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”Think about neutrinos” Peter Minkowski University of Bern, Switzerland 15. December 2004

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Page 1: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

”Think about neutrinos”

Peter MinkowskiUniversity of Bern, Switzerland

15. December 2004

Page 2: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

Topics

1 Base fermions and scalars in SO10

2 ’Mass from mixing’ in vacuo

3 Main neutrino oscillations in vacuo’low’ energy release in production’low’ energy ’gain’ in detection

4 Conclusions and outlook

Page 3: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

1 Base fermions and scalars in SO10The 3 families in the (full) chiral basis form each a 16 spinorrepresentation of SO10

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � " # $ � % & � ' ( � ( ) * � ' + , � - � � . � / ' - 0 . � 1 � � ) 2 3 � & ! . / . � ' & � . 4 * 50 ' ( ) * � ' 4 ) 5 ) * � 6 � 7 � ( 8

Figure 1: Key questions

9

why 3 ? why SO10 ?

Page 4: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

The left chiral notation shall be

� � �� � �� �

� � �

� � spin projection

� �

� family label

� � � � �

� � SO10 label

(1)

Lets call the above extension of the standard model the ’mini mal

nu-extended SM’ .

Page 5: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

The lepton flavors can be filtered out of the 16 representation s

� � � � � � � �

� � � � � �� � � �

� �� � � � � �

� ��

� �� � � � � �

(2)

The three neutrino flavors � are to be identified with the entriesin the figure as � � � � .

Page 6: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

The right-chiral base fields are then associated (by convent ion) to

� � ��

� �

� � � �

�� ��

� � ��

� �

� � � � � �� � �

(3)

The matrix � is the symplectic (� � �

) unit, as implicit in EttoreMajorana’s original paper

� �.

� � �

Ettore Majorana, ”Teoria (a) simmetrica dell’elettrone e del positrone”, Nuovo Cimento 14 (1937)

171-184.

Page 7: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

Yukawa interactions and mass termsThe doublet(s) of scalars are related to the ’tilt to the left ’ .key question

why ’tilt to the left’ ?

�� ��

� �

� � � �

� � � �

� �(4)

The green entries in eq. (4) denote singlets under

� � � .

The quantity z is associated with the quaternionic or octoni onicstructure inherent to the

� �

� �

representation of

� � � � � � � (outside of the electroweak gauge group

� � � � � )� � �

.

Page 8: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

The minimal scalar doublet is associated with the reality re strictionon z, in components

� �

� �

� � � � �

� � �

� � � � �

� � �

��

� � � �

� �� ���(5)

The reality condition, incompatible with susy ,translates using the components � �

� � �

�� �

to

� � �

� � � � �� � � � � �

(6)

� � �

F. Gursey and C.H. Tze , ”On the role of division- , jordan- an d related algebras in particle physics”,

Singapore: World Scientific (1996) 461.

Page 9: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

The Yukawa couplings are of the form (notwithstanding thequaternionic or octonionic structure of scalar doublets)

� � �

� � � � � �

� � � � � � � � � � �

��

� � ��

� � �

�� �

� � �

��

� �� ��

� � � � � � � ��

� � � � � � �� � � � � � �

(7)

The only allowed Yukawa couplings by

� � � � � invarianceare those in eq. (7) , with arbitrary complex couplings

� � � � .

Page 10: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

Spontaneous breaking of

� � � � � through the vacuumexpected value(s)

� � �

� � � �

� � � �

� � � � � � �

�� � � � � �

� � �� � �� �

� � �� ��

� ��

� � � �

��

� � �

� � �

� � ��

GeV

(8)

independent of the space-time point x ,

� � �

,

� � �

,

� � � �

Page 11: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

induces a neutrino mass term through the Yukawa couplings

� � � �

in eq. (7) a . We use the shorthand

aThe implied parallelizable nature of

��

�� �

is by far not trivial and relates in a wider context in-

cluding triplet scalar representations to potential (nona belian) monopoles and dyons.

� � �

Feza Gursey and Chia-Hsiung Tze, ”Octonionic torsion on S( 7) and Englert’s compactification of d =

11 supergravity” , Phys.Lett.B127 (1983) 191.

� � �

Feza Gursey, ”Charge space, exceptional observables and g roups” , In *Coral Gables 1976, Proceed-

ings, New Pathways In High-energy Physics, Vol.I*, New York 1976, 231-248.

� � �

Feza Gursey, Pierre Ramond and Pierre Sikivie, ”A universa l gauge theory model based on E6” ,

Phys.Lett.B60 (1976) 177.

Page 12: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

� � � � � � � � � �� �

� � � � � �

� �� �

� � � � � � � �� � � �

� �� � � � � � � � � � � �� ��

� � � �� �� � �

�� � �� ��

(9)

The matrix � defined in eq. (9) is an arbitrary complex

� � �

matrix, completely analogous to the similarly induced massmatrices of charged fermions, i.e. charged leptons and quar ks.In the setting of primary SO10 breakdown, a general (notsymmetric) Yukawa coupling

� � � � implies the existence in thescalar sector of at least two irreucible representations

� � � � � � �

.

key question

9

a ’drift’ towards unnatural comlexity ? It becomes even wors e including the heavy

neutrino mass terms : 256 (complex) scalars.

Page 13: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

2 ’Mass from mixing’ in vacuo a

� � �

-� � �

or ’Seesaw’ = ’gigampfi’ b

� � �

-

� � � �

The special feature, pertinent to (electrically neutral) n eutrinos is,that the � � extending degrees of freedom aresinglets under the whole SM gauge group

� � � � � ��

� � � � � , in fact remain singlets under

aThe documented discussions, but first for the general vectorlike situation, can be found in

� �

Harald Fritzsch, Murray Gell-Mann and Peter Minkowski, ”Vector - like weak currents and new elementary

fermions” , Phys.Lett.B59 (1975) 256,

� �

Harald Fritzsch and Peter Minkowski, ”Vector - like weak currents, massive neutrinos, and neutrino beam

oscillations” , Phys.Lett.B62 (1976) 72,

and for ’our world, tilted to the left’ in

� � �

Peter Minkowski, ” � 9 � at a rate of one out of 1-billion muon decays ?” , Phys.Lett.B67 (1977) 421.

bCorrect derivations were subsequently documented in

9

Page 14: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

the larger gauge group

� � � � � � .This allows an arbitrary (Majorana-) mass term, involving h oweveronly the bilinears formed from two -s.In the present setup (minimal �-extended SM) the full neutrinomass term is thus of the form

� � �

Murray Gell-Mann, Pierre Ramond and Richard Slansky, ”Complex spinors and unified theories” , pub-

lished in Supergravity, P. van Nieuwenhuizen and D.Z. Freedman (eds.), North Holland Publ. Co., 1979 and in

Stony Brook Wkshp.1979:0315 (QC178:S8:1979),

� � � �

Tsutomu Yanagida, ”Horizontal symmetry and masses of neutrinos” , published in the Proceedings of the

Workshop on the Baryon Number of the Universe and Unified Theories, O. Sawada and A. Sugamoto (eds.),

Tsukuba, Japan, 13-14 Feb. 1979, and in (QCD161:W69:1979) ,

and also in

� � � �

Shelley Glashow, ”Quarks and leptons” , published in Proceedings of the Cargese Lectures, M. Levy (ed.),

Plenum Press, New York, 1980,

� � � �

Rabindra Mohapatra and Goran Senjanovic, ”Neutrino mass and spontaneous parity violation” ,

Phys.Rev.Lett.44 (1980) 912.

Page 15: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

� � �

� � �

� � �� ��

��

� �

� � �

(10)

Again within primary SO10 breakdown the full extends thescalar sector to the representations

� � � � � � � � � � � �

.key questions

quo vadis ? is this a valid explanation of the ’tilt tothe left’ ? no , at least insufficcient !Especially the

entry needs explanation. It is an exclusive propertyof the minimal �-extension assumed here.

Page 16: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

Since the ’active’ flavors � � all carry

� � �

� terms of the form

� � � � � � � � � � �

� �

� � � � � �

(11)

cannot arise as Lagrangean masses, except induced by an

�-triplet of scalars, developing a vacuum expected value

independent from the doublet(s) .from ”The apprentice magician” by Goethe : ’The shadows I inv oked , I am unable to get rid of now !’

’Seesaw’The relative ’size’ of � and shall define the ’mass from mixing’situation and segregates 3 heavy neutrino flavors from the 3 l ightones :

� � � � � �(12)

Page 17: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

Figure 2: key questions 9 which is the scale of

?

� � � � � � �

GeV 9 isthere any evidence for this scale today ? hardly ! 9 and what about susy ?

Page 18: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

Diagonalization of

We shall use the generic expansion parameter

� �� � � � � �

. and determine a unitary� � �

matrix Uwith the property a

� � �

� � � � � �

� � � � � � � �

� � � �� � � � � � � � ��

� � � � � � � �

� � � �

(13)

aFor a recursive treatment to all orders see :

� � � �

Clemens Heusch and Peter Minkowski, ”Lepton flavor violation induced by heavy Majorana neutrinos” ,

Nucl.Phys.B416 (1994) 3, and continuation.

Page 19: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

and

� � � � � � � � � � � � �

� � ��

� � � ��

� � � � � � �

� ��

(14)

The matrix

in eq. (14) describes the mixing of light and heavyflavors, determined from a

� � �submatrix

.

� �

� � � � � � � � � � � � � � � � �

�� � � � � � � � � � � � � � � � � �

(15)

Page 20: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

The matrix

in eq. (15) is reduced to diagonal form through twounitary

� � �

matrices � and � a

� � � � � � � � � � � �� � � � � � � � � � � � � � � �� � � � � ��

� � � � � � � � � � � � �

for� �� � � � � �

(16)

is determined from the quadratic equation

� � �� � � �

� � � � �

(17)

which can be solved recursively, settinga

In eq. (16) � � � defines the three (real) heavy-light mixing angles � �� � � , which without loss of generality

can be chosen in the first quadrant, but which are small for

� � � � � � � � � � �

Page 21: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

�� � � � �� � � �

�� � �� � �

� � �

� � � �� � �

� � � � � �� � � � �� � � � �

� � �

� � �� � � �� � �

(18)

In order to control convergence we introduce the specific nor ms a

� � �

� � � � ��� � � � � � � � � � � �

� �� � � � � �

(19)

aThe sequence defined in eq.(18) is convergent for

� � �

.

Page 22: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

� � � the ’devil’ is in the details

� � in eq.(16) contain all 9 CP violating phases, pertaining to T .The above was intended to ’explain’ why the (un)observed lightneutrino masses are so much smaller than charged fermion one s.key question

does it ? wait .

� � � � � � � � � � � �� � � �

defined in eq. (16) and its determiningequation, repeated below

� � �� � � �

� � � � �

ensure block diagonal form of � ��

� � � �� .

� ��

� � � ��

� � � � � � � �� � ��

� � � ��

(20)

Page 23: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

� �

forming � ��

� � � �� defined in eq. (20) become

� �

� � � � � � � � ��

� � � �� � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �� � � � �

�� � � �

� � � � � � � � � �

(21)

Comparing � with

� � �

we find

Page 24: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

the relation a,

� � �

� � �� � �

(22)

It follows from the assumptions detailed in footnote a , that

� � � � � �

and hence the heavy-light mixing angles � ��

�� �

defined in eq. (16) are strictly bigger than 0.The lowest approximation,

� � � � and and �

, yieldsthe first nontrivial approximation of the light neutrino mas s matrix

aIn the scenario adopted here, we further assume

� � � � � �

and

� � � � � �

. This leaves no room

for light ’sterile’ neutrinos, which would imply a nonminimal � � extension of the standard model. This would be

mandatory, if the results of the LSND collaboration are correct.

� � � �

The LSND Collaboration (G.B. Mills for the collaboration), ”Results on neutrinos from LSND” , published in

*Stanford 1998, Gravity from the Hubble length to the Planck length* 467-475.

Page 25: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

in ’second order mixing’

� �

� �� � � �

� � � �(23)

Remaining dagonalization of � ��

� � � � �

We go back to eq. (14)

� � � � � :

� � diagonalizes the

� � �

blocks � a

� � � � � � �

� � ��

� � � ��� � ��

� � � ��

(24)

a � � is determined modulo diagonal (orthogonal ,

) matrices

� � � as shown in eq. (25) ,

representing the discrete abelian group

� � � �

.

Page 26: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

� � �

� �

� �

� � �

� � � � �� � � � �

� �

� � � � � � � � �� � � � � �� �

��

� � � � � � ��

� �� ���

� � �� � �

(25)

is constructed as a sequence (eq. 18) , convergent for

� �� � � � � � �

, as shown above , and thus unique.As a consequence of eq. (22) –

beeing determined (within

) –

Page 27: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

� and and hence also � � and � � are not independent ofeach other.We shall keep as independent variables � and

�(or equivalently

) .

Generic mixing and mass estimates1. Mixing

Here we introduce the arithmetic mean measure for

� � �

matricesA , not to be confused with the norms defined in eq.(19)

� � � �� � � � � � � �

(26)

Eq. (22) then implies

Page 28: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

�� � �

�� � �

��

��

� � � � ��

� � � � � �� � �

��

� � � ��

� �� � �

(27)

Lets introduce the arithmetic mean of the light and heavy neu trinomasses and the coorresponding ’would be’ masses if � and �

would be the only parts of the full� � �

mass matrix

� �

� � � � � �� � �� �

� �� � �

� � � � � � � � �� � � � � �� � � ��

� �

� � � � � �� � �

(28)

Page 29: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

Then beyond eq. (27) there is one more (exact) relation a

�� �

� � � � � � � � � � � � � � �� � �

� � �

��

�� �

� �

� � � �

�� � �

��

or equivalently

� �

�� �

��� � � �

seesaw ( of type I )

(29)

afor MSSM inspired seesaw of type II realizations see e.g.

� � � �

Page 30: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

2. MassThese estimates are based on the assumption that the scalardoublets (2) are part of a complex 10-representation of SO10 withthe Yukawa couplings of the form

� � � � � � �

� �

� � �

� � ��

� � � � �� ��

� � � � � � � � � �

(30)

It follows that at the unification scale we have a

aIn order to obtain a general (not a symmetric) heavy-light mass matrix � a combination of SO10 represen-

tations

� � � � � � � � �

is needed, which however would (could) ’destroy’ the mass relation in eq. (29) . key

question

9

is this relevant ? estimate shall be estimate .

Page 31: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

� � ��

� � �(31)

We shall use the relation at a scale near 100 GeV

� �

��

� � ��

(32)

The factor

�� accounts for the color rescaling reducing the

(colored) up-quark mass matrix from the unification scaledown to 100 GeV .It follows using the definitions in eq. (28) and the quarkmasses � � � �

MeV , � � �

GeV and � � � �

GeV

� � �

� � � � � � �

� � �

� �

GeV

� � �

� �GeV

(33)

Page 32: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

Further lets approximate the mass square differences, obta inedfrom the combined neutrino oscillation experiments, by

� �� � � �

eV

� � � � ��

� �

eV

(34)

Finally ’pour fixer les id ees’ I set the lowest light neutrino mass

meV and assume hierarchical (123) light masses. This implie s

� � �

meV � �

meV � � � �

meV

� � � �

meV(35)

Page 33: Peter Minkowski University of Bern, Switzerland 15. December …curtright/Miami2004/Minkowski.pdf · 2005. 2. 15. · The reality condition, incompatible with susy , translates using

It follows from eq. (29)

�� � � � � � ��

� � � � � � � � �� � �

� � �

GeV

��

� �

� � �

(36)

Light neutrino masses are indeed small. a

akey questions

9

is susy bringing down in ’small steps’ the B-L protecting mass scale

� � �

��� � � �

TeV to 1 TeV ? , or is

� � � ��� � � �

TeV in view of seesaw type II too small ? , � 9 � at a

rate of ?

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3 Main neutrino oscillations in vacuo

’low’ energy release in production’low’ energy ’gain’ in detection

In short ...3a) Production of neutrinos much below the energy scale of he avyflavors

First lets consider production processes, whereby the heav yneutrino flavor states are ’blocked’ , i.e. inhibited by ener gyconservation.Then we can eliminate the heavy flavors

� � �� �

�� �� � ��� � �� �(37)

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The production amplitude shall be characterized by a definit e time

� � , in the system of ’preference’ , yielding – in amplitude – a st atewe denote by

� ��

� � �� �

(38)

We use the local asymptotic fields associated with the masseigenfields

�� to We use the local asymptotic fields associated withthe mass eigenfields

�� to represent this state by means of the fullMajorana extended asymptotic fields as

� ��

� � �� �

� � � � � � � � � � �� � �

� � � �

� �

� �

� � � ��

� � �

� � � � �

(39)

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In eq. (39) the ’neutrino wave packet’ described by the funct ions

� �� � �

� �

is meant to define the full production process to asufficent approximation in amplitude .

��

� � �

� �

are the nonrelativistically normalized absorption

operators for the mode characterized by the spinor �� � �

� �

.They satisfy the anticommutation relations

� ��

� �

� �

�� � � � �

� � � �

� � � � � � � � � �

� � � �� � ��� � �

� �� � �

� ��

(40)

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The normalization is

� ��

� � �� � ��

� � �� �

� � � � � �� � � �

� � ��

� � �

� � � �� � �

� �(41)

We now return to the ’low’ production process of neutrinos, w hichshall derive from the charged current

� �� � � � �

��(42)

yielding the state

� ��

� � �� �

, described above , through the

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production amplitude

� � �

�� � �

� ��

� � ��� � � �� � � �

� � � � �� � � �

� �� � ��

� � ��

��

��

� �� � � � �� �

� � � � � �

� � � � � � � � � �� �

� � �

� � � � �

� �

� � � �

� �

� �� � �

� � � � � � � � � �� �

� � �

� � � � �

� �

� � � �

� �

� �� � �

no sum over

�(43)

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The neutrino flavor

is determined by tagging the charged(anti)lepton

� � , antilepton if produced together with � � .

The structure of

�� � is for

� � � � � ��� �

�� � �

� �� �

� � �

� � � � �

� �

� � � �

� �

� �� � �

� � � � �� �

� � � �

� �

� � �

� �� �

� � �

� � �

� �� � � �

� �

� �

� � � �

� � � � � � � � �

� �

� �� �

� �

� �

��

� � � � � � � � �

� �

��

� � � � � � � �

� ���

� � � �

� ��

� � � � � � �

(44)

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We will set

� � � � � � from here on , but retain, that� � � is only

exactly unitary if

� � �

, i.e. for vanishing light neutrino masses.

Then

� � � �� � in eq. (39) becomes a

� � � �� �

� � � �� �

� � � � ��� � � � �� � � �� �

� �

(45)

The functions

� � in eq. (45) describe the mixing independent wavepackets induced by the production of � � . The latter is assumed toinherit an intrinsic spread in momenta , with mean momentum

� �

. The latter could well depend on the lepton flavor

,characterizing the production process.The main helicity component – for neutrino production – is

� �

, although of course both helicities are actuallya

It is often assumed ’a priori’ , that

� � � is necessarily unitary , as is indeed the case for the CKM matrix.

This exact unitarity is impossible in the case of ’mass from mixing’ .

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produced from the charged current. The ratio of ’wrong’ to ri ghthelicity production is

������

� � � � � �� � ��

� � � � � �� �� �

������

���

���

� � � � � � � �

no

� � helicity suppression

� � � � � ���

for decay(46)

The ’wrong’ helicity component would dominantly appear asantineutrino in subsequent detection. a

Henceforth we set

� � � � � � � � � �

.

aThe neutrino

antineutrino transition between production and detection would prove overall lepton flavor

(B-L) violation. key questions�

why is this violation so much suppressed, or so well protected ?

is this a

consequence of the ’tilt to the left’ ?

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3b) From production to detection

We take the state

���

� � � �� � � � � , defined in eqs. (38 , 39 and 41)

���

� � � � � � � � � �

� � ��� � � � �� � � � � �

� � � � � � �� �� � �� � � �

no sum on F

(47)

and choose configuration space coordinates

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� � �

�� � � �� �

� �� � �� � � � � � ��

� � � �� �� �

� � � ���� � � � � � �� ��

(48)

� � �� � defined in eq. (48) designate the ’exact’ production time

(

� �) and average production point (�� �) respectively.

Next we propagate the state���

� � � � � � � � � to an arbitrary time

,

which will be identified with the detection time

� � .� � � � �

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���

� � � � � � � � � � ��� � � � � � � � � � �

� � � � �

�� �� � �� � � � � � � � � � � ��

� �

� � � � �� � � � � � � �� � � �

(49)

3c) Detection of neutrinos much below theenergy scale of heavy flavors

We proceed in a way analogous to the production amplitude

� � �

in eq.(43) and consider the detection amplitude

� � � yielding inthe final – detected – state a charged lepton with flavor G from t hecharged current ( contragredient to

� �� in eq.(42) )

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� � � � ��� � � � � � � � � ��

�� � � �� � �

� � � � � � � � � �� � � �� � ����

� � � � � ���

� � � ��

�� � �

(50)

As for production , characterised by the ’neutrino wave pack et inproduction’ defined by the functions

� � � � � �� �

in eq. (39) , thedetection process gives rise to ’neutrino wave packets in de tection’characterized by a sum over states we shall denote as

���

� � �� � � �� ���

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� � �� � � � � � � � � �� � �

� � � � � � � � ���

� � �� �

� � � �� ��� � � � �� ���

���

� � �� � � � ��� � �

� �� � � � � �

� ��� � � �� �� � �� � � �

(51)

In eq. (51) the scattering (in) state

� � � � � � � denotes the incomingstate within the detector, e.g. a neutron for the detecting r eaction

� � � � � � � .

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Each of the states

���

� � �� is determined by an associated wave

function

� �� � � �� ��

, in analogy with

���

� � � �� ���

, (eq.

49) characterising the x-propagated production process .The structure of both ’neutrino wave packets’ , a collection (densitymatrix) for detection and exactly one (wave function) for pr oduction

���

� � �� ���

� � � � � ��� �

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���

� � �� �

� � � ��� � �

� �� � � � � ��� � � �� �� � �� � � �

���

� � � � � ��� � � � ��� � � � �� � � � � �

� � � � �

�� �� � �� � � � � � � � � � � ��

� �

(52)

The neutrino wave packet projection in detection

� �� � �

���

� � �� � � �� ���

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is necessary in principle, to guarantee the geometric accep tance ofthe detector. a

Nevertheless we will assume in the following that the spread inposition and/or momentum of the production process is preci seenough, so that

� �� � � can be replaced by a complete sum over one

particle states

� �� � � � � � �

�� � �

� �� � � � � �

� � �� � � � � �� � � � � � � � ��

�� � �

(53)

An eventually necessary geometrical acceptance correctio n can beapplied ’classically’ , as is done for beam transport over sh ort beamlines conventionally.

aThe essential feature of quantum mechanics, that not both position and momentum can be determined

together has led to a ’propagation of errors’ , which shall not be documented here.

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3d) Summary

1) The remaining parts of the amplitudes

� � � for production

( eq. 43 ) and

� � � for detection ( eqs. 50-51 ) can be approximatedneglecting light neutrino masses.

2) Coherent and decoherent properties of ’low-low’ neutrin ooscillations

� � �

in vacuo and for the main helicitycomponents, are contained , using the approximation defined in eq.(52) – in amplitude – in the scalar products

� � ��

�� � � �� � � �

���

� � � � � ���

� � � � �� � � � � �

�� �� � � � ��

�� � � � � �� � � �

(54)

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We can proceed from eq. (54)in various steps of approximations, and here choose the foll owingpath :

3) Neglect of direct retardation

In short for

� �� � � �

� � � �

� � � � � � � � �

in general(55)

where L is the mean distance between detection and productio n .

4)

� � �

transition probabilities

In short – to lowest order in � �� – and for a Gaussian wavefunction we obtain

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� �� � �

� � � � � � � � � � ��

�� �� � �� � �

� � � � � � � � � � � � �� � � � � � � � � �

� �� � � � � � � � � � �� � � � � � �

(56)

The fluctuation distribution

�� � � � � � �becomes

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�� � � � � � �

��

�� � � � � � � � � � � �

� �� � � � ��

� � � � �

� �

� � � � ��

� � � � � � � � �

� � � � � � � � �

� � �

��

� � � � � � � �

� � �� �� � � �

� � � �� �

� �� �

(57)

Letting

� � � � �

assume large value we find asymptotically

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� �� � � � � � � � �

� � � � ��

� � � � � � � � �

� � � �

� � � � for

� � � � �

� � � � � �� � � �� � �

(58)

The

� � � � � � � �

behaviour shows the normal decay of a wave

packet, whereas decoherence sets in – for

�� � � �� � �

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for

� �

Ł � � � � � � �� � � �� �

Ł � � � � � � � � � � � � � � � � � � � Ł � � � � � � � � � � � � � � �

� � � � � � � ��

� �

km

�� � �

GeV�

� � � � � � � �

eV

� �

� � � � � � � � � � � � � � �

� � � �

� � � � � � �

� � � � � � � � a

(59)

aFor

� � � � � � � eV

�and � � �

MeV.

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4 Conclusions and outlook1) Neutrino properties are only to a very small extent open (u p to the

present) to deductions from oscillation measurements.

2) Notwithstanding this, a significant and admirable experi mentaleffort paired with theoretical analysis has revealed the ma in twooscillation modes. The matter effect due to Mikheev, Smirno v andWolfenstein demonstrates another clear form of quantumcoherence, over length scales of the solar radius.

3) Key questions remain to be resoloved : are all (ungauged orgauged) global charge-like quantum numbers violated ? (B-L ) , B ,L, individual lepton flavors.

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4) SO10 served fine (together with susy or without it) to guide ideas,but a genuine unification is as remote as the scales and nature ofheavy neutrino flavors, to name only these.

Yet, the quest for unification is wide open.