Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://nuclphys.sinp.msu.ru/conf/epp10/Akhmedov.pdf Äàòà èçìåíåíèÿ: Sun Sep 8 11:44:20 2013 Äàòà èíäåêñèðîâàíèÿ: Fri Feb 28 02:37:13 2014 Êîäèðîâêà:

Neutrino oscillations and oscillation coherence
Evgeny Akhmedov Max-Planck Institute fur Kernphysik, Heidelberg ¨

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 1


Pontecorvo's centennial
Idea of neutrino oscillations: First put forward by Pontecor vo in 1957

Bruno Pontecorvo 1913 - 1993
Evgeny Akhmedov XVI Lomonosov Conference Workshop Moscow, August 22-28, 2013 ­ p. 2


Oscillation probability in vacuum
Master formula:



P ( ; L) =

i

U i e

-i

m21 i 2p

L

2 Ui

What are the applicability conditions for this formula? When are the oscillations observable? ­ One answer to both questions!

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 3


Main assumption: Neutrinos produced and detected in weak interaction processes are flavour eigenstates ­ coherent linear superpositions of mass eigenstates:
fl |a = i m Uai |i a ss

( = e , µ , ,

i = 1 , 2 , 3)

Overall production ­ propagation ­ detection process: amplitudes with different m i ass in the intermediate state contribute coherently (cannot be distinguished)
Evgeny Akhmedov XVI Lomonosov Conference Workshop Moscow, August 22-28, 2013 ­ p. 4


When are neutrino oscillations observable?
Keyword: Coherence Neutrino flavour eigenstates e , µ and are coherent superpositions of mass eigenstates 1 , 2 and 3 oscillations are only observable if 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 measurements one can tell (through E = p2 + m2 ) which mass eigenstate is emitted, the coherence is lost and oscillations disappear! Production and detection coherence localization cond.: lpr
od

losc ,

ldet lo L < lco

sc

Usually satisfied with large margins. Propagation coherence:
Evgeny Akhmedov

h

vg vg

x =

2E 2 m2

vg x
Moscow, August 22-28, 2013 ­ p. 5

XVI Lomonosov Conference Workshop


Oscillations and QM uncertainty relations
Neutrino oscillations ­ a QM interference phenomenon, owe their existence to QM uncer tainty relations Neutrino energy and momentum are characterized by uncer tainties E and p related to the spatial localization and time scale of the production and detection processes. These uncer tainties allow the emitted/absorbed neutrino state to be a coherent superposition of different mass eigenstates (Kayser, 1981) determine the size of the neutrino wave packets govern decoherence due to wave packet separation (Nussinov, 1976) E ­ the effective energy uncer tainty, dominated by the smaller one between the energy uncer tainties at production and detection. Similarly for p .

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 6


When are neutrino oscillations observable?
Production and detection coherence: E E , p p

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 7


When are neutrino oscillations observable?
Production and detection coherence: E E , p p

If coherence is lost: Flavour transition can still occur, but in a non-oscillatory way. E.g. for µi decay with a subsequent detection of i with the emission of e: P
i

P

pro d

(µ i )P

det

(e i )
i

|Uµi |2 |Uei |

2

­ the same result as for averaged oscillations.

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 7


When are neutrino oscillations observable?
Production and detection coherence: E E , p p

If coherence is lost: Flavour transition can still occur, but in a non-oscillatory way. E.g. for µi decay with a subsequent detection of i with the emission of e: P
i

P

pro d

(µ i )P

det

(e i )
i

|Uµi |2 |Uei |

2

­ the same result as for averaged oscillations. How can the oscillations destroyed by precise measurements of neutrino energy and momentum? Suppose we can find the neutrino energy E and 2 momentum p with uncer tainties E and p . From Ei = p2 + m2 : i i m2 =
Evgeny Akhmedov

(2E E )2 + (2pp )

2 1/2

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 7


When are neutrino oscillations observable?
2 If m2 < m2 = |m2 - mk | ­ one can tell which mass eigenstate is emitted. i - m2 < m2 implies 2pp < m2 , or p < m2 /2p los1 . c

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 8


When are neutrino oscillations observable?
2 If m2 < m2 = |m2 - mk | ­ one can tell which mass eigenstate is emitted. i - m2 < m2 implies 2pp < m2 , or p < m2 /2p los1 . c

But: To measure p with the accuracy p one needs to measure the momenta of par ticles at production with (at least) the same accuracy uncer tainty of their coordinates (and the coordinate of production point) will be x,
pro d - p 1

> lo

sc

Oscillations washed out. Similarly for neutrino detection.

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 8


When are neutrino oscillations observable?
2 If m2 < m2 = |m2 - mk | ­ one can tell which mass eigenstate is emitted. i - m2 < m2 implies 2pp < m2 , or p < m2 /2p los1 . c

But: To measure p with the accuracy p one needs to measure the momenta of par ticles at production with (at least) the same accuracy uncer tainty of their coordinates (and the coordinate of production point) will be x,
pro d - p 1

> lo

sc

Oscillations washed out. Similarly for neutrino detection.

Natural necessary condition for coherence (observability of oscillations): Ls
ource

losc ,

Ldet lo

sc

No averaging of oscillations in the source and detector. For usual neutrinos: Satisfied with very large margins in most cases of practical interest
Evgeny Akhmedov XVI Lomonosov Conference Workshop Moscow, August 22-28, 2013 ­ p. 8


Wave packet separation
Wave packets representing different mass eigenstate components i have different group velocities vgi after time tcoh (coherence time) they separate Neutrinos stop oscillating! (Only averaged effect observable). Coherence time and length: vg · tco
h

x ;

lco

h

vg tco

h

vg

pi pk m2 = - Ei Ek 2E 2
vg vg

lcoh
The standard formula for P are negligible.
Evgeny Akhmedov

x =

2E 2 m2

vg x

osc

is obtained when the decoherence effects

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 9


Neutrino oscillations: Coherence at macroscopic distances ­
L

> 10,000 km in atmospheric neutrino experiments !

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 10


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 e emission and detection): Apr
o d/det

(1 ) U

e1

= cos ,

Apr

o d/det

(2 ) U

e2

= si n
i



A(e e ) =
i = 1 ,2

Aprod (i )Adet (i ) = cos2 + e-

si n 2

Phase difference vanishes at shor t L

P (e e ) = (cos2 + sin2 )2 = 1 If 1 and 2 were emitted and absorbed incoherently) to sum probabilities rather than amplitudes: P (e e )
i = 1 ,2
Evgeny Akhmedov XVI Lomonosov Conference Workshop Moscow, August 22-28, 2013 ­ p. 11



one would have

|Aprod (i )Adet (i )|2 cos4 + sin4 < 1


A universal oscillation probability?
Pf (k ) Df (k )

Pi (q )

Di (q )

Q.: When are the oscillations described by a universal (production and detection independent) oscillation probability P (E , L)?

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 12


A universal oscillation probability?
Pf (k ) Df (k )

Pi (q )

Di (q )

Q.: When are the oscillations described by a universal (production and detection independent) oscillation probability P (E , L)? A.: When neutrinos are relativistic or quasi-degenerate in mass and the conditions of coherent neutrino emission and detection are satisfied: E E , p p

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 12


A universal oscillation probability?
Pf (k ) Df (k )

Pi (q )

Di (q )

Q.: When are the oscillations described by a universal (production and detection independent) oscillation probability P (E , L)? A.: When neutrinos are relativistic or quasi-degenerate in mass and the conditions of coherent neutrino emission and detection are satisfied: E E , p p

Under these conditions the rate of the overall neutrino production-propagation-detection process can be factorized into the production rate dprod (E )/dE , propagation (oscillation) probability P and detection cross section (E ) P (E , L) can be extracted
(EA & J. Kopp, arXiv:1001.4815)
Evgeny Akhmedov XVI Lomonosov Conference Workshop



(E , L)

Moscow, August 22-28, 2013

­ p. 12


Are coherence constraints compatible?
Observability conditions for oscillations: Coherence of production and detection Coherence of propagation Both conditions put upper limits on neutrino mass squared differences m2 : (1) Ej m2k j E ; 2E (2) m2k j L x vg /E 2 2E

k

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 13


Are coherence constraints compatible?
Observability conditions for oscillations: Coherence of production and detection Coherence of propagation Both conditions put upper limits on neutrino mass squared differences m2 : (1) Ej m2k j E ; 2E (2) m2k j L x vg /E 2 2E

k

But: The constraints on E work in opposite directions: (1) Ej
k

m2k 2E 2 vg j E 2E m2k L j

(2)

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 13


Are coherence constraints compatible?
Observability conditions for oscillations: Coherence of production and detection Coherence of propagation Both conditions put upper limits on neutrino mass squared differences m2 : (1) Ej m2k j E ; 2E (2) m2k j L x vg /E 2 2E

k

But: The constraints on E work in opposite directions: (1) Ej
k

m2k 2E 2 vg j E 2E m2k L j

(2)

Are they compatible? ­ Yes, if LHS RHS 2
Evgeny Akhmedov

L lo
sc



vg ( 1) vg

­ fulfilled in all cases of practical interest
Moscow, August 22-28, 2013 ­ p. 13

XVI Lomonosov Conference Workshop


Are coherence conditions satisfied?
The coherence propagation condition: satisfied very well for all but astrophysical and cosmological neutrinos (solar, SN, relic 's ...)

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 14


Are coherence conditions satisfied?
The coherence propagation condition: satisfied very well for all but astrophysical and cosmological neutrinos (solar, SN, relic 's ...) Coherent production/detection: usually satisfied extremely well due to the tininess of neutrino mass

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 14


Are coherence conditions satisfied?
The coherence propagation condition: satisfied very well for all but astrophysical and cosmological neutrinos (solar, SN, relic 's ...) Coherent production/detection: usually satisfied extremely well due to the tininess of neutrino mass But: Is not automatically guaranteed in the case of "light" sterile neutrinos! msterile eV - keV - MeV scale heavy compared to the "usual" (active) neutrinos

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 14


Are coherence conditions satisfied?
The coherence propagation condition: satisfied very well for all but astrophysical and cosmological neutrinos (solar, SN, relic 's ...) Coherent production/detection: usually satisfied extremely well due to the tininess of neutrino mass But: Is not automatically guaranteed in the case of "light" sterile neutrinos! msterile eV - keV - MeV scale heavy compared to the "usual" (active) neutrinos Sterile neutrinos: hints from SBL accelerator experiments (LSND, MiniBooNE), reactor neutrino anomaly, keV sterile neutrinos, pulsar kicks, leptogenesis via oscillations, SN r -process nucleosynthesis, unconventional contributions to 2 0 decay ...

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 14


Are coherence conditions satisfied?
The coherence propagation condition: satisfied very well for all but astrophysical and cosmological neutrinos (solar, SN, relic 's ...) Coherent production/detection: usually satisfied extremely well due to the tininess of neutrino mass But: Is not automatically guaranteed in the case of "light" sterile neutrinos! msterile eV - keV - MeV scale heavy compared to the "usual" (active) neutrinos Sterile neutrinos: hints from SBL accelerator experiments (LSND, MiniBooNE), reactor neutrino anomaly, keV sterile neutrinos, pulsar kicks, leptogenesis via oscillations, SN r -process nucleosynthesis, unconventional contributions to 2 0 decay ... Production/detection coherence has to be re-checked ­ impor tant implications for some neutrino experiments!
Evgeny Akhmedov XVI Lomonosov Conference Workshop Moscow, August 22-28, 2013 ­ p. 14


Theory and phenomenology of oscillations

QM wave packet approach

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 15


QM wave packet approach
The evolved produced state:
fl | (x, t) = i

U

i

m |i ass (x, t) = i

U

i

m S (x, t)|i a i

ss

The coordinate-space wave function of the ith mass eigenstate (w. packet): (x, t) =
S i

d3 p S fi (p) ei (2 )3

px-iEi (p)t

Momentum distribution function fiS (p): sharp maximum at p = P (width of the peak pP P ). Ei (p) Ei (p) = Ei (P ) + p 1 2 Ei (p) (p - P ) + 2 p2 (p - P )2 + . . .
p
0

P

p Ei (p) = , vi = p Ei
Evgeny Akhmedov XVI Lomonosov Conference Workshop

2 Ei (p) m2 i = 2 p2 Ei
Moscow, August 22-28, 2013 ­ p. 16


Evolved neutrino state
S (x, t) e- i
S gi iEi (P )t+iP x S gi (x - vi t) p1 (x-vg t)

( 0)

(x - vi t)

d 3 p1 (2 )3

fiS (p1 ) ei

Center of the wave packet: x - vi t = 0. Spatial length: xP 1/p S (gi decreases quickly for |x - vi t| xP ). Detected state (centered at x = L):
fl | (x) = k

P

U

k

m D (x)|i a k

ss

The coordinate-space wave function of the ith mass eigenstate (w. packet): D (x ) = k d3 p D fk (p) ei (2 )3
p(x-L)

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 17


Oscillation probability
Transition amplitude:
flfl A (T , L) = | (T , L) = i Ui U i Ai (T , L)

Ai (T , L) =

d3 p S fi (p) fiD (p) e- (2 )3

iEi (p)T +ipL

Strongly suppressed unless |L - vi T | (L - vi T ) Ai (T , L) exp - 2 4x Oscillation probability: P ( ; T , L) = |A |2 =
i ,k

x . E.g., for Gaussian wave packets:
2

,

2 2 2 x xP + xD

Ui U i U

k

U

k

Ai (T , L)A (T , L) k

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 18


Oscillation probability in WP approach
Neutrino emission and detection times are not measured (or not accurately measured) in most experiments integration over T : P ( ; L) = dT P ( ; T , L) =
i ,k Ui U i U k
m2k i ¯ 2P

U

-i k e

L

~ Ii

k

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 19


Oscillation probability in WP approach
Neutrino emission and detection times are not measured (or not accurately measured) in most experiments integration over T : P ( ; L) = dT P ( ; T , L) =
i ,k Ui U i U k
m2k i ¯ 2P

U

-i k e

L

~ Ii

k

~ Iik = N

dq S fi (rk q - Eik /2v + Pi )fiD (rk q - Eik /2v + Pi ) 2
S D âfk (ri q + Eik /2v + Pk )fk (ri q + Eik /2v + Pk ) ei vi + v 2 v
v v

qL

Here:

v

k

, v vk - vi ,

r

i ,k



i,k

v

,

N 1/[2Ei (P )2Ek (P )v ]

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 19


Oscillation probability in WP approach
Neutrino emission and detection times are not measured (or not accurately measured) in most experiments integration over T : P ( ; L) = dT P ( ; T , L) =
i ,k Ui U i U k
m2k i ¯ 2P

U

-i k e

L

~ Ii

k

~ Iik = N

dq S fi (rk q - Eik /2v + Pi )fiD (rk q - Eik /2v + Pi ) 2
S D âfk (ri q + Eik /2v + Pk )fk (ri q + Eik /2v + Pk ) ei vi + v 2 v
v v

qL

Here:

v

k

, v vk - vi ,

r

i ,k



i,k

v

,

N 1/[2Ei (P )2Ek (P )v ]

~ For (v /v )p L 1 (i.e. L lcoh = (v /v )x ) Iik is approximately ~ independent of L; in the opposite case Iik is strongly suppressed

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 19


Oscillation probability in WP approach
Neutrino emission and detection times are not measured (or not accurately measured) in most experiments integration over T : P ( ; L) = dT P ( ; T , L) =
i ,k Ui U i U k
m2k i ¯ 2P

U

-i k e

L

~ Ii

k

~ Iik = N

dq S fi (rk q - Eik /2v + Pi )fiD (rk q - Eik /2v + Pi ) 2
S D âfk (ri q + Eik /2v + Pk )fk (ri q + Eik /2v + Pk ) ei vi + v 2 v
v v

qL

Here:

v

k

, v vk - vi ,

r

i ,k



i,k

v

,

N 1/[2Ei (P )2Ek (P )v ]

~ For (v /v )p L 1 (i.e. L lcoh = (v /v )x ) Iik is approximately ~ independent of L; in the opposite case Iik is strongly suppressed ~ Iik is also strongly suppressed unless Eik /v p , i.e. Eik E ­ coherent production/detection condition
Evgeny Akhmedov XVI Lomonosov Conference Workshop Moscow, August 22-28, 2013 ­ p. 19


The standard osc. probability?
~ The standard formula for the oscillation probability corresponds to Iik = 1.

~ If the two above conditions are satisfied, Iik is not suppressed and is L-, E and i, k -independent (i.e. a constant).

The standard probability is obtained when this constant is 1 (normalization necessary!) Normaliz. condition: d3 p S |fi (p)|2 |fiD (p)|2 = 1 (2 )3

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 20


How can neutrino wave packet be derived?
Pf (k ) Df (k )

Pi (q )

Di (q )

Amplitude of the production process (neutrino j with a momentum p ): j P (p) = d
4 ipx x1 e 1

[dq ] [dk ] fP i (q , Q)

fP f

(k , K ) e

-i(q -k)x 1

Mj P (q , k )

Detection amplitude: j D (p) = d
4 -ipx 2 x2 e

[dq ] [dk ] fDi (q , Q )









fDf

(k , K ) e





-i(q -k )x2

Mj D (q , k )

But: The probability amplitude that j is produced with momentum p is its momentum-space w.p.!
S fj (p) = j P (p) ,
Evgeny Akhmedov

D fj (p) = D (p) j
Moscow, August 22-28, 2013 ­ p. 21

XVI Lomonosov Conference Workshop


Coherence of production in different points
Neutrino production in extended sources: Amplitudes of neutrino emission in different points must be summed ­ a consistent QM procedure. The standard approach: calculate the probability that neutrino produced at a fixed point x oscillates, and then integrate over all x in the source (probability summation procedure ­ classical in nature). Both procedures give identical answers under realistic conditions! The two approaches lead to different results whenever the localization proper ties of the parent par ticles at neutrino production and of the detection process are such that they prevent the precise localization of the point of neutrino emission ­ difficult to realize in practice.

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 22


Graphical interpretation

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 23


Finite-width pion WP

Additional phase for the segment AB : = -[Ej (Pj ) - Ek (Pk )]t + (Pj - Pk )x . t and x: projections of AB on the t and x axes. vg x , x = x . t = vg - v vg - v m2k vg j - x · vg - v 2P
Evgeny Akhmedov XVI Lomonosov Conference Workshop Moscow, August 22-28, 2013 ­ p. 24


Finite-width pion WP ­ contd.
Are deviations between the results of the coherent amplitude summation and incoherent probability summation approaches experimentally observable? Requires extremely high energies of the parent pion: m2 2 (E x ) m2 1.

E.g. for x 10-4 cm and m2 1 eV2 would be 1 for pion energies E 103 TeV ­ not feasible, Another possibility: increase significantly the spatial width of w. packets of ancestor protons, which would increase the values of x . But: not clear how this could be achieved. Other possibilities...

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 25


Production coherence for some experiments
Unless otherwise specified, m2 = 2 eV2 . For -beams E0 = 2 MeV, 0 = 1s, = 100. Experiment LSND KARMEN MiniBo oNE NOMAD (20 eV2 ) CCFR(102 eV2 ) CDHS (20 eV2 ) K2 K T2K Minos NO A -beams E (MeV) 40 40 800 2.7 · 103 5 · 1 04 3000 1500 600 3300 2000 400 L(m) 30 17.7 541 770 891 130 300 280 1040 1040 1.3 · 10 lp (m) ldec (m) losc (m) 0 0 50 0 0 50 50 89 992 290 3009 33480 3348 352 5570 1240 52 334 3720 372 200 167 1861 96 66.4 744 675 368 4092 675 223 2480 2500 3 · 1010 496 lp /vP 3.8 2.24 3.43 0.56 0.145 0.1 1.45 0.1 4.51 0.06 0.22 0.155 2.2 0.155 1.01 1.2 2.36 1.45 1.6 1.84 2.64 3.03 1647 8.3 · 10- p 0 0 0.32 0.054 0.54 1.78 0.088 0.878 0.68 0.81 1.04 1.71 31.7 3 0 0 0.56 0.56 5.64 28.2 0.56 5.64 0.56 0.56 0.56 0.56 .8 · 10

5

8

8

Noticeable effects for MiniBooNE, NOMAD (20 eV2 ), CCFR (100 eV2), CDHS (20 eV2), K2K, T2K, MINOS, NO A, very large effects for -beams

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 26


Conclusions

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 27


Conclusions
The standard formula for osc. probability is stubbornly robust.

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 27


Conclusions
The standard formula for osc. probability is stubbornly robust. Validity conditions:

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 27


Conclusions
The standard formula for osc. probability is stubbornly robust. Validity conditions: Neutrinos are ultra-relativistic or quasi-degenerate in mass

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 27


Conclusions
The standard formula for osc. probability is stubbornly robust. Validity conditions: Neutrinos are ultra-relativistic or quasi-degenerate in mass Coherence conditions for neutrino production, propagation and detection are satisfied.

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 27


Conclusions
The standard formula for osc. probability is stubbornly robust. Validity conditions: Neutrinos are ultra-relativistic or quasi-degenerate in mass Coherence conditions for neutrino production, propagation and detection are satisfied. Gives also the correct result in the case of strong coherence violation (complete averaging regime).

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 27


Conclusions
The standard formula for osc. probability is stubbornly robust. Validity conditions: Neutrinos are ultra-relativistic or quasi-degenerate in mass Coherence conditions for neutrino production, propagation and detection are satisfied. Gives also the correct result in the case of strong coherence violation (complete averaging regime). Gives only order of magnitude estimate when decoherence parameters are of order one.

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 27


Conclusions
The standard formula for osc. probability is stubbornly robust. Validity conditions: Neutrinos are ultra-relativistic or quasi-degenerate in mass Coherence conditions for neutrino production, propagation and detection are satisfied. Gives also the correct result in the case of strong coherence violation (complete averaging regime). Gives only order of magnitude estimate when decoherence parameters are of order one. But: Conditions for par tial decoherence are difficult to realize

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 27


Conclusions
The standard formula for osc. probability is stubbornly robust. Validity conditions: Neutrinos are ultra-relativistic or quasi-degenerate in mass Coherence conditions for neutrino production, propagation and detection are satisfied. Gives also the correct result in the case of strong coherence violation (complete averaging regime). Gives only order of magnitude estimate when decoherence parameters are of order one. But: Conditions for par tial decoherence are difficult to realize They may still be realized if relatively heavy sterile neutrinos exist
Evgeny Akhmedov XVI Lomonosov Conference Workshop Moscow, August 22-28, 2013 ­ p. 27


Backup slides

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 28


Coherence production conditions
Coherence production conditions: |E | E , On t h e o t h e r h a n d : Constraint |E | E () |p| p .

m2 . E vg p + 2E m2 vg p + 1. E 2E E

(a) The two terms in E do not approximately cancel each other. vg |p| E p , i.e. for relativistic neutrinos |p| p follows from |E | E .

(b1) There is a strong cancellation, but both terms on the l.h.s. of (*) are small ­ see case (a). (b2) Strong cancellation, but both terms on the l.h.s. of (*) are 1: momentum condition is independent. But: the only known case ­ MÆssbauer neutrinos.
Evgeny Akhmedov XVI Lomonosov Conference Workshop Moscow, August 22-28, 2013 ­ p. 29


The paradox of

E

a nd

p

QM uncer tainty relations: p is related to the spatial localization of the production (detection) process, while E to its time scale independent quantities. On the other hand: Neutrinos propagating macroscopic distances are on the mass shell. For on-shell mass eigenstates E 2 = p2 + m2 means i E E = pp How can this be understood? The solution: At production, neutrinos are not on the mass shell. They go on shell only after they propagate x (a few)â De Broglie wavelengths. After that their energy and momentum get related by E 2 = p2 + m2 the i larger uncer tainty shrinks towards the smaller one to satisfy E E = pp . On-shell relation between E and p allows to determine the less cer tain of the two through the more cer tain one, reducing the error of the former.
Evgeny Akhmedov XVI Lomonosov Conference Workshop Moscow, August 22-28, 2013 ­ p. 30


What determines the length of w. packets?
The length of w. packets: x 1/p . For propagating on-shell neutrinos:
p p p p p min{p rod , (E /p)Erod } = min{p rod , (1/vg )Erod } p Which uncer tainty is smaller at production, p r od p or Er od

?

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 31


What determines the length of w. packets?
The length of w. packets: x 1/p . For propagating on-shell neutrinos:
p p p p p min{p rod , (E /p)Erod } = min{p rod , (1/vg )Erod } p Which uncer tainty is smaller at production, p r od p or Er od

?

Consider neutrino production in decays of an unstable par ticle localized in a box of size LS . Time between two collisions with the walls of the box: TS .

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 31


What determines the length of w. packets?
The length of w. packets: x 1/p . For propagating on-shell neutrinos:
p p p p p min{p rod , (E /p)Erod } = min{p rod , (1/vg )Erod } p Which uncer tainty is smaller at production, p r od p or Er od

?

Consider neutrino production in decays of an unstable par ticle localized in a box of size LS . Time between two collisions with the walls of the box: TS . If TS < ( ­ lifetime of the parent unstable par ticle) - E TS 1 (collisional broadening). Mom. uncer tainty: p L-1 . S Bu t: LS = vS T
S



E < p

(a consequence of vS < 1)

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 31


What determines the length of w. packets?
The length of w. packets: x 1/p . For propagating on-shell neutrinos:
p p p p p min{p rod , (E /p)Erod } = min{p rod , (1/vg )Erod } p Which uncer tainty is smaller at production, p r od p or Er od

?

Consider neutrino production in decays of an unstable par ticle localized in a box of size LS . Time between two collisions with the walls of the box: TS . If TS < ( ­ lifetime of the parent unstable par ticle) - E TS 1 (collisional broadening). Mom. uncer tainty: p L-1 . S Bu t: If T
S

LS = vS T

S



E < p

(a consequence of vS < 1) E
-1

> (quasi-free parent par ticle)
-1

= .

p [(p/E ) ]

[(p/E )E ]

-1

, i.e. E (p/E )p < p .

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 31


The length of w. packets ­ contd.
p Er od p < p r od

In both cases

also when s are produced in collisions.

=

p

eff



E , vg

x

vg E

In the stationary limit (E 0) one has p eff 0 even though p is finite! Therefore x and so the coherence length lcoh ­ a well known result.

Evgeny Akhmedov

XVI Lomonosov Conference Workshop

Moscow, August 22-28, 2013

­ p. 32