Download Neutrino oscillations, energy-momentum conservation and

Neutrino oscillations, energy-momentum
conservation and entanglement
Evgeny Akhmedov
MPI-K, Heidelberg & Kurchatov Inst., Moscow
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 1
4-mom. conservation and ν production
Neutrino oscillations and energy-momentum conservation
– an intricate relationship
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 2
4-mom. conservation and ν production
Neutrino oscillations and energy-momentum conservation
– an intricate relationship
Calculation of rates of processes in quant. theory (gen. Fermi’s Golden rule):
X
X 2
d3 pf (2π)4 δ 4
pi
M
pf −
Γ =
f
i
3
(2π) 2Ef
i
i
f
f
P
P
The factor δ 4
ensures energy-momentum conservation.
i pi
f pf −
Y
Evgeny Akhmedov
1
(2Ei )
Z Y
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 2
4-mom. conservation and ν production
Neutrino oscillations and energy-momentum conservation
– an intricate relationship
Calculation of rates of processes in quant. theory (gen. Fermi’s Golden rule):
X
X 2
d3 pf (2π)4 δ 4
pi
M
pf −
Γ =
f
i
3
(2π) 2Ef
i
i
f
f
P
P
The factor δ 4
ensures energy-momentum conservation.
i pi
f pf −
Y
1
(2Ei )
Z Y
Used to calculate neutrino production rates and detection cross sections.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 2
4-mom. conservation and ν production
Neutrino oscillations and energy-momentum conservation
– an intricate relationship
Calculation of rates of processes in quant. theory (gen. Fermi’s Golden rule):
X
X 2
d3 pf (2π)4 δ 4
pi
M
pf −
Γ =
f
i
3
(2π) 2Ef
i
i
f
f
P
P
The factor δ 4
ensures energy-momentum conservation.
i pi
f pf −
Y
1
(2Ei )
Z Y
Used to calculate neutrino production rates and detection cross sections.
If applied to neutrino production, implies that the neutrino 4-momentum
p = (E, p~) can be determined from the 4-momenta of all other particles
participating in the production process.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 2
4-mom. conservation and ν production
Neutrino oscillations and energy-momentum conservation
– an intricate relationship
Calculation of rates of processes in quant. theory (gen. Fermi’s Golden rule):
X
X 2
d3 pf (2π)4 δ 4
pi
M
pf −
Γ =
f
i
3
(2π) 2Ef
i
i
f
f
P
P
The factor δ 4
ensures energy-momentum conservation.
i pi
f pf −
Y
1
(2Ei )
Z Y
Used to calculate neutrino production rates and detection cross sections.
If applied to neutrino production, implies that the neutrino 4-momentum
p = (E, p~) can be determined from the 4-momenta of all other particles
participating in the production process. E.g. for π → µ + ν decay:
if 4-momenta of π and µ have well-defined values, then the neutrino
4-momentum is fully determined by energy-momentum conservation.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 2
A dichotomy
But: Due to the on-shell relation
E 2 = p~ 2 + m2 ,
if the neutrino energy and momentum are exactly known, so is its mass!
⇒
The emitted neutrino is a mass eigenstate rather than a flavor eigenstate
( = coherent superposition of different mass eigenstates).
⇒
Neutrino oscillations cannot occur! (Mass eigenstates do not oscillate in
vacuum).
A dichotomy:
On the one hand, energy-momentum conservation is an exact law of nature.
On the other hand, exact energy and momentum conservation at neutrino
production or detection would apparently make the oscillations impossible.
⇒
Significant confusion in the literature
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 3
Kinematic entanglement – way out?
A suggested way out (an attempt to base the theory of neutrino oscillations on
exact energy-momentum conservation):
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 4
Kinematic entanglement – way out?
A suggested way out (an attempt to base the theory of neutrino oscillations on
exact energy-momentum conservation):
Assumption: the neutrino is produced in an entangled state with the
accompanying particle(s), so that the 4-momentum of the entangled state
obeys exact energy-momentum conservation.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 4
Kinematic entanglement – way out?
A suggested way out (an attempt to base the theory of neutrino oscillations on
exact energy-momentum conservation):
Assumption: the neutrino is produced in an entangled state with the
accompanying particle(s), so that the 4-momentum of the entangled state
obeys exact energy-momentum conservation.
Example: for π → µ + ν decay. Take for the final-state wave function
|µ νi =
X
∗
Uµj
|µ(pµj )i|νj (pνj )i
j
For every value of neutrino 4-momentum pνj the corresponding value of
muon 4-momentum pµj is uniquely defined through pνj + pµj = pπ .
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 4
Kinematic entanglement – way out?
A suggested way out (an attempt to base the theory of neutrino oscillations on
exact energy-momentum conservation):
Assumption: the neutrino is produced in an entangled state with the
accompanying particle(s), so that the 4-momentum of the entangled state
obeys exact energy-momentum conservation.
Example: for π → µ + ν decay. Take for the final-state wave function
|µ νi =
X
∗
Uµj
|µ(pµj )i|νj (pνj )i
j
For every value of neutrino 4-momentum pνj the corresponding value of
muon 4-momentum pµj is uniquely defined through pνj + pµj = pπ .
The 4-momenta of mass eigenstates νj are correlated with 4-momenta of the
accompanying muon (entanglement). Only pπ is fixed; pµ and pν are
allowed to vary.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 4
Entanglement – contd.
Energy-mom. conservation is satisfied, yet the produced state contains a
superposition of different neutrino mass eigenstates.
But: because muon states of different 4-momenta are orthogonal,
energy-momentum conservation would still make the interference (oscillatory)
terms in Pαβ vanish – disentanglement is necessary.
Assumption: the muon is “measured”, either through a direct detection or
through interaction with environment, which leads to its localization ⇒
necessary energy and momentum uncertainties are created.
– Completely misses the point that in reality the parent pion is always
localized and so described by a wave packet rather than by a plane wave.
(Btw: without this localization the production coordinate would be completely
undefined ⇒ the oscillations would be unobservable).
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 5
Entanglement – contd.
The pion momentum has a spread σpπ around the mean momentum pπ
⇒ there is no strict correlation between 4-momenta of ν and µ.
For a given value pνj the value of pµ is no longer uniquely determined by
pνj + pµ = pπ ; can take any value within a range of width ∼ σpπ .
In other words, instead of pνj + pµj = pπ we now have
pνj + pµj = pπj
where pπj is no longer uniquely fixed.
E.g., in the case when the 4-momenta of the components of the muon
accompanying different νj are the same, pµ1 = pµ2 = pµ3 ≡ pµ :
energy-momentum conservation is satisfied if
|pνj − pνk | . σpπ .
No entanglement takes place!
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 6
NB: |pνj − pνk | . σpπ is just the condition of coherent neutrino
production/detection (necessary for observability of ν oscillations).
⇒
Due to localization of the neutrino production and detection processes,
kinematic entanglement is completely irrelevant to neutrino oscillations.
Entanglement/disentanglement approach predicts no oscillations for ν’s from
π → µ + ν decay if the muons is not detected and do not interact with the
environment; wave packet approach predicts that the oscillations should occur.
QM energy and momentum uncertainties play a crucial role!
Energy and momentum uncertainties are in no contradiction with exact
energy-mom. conservation! In describing the oscillations we have to deal with
localized states which are not eigenstates of E and p but rather linear
superpositions of their eigenstates.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 7
QM energy and mom. uncertainties
Neutrino oscillations are a QM interference phenomenon ⇒ owe their very
existence to QM uncertainty relations.
Coordinate-momentum and time-energy uncertainty relations are implicated in
the oscillations phenomenon in a number of ways:
It is the E and p uncertainties of the emitted ν state that allow it to be a
coherent superposition of the states of well-defined and different mass.
Similarly, for neutrino detection to be coherent, the E and p uncertainties
inherent in the detection process should be large enough to prevent a
determination of the absorbed neutrino’s mass in this process.
QM uncertainty relations determine the size of the neutrino wave packets
⇒ are crucial to the issue of the loss of coherence due to the wave
packet separation.
Are important for understanding how the produced and detected neutrino
states are disentangled from the accompanying particles.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 8
When are neutrino oscillations observable?
Keyword: Coherence
Neutrino flavour eigenstates νe , νµ and ντ are coherent superpositions of
⇒ oscillations are only observable if
mass eigenstates ν1 , ν2 and ν3
neutrino production and detection are coherent
coherence is not (irreversibly) lost during neutrino propagation.
Possible decoherence at production (detection): If by accurate E and p
p
measurements one can tell (through E = p2 + m2 ) which mass eigenstate
is emitted, the coherence is lost and oscillations disappear!
Full analogy with electron interference in double slit experiments: if one can
establish which slit the detected electron has passed through, the interference
fringes are washed out.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 9
When are neutrino oscillations observable?
Another source of decoherence: wave packet separation due to the difference
of group velocities ∆v of different mass eigenstates.
If coherence is lost: Flavour transition can still occur, but in a non-oscillatory
way. E.g. for π → µνi decay with a subsequent detection of νi with the
emission of e:
P ∝
X
Pprod (µ νi )Pdet (e νi ) ∝
X
|Uµi |2 |Uei |2
i
i
– the same result as for averaged oscillations.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 10
A consistent approach – WP formalism
The evolved produced state:
|ναfl (~x, t)i
=
X
∗
Uαj
|νjmass (~x, t)i
=
X
∗
Uαj
ΨSj (~x, t)|νjmass i
j
j
The coordinate-space wave function of the jth mass eigenstate (w. packet):
ΨSj (~x, t)
=
Z
d3 p S
i~
p~
x−iEj (p)t
f
(~
p
)
e
(2π)3 j
p): sharp maximum at p~ = P~ (width of the
Momentum distribution function fjS (~
~
peak σpP ≪ P ). Detected state (centered at ~x = L):
X
fl
∗
|νβ (~x)i =
Uβk
ΨD
x)|νkmass i
k (~
k
The coordinate-space wave function of the kth mass eigenstate (w. packet):
ΨD
x)
k (~
Evgeny Akhmedov
=
Z
d3 p D
~
i~
p(~
x−L)
f
(~
p
)
e
(2π)3 k
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 11
Oscillation probability
Transition amplitude:
~ = hνβfl |ναfl (T, L)i
~ =
Aαβ (T, L)
X
∗
~
Uαj
Uβj Aj (T, L)
j
~ =
Aj (T, L)
Z
d3 p S
~
D∗
−iEj (p)T +i~
pL
f
(~
p
)
f
(~
p
)
e
j
(2π)3 j
~ − ~vj T | . σx . E.g., for Gaussian wave packets:
Strongly suppressed unless |L
"
~ − ~vj T )2
(
L
~ ∝ exp −
Aj (T, L)
4σx2
#
,
2
2
σx2 ≡ σxP
+ σxD
Oscillation probability:
♦
2
X
∗
∗
~ = |Aαβ | =
~ ∗k (T, L)
~
P (να → νβ ; T, L)
Uαj
Uβj Uαk Uβk
Aj (T, L)A
j,k
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 12
Oscillation probability
Neutrino emission and detection times are not measured (or not accurately
measured) in most experiments ⇒ integration over T :
Z
∆m2
X
∗
∗
−i 2P̄jk L ˜
Ijk
dT P (να → νβ ; T, L) =
P (να → νβ ; L) =
Uαj Uβj Uαk Uβk e
j,k
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 13
Oscillation probability
Neutrino emission and detection times are not measured (or not accurately
measured) in most experiments ⇒ integration over T :
Z
∆m2
X
∗
∗
−i 2P̄jk L ˜
Ijk
dT P (να → νβ ; T, L) =
P (να → νβ ; L) =
Uαj Uβj Uαk Uβk e
j,k
I˜jk = N
Z
dq S
fj (rk q − ∆Ejk /2v + Pj )fjD∗ (rk q − ∆Ejk /2v + Pj )
2π
×fkS∗ (rj q
Here:
v≡
vj +vk
2
Evgeny Akhmedov
+ ∆Ejk /2v +
, ∆v ≡ vk − vj ,
Pk )fkD (rj q
rj,k ≡
TAUP-2011
vj,k
v
,
i ∆v
v qL
+ ∆Ejk /2v + Pk ) e
N ≡ 1/[2Ei (P )2Ek (P )v]
Munich, Sept. 5-9, 2011
– p. 13
Oscillation probability
Neutrino emission and detection times are not measured (or not accurately
measured) in most experiments ⇒ integration over T :
Z
∆m2
X
∗
∗
−i 2P̄jk L ˜
Ijk
dT P (να → νβ ; T, L) =
P (να → νβ ; L) =
Uαj Uβj Uαk Uβk e
j,k
I˜jk = N
Z
dq S
fj (rk q − ∆Ejk /2v + Pj )fjD∗ (rk q − ∆Ejk /2v + Pj )
2π
×fkS∗ (rj q
Here:
v≡
vj +vk
2
+ ∆Ejk /2v +
, ∆v ≡ vk − vj ,
Pk )fkD (rj q
rj,k ≡
vj,k
v
,
i ∆v
v qL
+ ∆Ejk /2v + Pk ) e
N ≡ 1/[2Ei (P )2Ek (P )v]
For (∆v/v)σp L ≪ 1 (i.e. L ≪ lcoh = (v/∆v)σx ) I˜jk is approximately
independent of L; in the opposite case I˜jk is strongly suppressed
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 13
Oscillation probability
Neutrino emission and detection times are not measured (or not accurately
measured) in most experiments ⇒ integration over T :
Z
∆m2
X
∗
∗
−i 2P̄jk L ˜
Ijk
dT P (να → νβ ; T, L) =
P (να → νβ ; L) =
Uαj Uβj Uαk Uβk e
j,k
I˜jk = N
Z
dq S
fj (rk q − ∆Ejk /2v + Pj )fjD∗ (rk q − ∆Ejk /2v + Pj )
2π
×fkS∗ (rj q
Here:
v≡
vj +vk
2
+ ∆Ejk /2v +
, ∆v ≡ vk − vj ,
Pk )fkD (rj q
rj,k ≡
vj,k
v
,
i ∆v
v qL
+ ∆Ejk /2v + Pk ) e
N ≡ 1/[2Ei (P )2Ek (P )v]
For (∆v/v)σp L ≪ 1 (i.e. L ≪ lcoh = (v/∆v)σx ) I˜jk is approximately
independent of L; in the opposite case I˜jk is strongly suppressed
I˜jk is also strongly suppressed unless ∆Ejk /v ≪ σp , i.e. ∆Ejk ≪ σE
– coherent production/detection condition
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 13
Conservation of hEi and h~pi
For the initially produced flavour state |να i:
Z
X
|Uαj |2
hEα i ≡ hνα |H|να i =
i
h~
pα i ≡ hνα |~
p|να i =
X
|Uαj |
2
j
For the evolved state
~ t)i =
|ν(L,
X
∗
Uαj
j
Z
d3 p
S
2
|f
(~
p
)|
Ej (~
p)
j
(2π)3
d3 p
S
2
|f
(~
p
)|
p~
j
3
(2π)
d3 p S
~
i~
pL−iE
j (p)t
f
(~
p
)
e
|νj i
j
3
(2π)
Z
E and p expectation values are
hEi ≡ hν|H|νi = hEα i ,
⇒
h~
pi ≡ hν|~
p|νi = h~
pα i
hEi and h~
pi do not change in the course of the oscillations.
In neutrino oscillations energy-momentum conservation manifests itself as
conservation of the mean values of neutrino energy and momentum.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 14
Flavor independence of hEi and new physics
Ahluwalia & Schritt, arXiv:0911.2965: Assume that mean neutrino energies
hEα i measured in ν oscillation experiments are flavor independent:
hEe i = hEµ i = hEτ i .
Then: either sterile neutrinos must exist, or neutrinos must have some new
interactions beyond those of the standard model. (Flavour independence of
hEα i is inconsistent with what is known about the usual 3-f neutrino mixing).
A 2-f example:
hEe i = hE1 i cos2 θ + hE2 i sin2 θ ,
hEµ i = hE1 i sin2 θ + hE2 i cos2 θ ,
For m1 6= m2 one has hE1 i =
6 hE2 i
⇒
the condition hEe i = hEµ i would exclude all values of the mixing angle
except θ = 45◦ corresponding to the maximal leptonic mixing!
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 15
Flavour independence of hEi – contd.
There is no physical reasons to expect flavour independence of hEα i !
~ can be represented as
The oscillated state |ν(t, L)i
~ = Ae (t, L)|ν
~ e i + Aµ (t, L)|ν
~ µ i + Aτ (t, L)|ν
~ τi,
|ν(t, L)i
~ the probability amplitudes of finding the flavour states να
with Aα (t, L)
~ The expect. value hEi in the state |ν(t, L)i
~
(α = e, µ, τ ) at x = (t, L).
remains constant and equal to that in the initially produced flavour state.
But: This does not mean that each individual flavour component of the state
has the same energy expectation value!
The conjecture of Ahluwalia & Schritt actually contradicts the observability
conditions for neutrino oscillations.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 16
Flavour independence of hEi – contd.
An assumption of Ahluwalia & Schritt: The detection process does not distort
the measured neutrino energy, because “by choosing an appropriate
kinematical setting, this ‘back reaction’ can be made average to zero”.
But: this condition would actually make neutrino oscillations unobservable!
In practical terms the condition of “no back reaction” actually means
|hEα i − hEβ i| ≫ σE
Notation: hEj i ≡ Ēj ;
X
X
2
2
(|Uαi |2 − |Uβi |2 )(Ēi − Ēk + Ēk )
(|Uαi | − |Uβi | )Ēi =
hEα i − hEβ i =
i
i
=
X
(|Uαi |2 − |Uβi |2 )(Ēi − Ēk ) ,
i
(Ēk is the mean energy of any of the neutrino mass eigenstates;
unitarity of U used).
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 17
Flavour independence of hEi – contd.
Taking into account |Uai |2 ≤ 1, from |hEa i − hEb i| ≫ σE
|Ēi − Ēk | ≫ σE
⇒
(i 6= k) .
But: This is just the opposite of the coherent detection condition which is a
necessary condition for observability of neutrino oscillations!
Coherent det. condition ensures that the detection process cannot discriminate
between different neutrino mass eigenstates. If it is violated, the neutrino
cannot be detected as a coherent superposition of different mass eigenstates.
⇒
The fact that the oscillations have been observed completely rules out
the conjecture of Ahluwalia & Schritt.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 18
Summary
Attempts of simple-minded application of energy-momentum conservation
to neutrino oscillations are inconsistent and lead to no oscillations at all.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 19
Summary
Attempts of simple-minded application of energy-momentum conservation
to neutrino oscillations are inconsistent and lead to no oscillations at all.
Kinematic entanglement is irrelevant to neutrino oscillations.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 19
Summary
Attempts of simple-minded application of energy-momentum conservation
to neutrino oscillations are inconsistent and lead to no oscillations at all.
Kinematic entanglement is irrelevant to neutrino oscillations.
Energy-momentum conservation manifests itself in ν oscillations as
conservation of the expectation values of ν energy amd momentum in
the course of the oscillations.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 19
Summary
Attempts of simple-minded application of energy-momentum conservation
to neutrino oscillations are inconsistent and lead to no oscillations at all.
Kinematic entanglement is irrelevant to neutrino oscillations.
Energy-momentum conservation manifests itself in ν oscillations as
conservation of the expectation values of ν energy amd momentum in
the course of the oscillations.
Neutrino oscillations are a QM interference phenomenon; owe their
existence to QM uncertainty relations. The coord.-mom. and time-en.
uncertainty relations play a crucial role.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 19
Summary
Attempts of simple-minded application of energy-momentum conservation
to neutrino oscillations are inconsistent and lead to no oscillations at all.
Kinematic entanglement is irrelevant to neutrino oscillations.
Energy-momentum conservation manifests itself in ν oscillations as
conservation of the expectation values of ν energy amd momentum in
the course of the oscillations.
Neutrino oscillations are a QM interference phenomenon; owe their
existence to QM uncertainty relations. The coord.-mom. and time-en.
uncertainty relations play a crucial role.
QM uncertainties are in no conflict with energy-momentum conservation!
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 19
Backup slides
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 20
When are neutrino oscillations observable?
Another source of decoherence: wave packet separation due to the difference
of group velocities ∆v of different mass eigenstates.
If coherence is lost: Flavour transition can still occur, but in a non-oscillatory
way. E.g. for π → µνi decay with a subsequent detection of νi with the
emission of e:
P ∝
X
Pprod (µ νi )Pdet (e νi ) ∝
X
|Uµi |2 |Uei |2
i
i
– the same result as for averaged oscillations.
How are the oscillations destroyed? Suppose by measuring momenta and
energies of particles at neutrino production (or detection) we can determine its
energy E and momentum p with uncertainties σE and σp . From
Ei2 = p2i + m2i :
2
2 1/2
σm2 = (2EσE ) + (2pσp )
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 21
When are neutrino oscillations observable?
If σm2 < ∆m2 = |m2i − m2k | – one can tell which mass eigenstate is emitted.
−1
σm2 < ∆m2 implies 2pσp < ∆m2 , or σp < ∆m2 /2p ≃ losc
.
But: To measure p with the accuracy σp one needs to measure the momenta
of particles at production with (at least) the same accuracy ⇒ uncertainty
of their coordinates (and the coordinate of ν production point) will be
σx, prod & σp−1 > losc
⇒
Oscillations washed out. Similarly for neutrino detection.
Natural necessary condition for coherence (observability of oscillations):
Lsource ≪ losc ,
Ldet ≪ losc
No averaging of oscillations in the source and detector
Satisfied with very large margins in most cases of practical interest
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 22
Wave packet separation
Wave packets representing different mass eigenstate components have
different group velocities vgi ⇒ after time tcoh (coherence time) they
separate ⇒ Neutrinos stop oscillating! (Only averaged effect observable).
Coherence time and length:
∆v · tcoh ≃ σx ;
lcoh ≃ vtcoh
pi
pk
∆m2
∆v =
−
≃
Ei
Ek
2E 2
lcoh ≃
v
σ
∆v x
=
2E 2
∆m2
vσx
The standard formula for Posc is obtained when the decoherence effects
are negligible.
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 23
A manifestation of neutrino coherence
Even non-observation of neutrino oscillations at distances L ≪ losc is a
consequence of and an evidence for coherence of neutrino emission and
detection! Two-flavour example (e.g. for νe emission and detection):
Aprod/det (ν1 ) ∼ cos θ ,
A(νe → νe ) =
X
Aprod/det (ν2 ) ∼ sin θ
⇒
Aprod (νi )Adet (νi ) ∼ cos2 θ + e−i∆φ sin2 θ
i=1,2
Phase difference ∆φ vanishes at short L
⇒
P (νe → νe ) = (cos2 θ + sin2 θ)2 = 1
If ν1 and ν2 were emitted and absorbed incoherently)
to sum probabilities rather than amplitudes:
P (νe → νe ) ∼
X
⇒
one would have
|Aprod (νi )Adet (νi )|2 ∼ cos4 θ + sin4 θ < 1
i=1,2
Evgeny Akhmedov
TAUP-2011
Munich, Sept. 5-9, 2011
– p. 24