperiments
tooelectron
heavy
tomuch less
e4 |constrast
e4 |between
e5
e5 | absence
In
with
appearance
experiments,
in the
matter
effects,
are
The tension
electron
antineutrino
appearance
at LSND
and short baseline
be produced
in pion
or muon
decay.
In this
case,
only
4
ofbethe
5 mass
eigenstates
can
be µproduced
in neutrino
a muon
and the probability for muon neutrino
or muon
antineutrino
disappearance
isfermions.
obtained
by
replacing
eoscillation
→
the
muontoneutrino
disappearance
experiments
may
reduced
by allowing
r to short
be greater
than
1. in
Only
electron
sensitive
mixing
with
heavy
neutral
Averaging
over
the
length
associated
with ∆m251 ,
appearance
experiments
will in
constrain
r.ofThe
strongest
such
constraint
comes
from
the
E776
[?the
] . Theneutrino or
and the probability
for muon
or
antineutrino
disappearance
is obtained
by probability
replacing
eexperiment
→
µ inelectron
neutrinobounds
beam.
This
situation
canmuon
be
described
terms
athe
non-unitary
mixing
matrix
forerrors
the
light
states.
However
and
neglecting
the
mass
differences
among
three light
neutrinos,
the
for
vacuum
preceding formula. Typically,
stringent
onneutrino
disappearance
are
obtained
by
cancelling
systematic
using
absence
ofneutrino
very short
baseline electron
appearance implies
electron
anti
is neutrino
preceding formula. Typically, stringent
bounds
ondisappearance
disappearance
are obtained
by cancelling systematic errors using
the L/E dependence. Thus
Ue5 dependence.
and Uµ5 are
only
in
the limit
2 the oscillation length
2 where
the L/E
Thus
Ue5weakly
and Uµ5constrained,
are only weaklyand
constrained,
and
where
length (10)
|U |2 |U |22((1
− r)in
+
4r sin
β) < 0.0008
.2 the oscillation
2 limit
2 the
2
4|Ue4 |2 (1e4− |Uµ4
(9)
e4 | − |Ue5 | ) sin x41 + 2|Ue5 | (1 − |Ue5 | )
2
associated with ∆m251 isassociated
too short
to
give
any
measurable
L/E
dependence,
disappearance
experiments
strongly
with ∆m51 is too short toTherefore
give any
measurable
L/E
dependence,
disappearance
experiments
strongly
the existence of the heavy fifth neutrino makes it is possible to obtain much larger electron neutrino
and the
probability
for muon
neutrino
or muon
antineutrino
disappearance
is obtained
by for
replacing
e → µ in the
appearance
probabilities
at non
zero L/E
than would
be possible
in a 3+1 model.
For instance
β = 0, and
bound only Ue4 and Uµ4 .bound only Ue4 and Uµ4 .
2
preceding
formula.
Typically,
stringent
bounds
on
disappearance
aremodel
obtained
errors using
|Ue4 |2 |U
2×
10−4 , the maximum
allowed
value
of
r is 3. For aand
3+1
with
δby
= 0cancelling
andelectron
r = 1, systematic
the and
maximum
µ4 | =appearance
The
tension
between
electron
antineutrino
at
LSND
and
MiniBooNE
short
baseline
The tension between electron antineutrino appearance
at LSND
and
MiniBooNE
and
short
electron
and
probability
of electron
neutrino
appearance
as aonly
function
of
L/E
is 8baseline
× 10−4 .and
In contrast,
for r =
3, thethe
probability
the L/E
dependence.
Thus
Ue5
and Uµ5 are
weakly
constrained,
in
the limit
where
oscillation length
−3
muon
neutrino
disappearance
experiments
may
be
reduced
by
allowing
r
to
be
greater
than
1.
Only
electron
neutrino
of electron
atany
a much
larger 1.
4 ×L/E
10
. dependence,
In electron
the next section
we will findexperiments
even larger strongly
associated
with
∆m251 appearance
is toorshort
to give
measurable
disappearance
muon neutrino disappearance experiments may be reduced
byneutrino
allowing
tomaximizes
be
greater
than
Only
neutrino
enhancements
possible
whensuch
neutrinos
mix with fermions
too heavy
be produced. [? ] . The
appearance experiments will constrain
r. UThe
strongest
constraint
comeswhich
fromarethe
E776toexperiment
bound
only
e4 and Uµ4 .
appearance experiments absence
will constrain
r. The
strongest
such
constraint
comes
from
the E776 experiment [? ] . The
The tension
between
electron antineutrino
appearance at LSND and MiniBooNE and short baseline electron and
of very short
baseline
electron
neutrino
appearance
implies
absence of very short baseline electron neutrino appearance
implies
muon neutrino
disappearance
experiments
may
be
reduced by IN
allowing
to be greater
Only electron neutrino
IV. CP VIOLATION IN NEUTRINO OSCILLATIONS
A 3+2rMODEL
WITHthan
ONE1.FERMION
2
mµ
appearance
will
such
comes from the E776 experiment
[? ] . The
|Ue4 |2experiments
|Uµ4 |2 ((1 −
r)constrain
+ 4r sinr.2 The
β) HEAVIER
<strongest
0.0008THAN
. constraint
(10)
absence2of very short2baseline electron neutrino appearance implies
|Ue4 | |Uµ4 | ((1 − r)In this
+ 4r
sinweβ)
< 0.0008
(10)
section
consider
in detail a. model with 2 additional sterile neutrinos, one of which
is too heavy to
2
2
Therefore the existence of the heavy
fifth in
neutrino
makes
it
is|2this
possible
larger. can
electron
neutrino
be produced
pion or muon
decay.
In
of
thesin
5 2much
mass
be produced
in a muon
|Ue4
|U
|2 ((1only
−to
r)42obtain
+ 4r
β) <eigenstates
0.0008
(10)
µ4case,
neutrino
situation
be described
non-unitary
mixing
matrix forfor
the β
light
However
appearance probabilities at non zero
L/Ebeam.
thanThiswould
becanpossible
ininaterms
3+1of amodel.
For
instance
=states.
0, and
Therefore the existence
of
makes
it
is
possible
to
obtain
much
larger
electron
neutrino
2 the 2heavy fifth
−4 neutrino
Therefore
the existence
neutrino
makes
it δ
is =
possible
much
larger electron neutrino
|Ue4 | |Uµ4 | = 2 × 10 , the maximum
allowed
value ofof rthe
is heavy
3. Forfifth
a 3+1
model
with
0 andtor obtain
= 1, the
maximum
appearance
probabilities
at
non
zero L/E
than would
be instance
possible in afor
3+1βmodel.
For
instance for β = 0, and
appearance probabilitiesprobability
at non zero
L/E neutrino
than would
be
possible
in
a
3+1
model.
For
=
0,
and
−4
of electron
appearance
as
a
function
of
L/E
is
8
×
10
.
In
contrast,
for
r
=
3,
the
probability
|Ue4 |2 |Uµ4 |2 = 2 × 10−4 , the maximum allowed
of r is 3. For a 3+1 model with δ = 0 and r = 1, the maximum
−3 value
|Ue4 |2 |Uµ4 |2 = 2 × 10−4 , of
theelectron
maximum
allowed
value
of r isof3.electron
a 3+1
model
δ=
and ris =
1,we
the
maximum
neutrino
appearance
maximizes
atFor
a much
larger
4 × 10with
In
the0next
find even
−4
probability
neutrino
appearance
as .a function
of L/Esection
8 × 10
.will
In contrast,
for larger
r = 3, the probability
−4
enhancements
possible when
mix
with appearance
fermions
which
too
tofor
be4 r×
produced.
probability of electron neutrino
appearance
as a neutrinos
function
ofneutrino
L/E
is 8 × 10
. are
In at
contrast,
=
the
of electron
maximizes
a heavy
much larger
10−33,
. In
theprobability
next section we will find even larger
enhancements possible when
with fermions which are too heavy to be produced.
of electron neutrino appearance maximizes at a much
larger 4 × 10−3 .neutrinos
In themixnext
section we will find even larger
enhancements possible when neutrinos mix with fermions which are too heavy to be produced.
µ5 wave packets separate, as interference
the oscillation formulaare
whether
the
heavy
enough
so
that the
much
than thebecomes
W
boson.
φ ≡OR
argANTINEUTRINO
(4)
III.lighter
CPneutrino
VIOLATING
ELECTRON
NEUTRINO
APPEARANCE IN A 3+2
∗
2
2
U
U
For
anti
neutrinos
the
matrix
elements
are
complex
conjugated.
This
probability
can be written
[? ? ]
e4
MODEL
WITH
ONE
∆m
IN
THE
RANGE
BETWEEN
100
EV2 formula
AND 1000
MEV
µ4
effects become unobservable
when the
the mass
rapiddifferences
oscillations
are averaged
The
appearance
probability
then
becomes
Neglecting
among
the 3 lightover.
eigenstates,
the
for
the probability
ofaselectron
neutrino
|2 |Uµ4 |2 sin2 x41 + 4|Ue5 |2 |Uµ5 |2 sin2 x51 + 8|Ue5 ||Ue4 ||Uµ4 ||Uµ5 | sin x51 sin x41 cos(x51 − x41 ± φ)
appearance
in the
a2observable
muon
neutrino
beamenough
is 4|Ue4and
2
2are small2 phase
Assuming
mass differences
so that
difference
of the wavefunction do not
is 2a physically
CP violating
thethe
+(-)
sign ismass
usedcomponents
for neutrinos(antineutrinos).
In the limit where
|Ue4 | separate
|Uµ4 | spatially,
sin x41and+neglecting
2|Ue5 | |U
| +
4|Ue5 ||U
||U eigenstates,
| sin x41 the
sin(x
e4 ||U
41 ∓forφ)
where
theµ5
mass
differences
among
theµ4
3 lightµ5
formula
the probability
(5)
(3)
the 5th neutrino is heavy x51 varies very rapidly and should be averaged over.� In this
limit it makes no difference to
�
electron neutrino appearance in aFor
beam
is
∗
�the matrix
�This
Uwave
antineutrino
neutrinos
elements
aresocomplex
conjugated.
written as [? ? ]
theofoscillation
formula whether themuon
neutrino
becomes
heavy
enough
that
the
separate,can
as be
interference
e5 Uµ5 packets
−2ix
14
φ ≡ arg
(4) (11)
�2 probability
We can simplify this expression
by defining the CPPodd
βµ4are
∗
= �Ue4 U
e averaged
− Uover.
−U
Uappearance
UU
e4 Uµ4The
e5
µ5
νµrapid
→νquantity
e4
e oscillations
µ4
effects become unobservable when� the
probability
then
becomes
2
2
2
2
2
2
�
where
β≡
Disappearance vs appearance, very
4|U | |U | sin x + 4|U | |U | sin x + 8|U ||U ||U ||U | sin x sin x cos(x − x ± φ)
� 2
−2iβ
�= �U isUa physically
�
P
e
U U|2 |U
eCP violating
−
U and
−+Uthe
(1)
=+
|U
|2m
|eU−2ix
reU+(-)
| is used for neutrinos(antineutrinos).
observable
phase
sign
In the limit where (12)
e4
2
2
2large
limit
sin
φ|U
||U
|µ4varies
1|Ue4 |2 |U
sin the
x415th
+neutrino
2|U
|2x +
4|U
x41 sin(x
∓this
φ) limit it makes no difference
(5) to
−1µ4 | where
e5 |e5
µ5µ5
e5
e4 ||Uµ4
µ5 | sin
41 In
� averaged
�5||Urapidly
is |U
heavy
very
and||U
should
be
over.
νµ →νe
2
tan
e4
e4 41 µ4
∗ −2ix
e5
µ4
∗ 41
−2ix51
µ5
e5
e4
µ5
∗14
µ4
e5
51
∗ 2
µ5
e5
e4
µ4
µ5
51
41
51
(6)
2
2 51
oscillation
formula
whether
the
neutrino
heavy
� . so that
� the wave packets separate, as interference (13)
=
4|U
|
|Uµ4
|2µ5
r| sinbecomes
(x41 −
β))enough
e4
|Ue4the
||U
|
+
cos
φ|U
||U
∗
Ue5 Uµ5
effects µ4
become unobservablee5
when the
rapid oscillations are averaged
over. The appearance probability then becomes
We can simplify this expression by defining the
quantity β
2
(m2i CP
− modd
L/E
j)
φ ≡ arg
2
2
2
2
2
Define
.
(2)
x ≡ 1.27
e4 Uµ4
4|Ue5U||U
µ4 | sin x41 + 2|Ue5 | |Uµ5 | +�
e4 ||Uµ4 ||Uµ5 | sin x41 sin(x41 ∓ φ)
and the mixing ratio rWe can define the CP odd quantity β ij 1 −1 �eV|U2e4 | |Ukm/GeV
sin φ|Ue5 ||Uµ5 |
and the mixing ratio r
and the mixing ratio r
r≡
and get oscillation probability
41
∗
2
(3)
(4)
(5)
β We
≡ cantan
simplify
this expression
by defining
quantity
is a physically
observable
CP violating
phase the
andCP
theodd
+(-)
sign isβ used for neutrinos(antineutrinos).(6)
In the limit where
−1 ||Uµ4 | + cos φ|Ue5 ||Uµ5 |
|U
�
�over. In this limit it makes no difference to
∗ 2 β ≡is tan
∗ e4x(r
sin
φ/((1
+
r
cos
φ)))
the
5th
neutrino
heavy
varies
very
rapidly
and
should
be
averaged
|Ue5 Uµ5 + Ue4 Uµ4 | 51
sin
φ|U
||U
|
1
e5
µ5
(6)interference
β ≡ tan−1
the oscillation formula whether the neutrino
becomes heavy enough so that the wave packets
(7)separate, as
2
|Ue4 ||Uµ4 | + cos φ|Ue5 ||Uµ5 |
∗ |
effects become
when the rapid oscillations
are averaged over. The appearance probability then becomes
|Ue4 Uunobservable
µ4
and the mixing ratio r
∗ 2
∗
Uµ4
Uµ4
| e5 |2 |Uµ5∗|2 + 4|Ue5
|U|U
|2 |U
sinU2e4
x41
+
2|U
||Ue4 ||Uµ4 ||Uµ5 | sin x41 sin(x41 ∓ φ)
(5)
e4e5
µ5| +
∗
(7)
|Ue5 Uµ5 + Ue4 Uµ4 |
rr≡≡ |Ue5 ||Uµ5 |/(|U
∗
r ≡µ4 |)
e4 ||U
∗ |
|Ue4 Uby
µ4 |defining the CP|Uodd
We can simplify this expression
quantity β
e4 Uµ4
�
�
2
2
2
2
2
and
get
oscillation
probability
1 41−1
and
get
electron
appearance
probability
P
= neutrino
2|U
+ 2r sin β + 2r sin
(x
− β)] . sin φ|Ue5 ||Uµ5 |
and
oscillation
probability
νget
e4 | |U
µ4 | [(1 − r)
µ →ν
e
β ≡ tan
|Ur)
| +2 cos
| β)] .
e42||U
µ4sin
e52||U
µ5−
Pνµ →νe = 2|Ue42|2 |Uµ4 |2 [(1 −
+ 2r
β + φ|U
2r sin
(x41
(7)
(8)
(6)
(8)
2 2
2 2
2
22r sin2 β +2 2r sin2 (x − β)] .
Pνµ →νeand
= the
2|U
|
|U
|
[(1
−
r)
+
(8)
|U
|
|U
|
[(1
−
r)
+
4r
sin
(x
−
β)]
e4
µ4
41
e4
µ4
41
mixing
ratio we
r replace β → −β. Note that CP violation remains observable in the5limit of heavy m5 . Note
For anti
neutrinos
also that the amplitude of oscillations associated with m4 may be enhanced by mixing with a heavy 5th neutrino.
∗
∗
4
For βanti
we replace β → In
−β.
Note
that
CPbound
violation
remains
observable
limit
of heavy
m5in
. are
Note
|Ue5 U
+ Uin
Uthe
In
constrast
withthe
appearance
experiments,
absence
ofµ5
matter
effects,
experiments
muchprevious
less
e4
µ4 | disappearance
with
→ neutrinos
−β for antineutrinos.
this
case
on fermions.
r isin the
weakened
compared
to the
bound
the
rAveraging
≡
(7)
sensitive
to
mixing
with
heavy
neutral
over
the
short
oscillation length associated with ∆m251 ,
∗
also
that
the
amplitude
of
oscillations
associated
with
m
may
be
enhanced
by
mixing
|U
4
e4 Uµ4 | with a heavy 5th neutrino.
section, to
and neglecting the mass differences among the three light neutrinos, the probability for vacuum electron
neutrino
or
2
In constrast with appearance experiments,
the absence
of matter
effects, disappearance experiments are much
antiinneutrino
disappearance
is
51 less
and electron
get oscillation
probability
�
�
5
sensitive to mixing with heavy neutral fermions. Averaging
over
the
short
oscillation
length
associated
with
∆m251 ,
120
2
4|Ue4 |2 (1 − |Ue42|22− |U2e5 |2 ) sin2 x241 + 2|Ue52|2 (1 − |Ue5 |2 )2
(9)
2
2 (1 − r)
For anti neutrinos we replace β → −β. Note that CP violation remains observable in the limit of heavy m . Note
also that the amplitude of oscillations associated with m may be enhanced by mixing with a heavy 5th neutrino.
In constrast with appearance experiments,large
in therabsence
of matter effects,
disappearance
experiments
gives enhancement
of μ→e
oscillations,
not are much less
sensitive to mixing with heavy neutral fermions. Averaging over the short oscillation length associated with ∆m ,
constrained by disappearance in large m limit
and neglecting the mass differences among the three light neutrinos, the probability for vacuum electron neutrino or
and neglecting the
light
neutrinos,
for
electron
neutrino
or
Pν →ν
=
|the
|Uβ
| [(1
r) + 2r
β + 2r sin
(x41 − β)]
.
|Ue4 | the
|Uµ4three
|
+ 2|U
4r e4
sin
< −0.0008
.sinvacuum
(14)
µ4probability
electron anti neutrino disappearance
is mass differences among
2 neutrino or muon antineutrino disappearance is obtained by replacing e → µ in the
and the probability for muon
electron anti neutrino disappearance is
µ
e
(8)
preceding
formula.
disappearance
are obtained
cancelling in
systematic
errors
using m5 . Note
For anti
neutrinos
weTypically,
replace βstringent
→ −β.bounds
Note on
that
CP violation
remains by
observable
the limit
of heavy
2
2 L/E dependence.
the
Thus U 2and U are only2weakly constrained, and in the limit where the oscillation length
e5 2
µ5
also
the
of oscillations
associated
with m4 may2 be enhanced by mixing with
a heavy 5th neutrino.
2that
2)
2 amplitude
−4
4|Ue4
| (1
− 0,
|Uand
|U
|(1µ4)−|sin
x2|241
+
2|U
| (1
| |allowed
(9)
e4 | −
e5
e5e5
2 e5
4|U
||U
|U
−appearance
|U
|2,)e5
sin
xto41−
+|U
2|U
(1 − |Ue5
| dependence,
) ofeffects,
(9)
For instance
for
β=
|U
|2In
=e4
×
10
the
maximum
value
r is
3.8, andexperiments
the
probability
of
e4e4
associated
with
give
any
disappearance
strongly
51 is too short
constrast
with∆m
experiments,
in measurable
the absenceL/E
of matter
disappearance
experiments
are much less
−3
bound
only
U
and
U
.
e4
µ4
electron neutrino appearance maximizes
8 ×with
10 heavy
. neutral fermions. Averaging over the short oscillation length associated with ∆m251 ,
sensitive to at
mixing
2
2
Theortension
between
electron antineutrino
appearance is
at obtained
LSND and MiniBooNE
and short
baseline
the neutrino
probabilityor
formuon
muon antineutrino
neutrino
muon
antineutrino
disappearance
by replacing
→
µ inelectron
the and neutrino or
and the probability for and
muon
disappearance
is may
obtained
by
replacing
e →
µe in
the
and muon
neglecting
the
mass
differences
among
thebethree
light
neutrinos,
probability
vacuum
neutrino
disappearance
experiments
reduced
by allowing
r to the
be greater
than 1.for
Only
electronelectron
neutrino
preceding
formula.
Typically,
stringent
bounds
on
disappearance
are
obtained
by
cancelling
systematic
errors
using
electron
anti
neutrino
disappearance
is
appearance experiments
constrain r. The
such constraint
comes from errors
the E776 experiment
preceding formula. Typically, stringent bounds on disappearance
arewillobtained
by strongest
cancelling
systematic
using [? ] . The
of very
shortweakly
baseline electron
neutrino appearance
implies
the L/E dependence. Thus Ue5 andabsence
Uµ5 are
only
constrained,
and in 2the
limit where2 the oscillation
length
2
V. LSND AND
MINIBOONE
4|Ue4 |2in
(1
−
|Ue4limit
|2 − |Ue5where
| ) sin2 x41
+ 2|U
)
(9)
the L/E dependence. Thus
Ue5 and
U
onlyshort
weakly
constrained,
and
the
the
oscillation
e5 | (1 − |Ue5 | length
µ52 are
2
2 dependence,
2
2
associated
with
∆m
is
too
to
give
any
measurable
L/E
disappearance
experiments
strongly
|Ue4 | |Uµ4 | ((1 − r) + 4r sin β) < 0.0008 .
(10)
51
associated with ∆m251 isbound
too only
short
to
give
any
measurable
L/E
dependence,
disappearance
experiments
strongly
and the probability for muon neutrino or muon antineutrino disappearance is obtained by replacing e → µ in the
Ue4 and Uµ4 .
Therefore
the existence
of the
heavy fifth
neutrino
makes it is possible
to
obtain much
larger electron
neutrinoerrors using
preceding
formula.
Typically,
stringent
bounds
on disappearance
are
obtained
by cancelling
systematic
VI.
EFFECTS
OF MIXING
WITH
HEAVY
NEUTRAL
FERMIONS
ON
OSCILLATIONS
IN
MATTER:
bound only Ue4 and Uµ4
. The
tension between
electron antineutrino
appearance
at
LSND
MiniBooNE
baseline
electron
appearance probabilities
at non
zero
L/E and
than would
be possibleand
in a short
3+1 model.
For instance
for βand
= 0, and
the L/E
Thus
U
and
U
are
only
weakly
constrained,
and
in
the
limit
where
the
oscillation length
2 dependence.
2
−4
e5
µ5
INDUCED
NONSTANDARD
INTERACTIONS
AND
APPARENT
CPT
VIOLATION
|Ue4 | |U
| LSND
= 2be
× 10
, the maximum
allowed value
of
3. For a baseline
3+1 model
with
δ=
0 and r =
1, the maximum
muon neutrino
disappearance
experiments
may
by allowing
r to
ber isgreater
than
1. Only
electron
neutrino
The tension between electron
antineutrino
appearance
atµ4
and
MiniBooNE
and
short
electron
and
2reduced
−4
associated
with
too short
to giveas any
measurable
L/E
dependence,
disappearance
strongly
51 is
probability
of ∆m
electron
neutrino
appearance
a function
of L/E is
8 × 10
. In contrast,
for r = 3, the experiments
probability
appearance
experiments
willbe
constrain
r. The
strongest
such
constraint
comes
from
the
experiment
[? ] . The
−3 E776
bound
only
Ue4
and
Uappearance
muon neutrino disappearance
experiments
may
reduced
by
allowing
r tomaximizes
be greater
than
Only
neutrino
µ4 .
of electron
neutrino
at a much
larger 1.
4 × 10
. In electron
the next section
we will find even larger
of very short baseline electron
neutrino
appearance
implies
The
tension
between
electron
antineutrino
at are
LSND
and MiniBooNE
short baseline electron and
enhancements
possible
when
neutrinos
mix withappearance
which
too heavy
to be produced.
VII.
CONCLUSIONS
appearance experimentsabsence
will constrain
r. The strongest
such
constraint
comes
fromfermions
the E776
experiment
[? ] and
. The
muon neutrino disappearance experiments may be reduced by allowing r to be greater than 1. Only electron neutrino
absence of very short baseline electron neutrino appearance
implies
appearance
experiments
constrain
The
strongest
[? ] . The
|Ue4
|2 |U
|2 ((1 −will
r)2IN
+ NEUTRINO
4r sin2r.β)
<
0.0008 . such constraint comes from the E776 experiment
(10)
µ4VIOLATION
IV. CP
OSCILLATIONS IN A 3+2 MODEL WITH ONE FERMION
absence the
of very
short baseline electron neutrino
A major difficulty with explaining
LSND/MiniBooNE
dataHEAVIER
inappearance
terms
ofimplies
neutrino oscillations via mixing with
THAN
m
2electron
2 difficulty
2 the heavy
2 fifth neutrino
Therefore
the|U
of
makes
it
is
possible
to
obtain
much
larger electron(10)
neutrino
2
2
2
sterile
neutrinos
isexistence
the
of
reconciling
neutrino
or
anti
neutrino
baseline with (10)
e4 | |Uµ4 | ((1 − r) + 4r sin β) < 0.0008
|U .| |Uµ4 | ((1 − r) + 4r sin2 appearance
β) < 0.0008 . at short
In
this section
we
consider
inpossible
detail e4
a model
with
2 additional
sterile
neutrinos,
onefor
of which
is0,tooand
heavy to
appearance
probabilities
at
non
zero
L/E
than
would
be
in
a
3+1
model.
For
instance
β
=
the absence
of evidince
of electron and
muonin neutrino
baseline.
However
disappearance
be produced
pion or muondisapeparance
decay. In this case, at
onlyshort
4 of the
5 mass eigenstates
can bethese
produced
in a muon
2
2
−4
the existence
fifth
neutrino
makes
itδ is=possible
much
largerHowever
neutrino
|Ue4of
| |Uthe
= 2×
, the
maximum
allowed
ofofr the
iscan3.heavy
For
aL/E
3+1
model
0electron
andmatrix
rto=obtain
1,neutrino
the
maximum
µ4 | rely
neutrino
beam.
This
situation
beto
described
in terms
of a with
non-unitary
mixing
for
the
light
states.
expeirments
on10fifth
dependence
ofTherefore
the
detection
on
to
cancel
out
systematics.
Mixing
withelectron
fermions
Therefore the existence
heavy
neutrino
makes
itvalue
is probability
possible
obtain
much
larger
−4
appearance
probabilities
at non
zero
L/E
than
would
becontrast,
possible in
a 3+1
model.
For instance for β = 0, and
probability
of
electron
neutrino
appearance
as
a
function
of
L/E
is
8
×
10
.
In
for
r
=
3,
the
probability
2
which
than than
30 eVwould
does
not
L/E allowed
dependence
so ais3+1
less
constrained.
this
paper
|Ue4 |2 |U
=
2 × observable
10−4 , the
maximum
valueFor
of rand
isinstance
3. For
model
δ =and
0 andInr =
1, the
maximum
appearance probabilities
at are
nonheavier
zero L/E
be
possible
inlarger
a 3+1
model.
for
β with
= 0,
µ4 | give
of electron neutrino appearance probability
maximizes
a much
4 × 10−3as. aInfunction
the next
section
we
evenforlarger
of at
electron
neutrino
appearance
of L/E
is 8 ×
10−4will
. In find
contrast,
r = 3,
the allow
probability
2
2
−4
we
have
shown
how
mixing
with
such
neutral
heavy
fermions,
in
combination
with
neutrino
oscillations,
can
|Ue4 | |Uµ4 | = 2 × 10 ,enhancements
the maximum
allowed
of rmix
isneutrino
3.
For
a 3+1which
model
δ = to
0 and
r = 1, the maximum
possible
whenvalue
neutrinos
with
fermions
are with
too
be 4produced.
of electron
appearance
maximizes
at aheavy
much larger
× 10−3 . In the next section we will find even larger
both CPappearance
violation and
enhancement
of flavor
interference.
also have shown that in
probability of electronfor
neutrino
as an
a function
of L/E
iswhen
8 ×change
10−4 . via
In constructive
contrast,
r are
= too
3, heavy
the Iprobability
enhancements
possible
neutrinos
mix
with
fermions for
which
to be produced.
general,
the
effects
of
mixing
with
heavy
fermions
can
be
incorporated
by
adding
a
small
number
of new parameters
of electron neutrino appearance maximizes at a much larger 4 × 10−3 . In the next section we will find even larger
neutrino
oscillation
For exampleOSCILLATIONS
3+1 model to with
new
to incorporate
the effects of heavy
IV. CP
VIOLATION
NEUTRINO
IN 2
3+2parameters
MODEL WITH
ONE FERMION
enhancements possibletowhen
neutrinos
mixformulae.
withIN
fermions
which
are tooINheavy
to Abe
produced.
IV. CP
VIOLATION
NEUTRINO
OSCILLATIONS
IN A 3+2 MODEL WITH ONE FERMION
HEAVIER
THAN m
µ HEAVIER
fermions can give a good fit to all short baseline
data including
LSND,
MiniBooNE,
short
baseline
disappearance
THAN mand
µ
µ
Evidence for neutrino mixing with heavy
fermion?
• 3+2 +CPV with all Δm < 100 eV
2
fits MB/LSND
appearance, does not fit disappearance well
• 3+1 (+ 1)
2
can fit MB/LSND appearance and all
disappearance well, CPV improves fit
121
What about
cosmology?
122
“Cosmology seeking friendship with sterile neutrinos”
arXiv1006.527v6
Jan Hamann, Steen Hannestad, Georg G. Raffelt, Irene Tamborra, and
Yvonne Y. Y. Wong
Precision cosmology and big-bang nucleosynthesis mildly favor extra
radiation in the universe beyond photons and ordinary neutrinos, lending
support to the existence of low-mass sterile neutrinos. We use the
WMAP 7-year data release, small-scale CMB observations from ACBAR,
BICEP and QuAD, the 7th data release of the Sloan Digital Sky Survey,
and measurement of the Hubble parameter from Hubble Space
Telescope observations to derive credible regions for the assumed
common mass scale ms and effective number Ns of thermally excited
sterile neutrino states. Our results are compatible with the interpretation
of the LSND and MiniBooNE anomalies in terms of 3 active + 2 sterile
neutrinos if ms is in the sub-eV range.
123
Cosmology and
sterile ν’s
•
CMB, structure formation
consistent with up to 4 light
sterile ν’s in thermal
equilibrium
•
primordial Helium now
consistent with up to
2 sterile ν’s
•
stable sterile ν’s in thermal
equilibrium should be
lighter than ~1 eV
from arXiv1006.527v6
124
•
•
•
•
•
Summary
Neutrino portal is sensitive to new sectors containing fermions
Is there any evidence?
surprising LSND/MB results
➡ LSND may be systematic, not compelling
➡ MB is statistics limited
more MB antineutrino data will be interesting
CMB, Helium, structure now seem consistent with a few
additional light states in thermal equilibrium
• propose analyzing short baseline ν flavor appearance expts with 2
new parameters
125