1

A review of neutrino propertiesa historical overview and its projection

Peter Minkowski AEC-ITP, University of Bern, Switzerland a ndTH-Div , CERN

AbstractIn a first part neutrino properties are presented from the beg innings through the letter of Wolfgang Pauli

om 4. Deember 1930 suggesting a new neutral fermion to the ”Ra opactive Ladies and Gentlemen” at a

Conference in Tubingen to the first full oscillation cycle, based on my review at the meeting

”Neutrino Telescope in Venice” in 2005 .

In the remaining time I shall present a selection of neutrino properties featuring the structure of mass

and mixing of the light and heavy neutrino flavors, a symmetri c , complex 6 by 6 matrix within the

unifying SO10 gauge group, and present prospects for the det ection of the leptonflavor violating

processes B s → µ e ; B → K µ e, also such involving b-flavored baryons .

Kolymbari, 26. August 2017

– p. 1

2

The first part of this outline is based on my contribution to th eConference ”Telescopes in Venice” -2005- in ref. [ -1] .

1. The beginnings of neutrinos - Wolfgang Pauli’s idea 1930

The spin 1/2 fermion was called’neutron’. The neutron of today wasdiscovered by James Chadwickin 1932 [ -2] .→ Nobel prize for physics 1935.

The name neutrino was coinedlater by Oscar D’Agostino, Emilio Segr e,Edoardo Amaldi, Franco Rasetti,Ettore Majorana and Enrico Fermi→ ”I ragazzi di via Panisperna”.

– p. 2

3

– p. 3

4

Topics

1 Base fermions and scalars in SO10neutrinos are unlike charged fermions - Ettore Majorana

2 Neutrino ’mass from mixing’ in vacuo and matterneutrinos oscillate like neutral Kaons (yes, but how ?) - Bru no Pontecorvo

– p. 4

5

– p. 5

6

Key questions → why 3 ? why SO10 ?

I shall cite two sentences from ref. [1] :

”Per quanto riguarda gli elettroni e i positroni, da essa (vi a) si pu overamente attendere soltanto un progresso formale ...Vedremo infatti che e perfettamente possibile costruire, nellamaniera piu naturale,

una teoria delle particelle neutre elementarisenza stati negativi.”

( ”As far as electrons and positrons are concerned from this ( path) one may expect only a formal

progress ... We will see in fact that it is perfectly possible to construct, in the most general manner,

a theory of neutral elementary particles

without negative states.” a )

a... upon normal ordering .

– p. 6

7

But the real content of the paper by E. M. (1937) is in the formu lae,exhibiting the ’oscillator decomposition’ of spin 1/2 ferm ions

as seen and counted by gravity, 1 by 1 and doubled through the ’ external’ SO2 symmetry associated with

electric charge a

The left chiral notation shall be

( f k )γF ; γ = 1, 2 : spin projection

F = I, II, III : family label

k = 1, · · · , 16 : SO10 label

(1)

a[ 2 ] P. A. M. Dirac, Proceedings of the Cambridge Philosophical S ociety, 30 (1924) 150. Paul Dirac

shall be excused for starting the count at 2 for ’elettrone e p ositrone’ .

– p. 7

8

Lets call the above extension of the standard model the ’mini malnu-extended SM’ . a

• • • ν | N • • •

• • • ℓ | ℓ • • •

γ

F = e , µ , τ

↓

ν N

ℓ ℓ

γ

F = e , µ , τ

(2)

a [ 3 ] Harald Fritzsch and Peter Minkowski, ”Unified interactions of leptons and hadrons”, Annals

Phys.93 (1975) 193 and Howard Georgi, ”The state of the art - gauge theories”, AIP Co nf.Proc.23 (1975)

575.

– p. 8

9

The right-chiral base fields are then associated to 1 for 1

(f ∗k

)F α

= ε αγ

[( f k )

γF

] ∗

( ε = i σ 2 ) αγ =

0 1

−1 0

(3)

The matrix ε is the symplectic ( Sp (1)) unit, as implicit in EttoreMajorana’s original paper [1] .The local gauge theory is based on the gauge (sub-) group

SL ( 2 , C ) × SU3 c × SU2 L × U1 Y(4)

... why ? why ’tilt to the left’ ? we sidestep a historical overview here !

– p. 9

10

1 a) Yukawa interactions and mass terms

The doublet(s) of scalars are related to the ’tilt to the left ’ .

ν N

ℓ ℓ

F↔

ϕ 0 Φ +

ϕ − Φ 0

= z(5)

The green entries in eq. (4) denote singlets under SU2 L .

The quantity z is associated with the quaternionic or octoni onicstructure inherent to the ( 2 , 2 ) representation ofSU2 L ⊗ SU2 R (beyond the electroweak gauge group ) [ 4 ] a .

ae.g. [ 4 ] F. Gursey and C.H. Tze , ”On the role of division- , jordan- an d related algebras in particle

physics”, Singapore, World Scientific (1996) 461.

– p. 10

11

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

H Y =[( ϕ 0 ) ∗ , ( ϕ − ) ∗

]λ F ′ F ×

×

εγδ N δ

F ′

ν γ

ℓ γ

F

+ h.c.

N γ F ′ = εγδ N δF ′ ; εγδ = εγδ = εγδ

(6)

The only allowed Yukawa couplings by SU2 L ⊗ U1 Y invarianceare those in eq. (6) , with arbitrary complex couplings λ F ′ F .

– p. 11

12

Spontaneous breaking of SU2 L ⊗ U1 Y through the vacuumexpected value(s)

〈 Ω |

ϕ 0 Φ +

ϕ − Φ 0

( x ) | Ω 〉 =

= 〈 z ( x ) 〉 =

v ch (v uch) 0

0 v ch (v dch)

v ch = 1√2

(√2 G F

) −1/2= 174.1 GeV

(7)

independent of the space-time point x a ,a

The implied parallelizable nature of 〈 z ( x ) 〉 is by far not trivial and relates in a wider context

including triplet scalar representations to potential (no nabelian) monopoles and dyons. (no h.o.)

– p. 12

13

induces a neutrino mass term through the Yukawa couplings λ F ′ F

in eq. (6)

F ′ N ν F = N γ F ′ νγF = ν γ F N γ

F ′

µ F ′ F = v ch λ F ′ F

→ H µ = F ′ N µ F ′ F ν F + h.c. = ν T µ T N + h.c.

(8)

The matrix µ defined in eq. (8) is an arbitrary complex 3 × 3matrix, analogous to the similarly induced mass matrices ofcharged leptons and quarks. In the setting of primary SO10breakdown, a general (not symmetric) Yukawa coupling λ F ′ F

implies the existence in the scalar sector of at least two irr euciblerepresentations (16)⊕ (120) a.

akey question → a ’drift’ towards unnatural complexity ? It becomes even wor se including the

heavy neutrino mass terms : 256 (complex) scalars.

– p. 13

14

2 a) ’Mass from mixing’ in vacuo [ 5 ] - [ 7 ] (→ )or ’Seesaw’ of type 1 [ 8 ] - [ 11 ] (→ )

neutrinos oscillate like neutral Kaons (yes, but how ?) - Bru no Pontecorvo a

The special feature, pertinent to (electrically neutral) n eutrinos is,that the ν− extending degrees of freedom N are singlets under thewhole SM gauge group G SM = SU3 c ⊗ SU2 L ⊗ U1 Y , infact remain singlets under the larger gauge group SU5 ⊃ G SM .This allows an arbitrary (Majorana-) mass term, involving t hebilinears formed from two N -s.In the present setup (minimal ν-extended SM) the full neutrinomass term is thus of the form →

aWe will come back to the clearly original idea in 1957 of Bruno Pontecorvo [ 12− ] but let me first

complete the ’flow of thought’ embedding neutrino masses in S O10.

– p. 14

15

HM = 12 [ ν N ] M

ν

N

+ h.c.

M =

0 µ T

µ M

; M = M T → M = M T

(9)

Again within primary SO10 breakdown the full M extends thescalar sector to the representations (16) ⊕ (120) ⊕ (126) a .

aIt is from here where the discussion – to the best of my knowled ge – of the origin and magnitude

of the light neutrino masses (re-) started in 1974 as documented on the next slide.

– p. 15

16

[ 5 ] Harald Fritzsch, Murray Gell-Mann and Peter Minkowski, ”Ve ctor - like weak currents and new

elementary fermions” , Phys.Lett.B59 (1975) 256.

[ 6 ] Harald Fritzsch and Peter Minkowski, ”Vector - like weak cur rents, massive neutrinos, and neutrino

beam oscillations” , Phys.Lett.B62 (1976) 72. ( [ 5 ] and [ 6 ] in the the general vectorlike situation.)

and for ’our world, tilted to the left’

[ 7 ] Peter Minkowski, ” µ → eγ at a rate of one out of 1-billion muon decays ?” , Phys.Lett.B6 7 (1977)

421.

Correct derivations were subsequently documented in [ 8 ] - [ 11 ]

[ 8 ] Murray Gell-Mann, Pierre Ramond and Richard Slansky, ”Comp lex spinors and unified theories” ,

published in Supergravity, P. van Nieuwenhuizen and D.Z. Fr eedman (eds.), North Holland Publ. Co., 1979

and in Stony Brook Wkshp.1979:0315 (QC178:S8:1979).

[ 9 ] Tsutomu Yanagida, ”Horizontal symmetry and masses of neutr inos” , published in the Proceedings

of the Workshop on the Baryon Number of the Universe and Unifie d Theories, O. Sawada and A.

Sugamoto (eds.), Tsukuba, Japan, 13-14 Feb. 1979, and in (QC D161:W69:1979) .

[ 10 ] Shelley Glashow, ”Quarks and leptons” , published in Procee dings of the Carg ese Lectures, M.

Levy (ed.), Plenum Press, New York, 1980.

[ 11 ] Rabindra Mohapatra and Goran Senjanovi c, ”Neutrino mass and spontaneous parity violation” ,

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

– p. 16

17

We resume the discussion of the mass term in eq. (9).Especially the 0 entry needs explanation. It is an exclusive propertyof the minimal ν-extension assumed here.Since the ’active’ flavors ν F all carry I 3 w = 1

2 terms of the form

12 F ′ ν χ F ′ F ν F = 1

2 ν T χ ν ; χ = χ T(10)

cannot arise as Lagrangean masses, except induced by anI w-triplet of scalars, developing a vacuum expected valueindependent from the doublet(s) a .from ”The apprentice magician” by Goethe : ’The shadows I inv oked , I am unable to get rid of now !’,

[ -3].

akey questions → quo vadis ? is this a valid explanation of the ’tilt to the left ’ ? no , at least

insufficcient !

– p. 17

18

2 0) Neutrino oscillations - historical overview

The idea that light neutrinos have mass and oscillate goes ba ck toBruno Pontecorvo, but starting with (para-) muonium -

antimuonium oscillations [ 12 ] a - like K 0 ↔ K 0 [ 13 ]b .

Assuming CP conservation there are two equal mixtures of µ − e+

and µ + e− with opposite CP values ± (at rest and using asemiclassical description of quantum states)

| ( eµ ) ± ; τ = 0 〉 = 1√2

( ∣∣ ( e −µ + )⟩∓

∣∣ ( e +µ − )⟩ )

τ=0

∣∣ ( e +µ − )⟩= C

∣∣ ( e −µ + )⟩

(11)a

[ 12 ] Bruno Pontecorvo, ”Mesonium and antimesonium”, JETP (USSR ) 33 (1957) 549, english

translation Soviet Physics, JETP 6 (1958) 429.b

[ 13 ] Murray Gell-Mann and Abraham Pais, Phys. Rev. 96 (1955) 1387 , introducing τ .

– p. 18

19

For the leptonium case the rest system is a good appoximation .The evolution of the CP ± states is then characterized by

m α = m α − i2 Γ α ; α = ± with

∣∣ ( e −µ + )⟩

= | 1 〉 → τ

1√2

(| + ; τ = 0 〉 e −i m + τ + | − ; τ = 0 〉 e −i m − τ

)

∣∣ ( e +µ − )⟩

= | 2 〉 → τ

1√2

(− | + ; τ = 0 〉 e −i m + τ + | − ; τ = 0 〉 e −i m − τ

)

(12)This reconstructs to →

– p. 19

20

| 1 〉 → τ E + ( τ ) | 1 〉 − E − ( τ ) | 2 〉

| 2 〉 → τ − E − ( τ ) | 1 〉 + E + ( τ ) | 2 〉

E ± ( τ ) = 12

(e −i m + τ ± e −i m − τ

)(13)

and leads to the transition relative probabilities indeed identical to

the K 0 → | 1 〉 ;K 0 → | 2 〉 system.

d p 1 ← 1 = d p 2 ← 2 = | E + ( τ ) | 2 d τ

d p 2 ← 1 = d p 1 ← 2 = | E − ( τ ) | 2 d τ

| E ± ( τ ) | 2 =

= 12 e− 1

2( Γ + + Γ − ) τ

(cosh 1

2 ∆ Γ τ ± cos ∆m τ)→

(14)

– p. 20

21

The term cos ∆m τ in eq. (14) indeed signals ( eµ ) ↔ ( eµ )oscillations, with

∆m = m + − m − ; ∆ Γ = Γ + − Γ −

∆m = O

α m e m νeνµ

v 2

2

m µ

∼ 4 . 10 −41 MeV

τ osc = ( 2π ) /∆m ∼ 10 20 sec = 3.3 . 10 12 y(15)

”Erstens kommt es anders, zweitens als man denkt.” a”First it happens differently, second as one thinks.”

aNot only this is clearly unobservable, but eq. (14) ignores C P violation (no h.o.) , and details of

neutrino mass and mixing, which induces ∆m in eqs. (14-15) .

– p. 21

22

From mesonium to neutrino’s [ 14 ] a

What is to be remembered from ref. [ 14 ] is the i d e a of neutrinooscillations, expressed in the corrected sentence :

”The effects due to neutrino flavor transformations may not beobservable in the laboratory, owing to the large R, but they w ill takeplace on an astronomical scale.”

The ( V − A )× ( V − A ) form of the Fermi interaction [ 15 ] b

a [ 14 ] Bruno Pontecorvo, ”Inverse β processes and nonconservation of lepton charge”, JETP

(USSR) 34 (1957) 247, english translation Soviet Physics, J ETP 7 (1958) 172.b [ 15 ] Richard Feynman and Murray Gell-Mann, ”Theory of Fermi inte raction”, Phys.Rev.109 1. Jan-

uary (1958) 193; E.C.G. Sudarshan, R.E. Marshak (Rochester U.), ”The nature of the four-fermion interac-

tion”, first published in 1957, (no h.o.) , which subsequently clarified the structure of neutrino emi ssion

and absorption, was documented almost contemporaneously [ -4] .

– p. 22

23

∆m τ from rest system to beam system [ 16 ] a

There is time dilatation from rest system to beam system, and alsowe express time in the beam system by distance ( c = 1 )

τ →d

γ β; γ −1 =

√1 − v 2 ; β = v b(16)

Then we replace ∆m , for 12 beam oscillations

∆m =∆m 2

2 〈m 〉;

∆m 2 = m 21 − m 2

2

〈m 〉 = 12 (m 1 + m 2 )

(17)

a [ 16 ] In notes to Jack Steinberger, lectures on ”Elementary parti cle physics”, ETHZ, Zurich WS

1966/67, not documented (again zigzag in time) .b

key question → which v ? →

– p. 23

24

Thus we obtain , for any 12 oscillation phenomenon

∆m τ =∆m 2

2 〈m 〉 β γd ; 〈m 〉 β γ = 〈 p 〉(18)

It is apparently clear that 〈m 〉 β γ = 〈 p 〉 represents the averagebeam momentum, yet this is not really so. Lets postpone thequestions ( which 〈 p 〉 ? - which d ? ) . From eq. (18) it follows

∆m τ =∆m 2

2 〈 p 〉d → cos

∆m 2

2 〈 p 〉d

(19)

The oscillation amplitude in vacuo (eq. 19) is well known, ye t it contains ’subtleties’ . →

– p. 24

25

∆m 2 d / ( 2 〈 p 〉 ) : what means what ?

The semiclassical intuition from beam dynamics and opticalinterference is obvious. A well collimated and within∆ | ~p | / | 〈 ~p 〉 | a ’monochromatic’ beam is considered as aclassical line, lets say along the positive z-axis, defining the meandirection from a definite production point ( ~x = 0 ) towards adetector, at distance d .But the associated operators for a single beam quantum

p z , z → ∆ p z ∆ z ≥ 12

(20)

are subject to the uncertainty principle ( using units ~ = 1 ) .The same is true for energy and time. Yet we are dealing →

a– or any similar definition of beam momentum spread –

– p. 25

26

in oscillations – with single quantum interference – and thus thespread from one beam quantum to the next is only yielding a ’go odguess’ of the actual expectation values, e.g. appearing in e q. (20) .The quantity 〈 p 〉 in the expression for the phase

∆m 2 d / ( 2 〈 p 〉 )(21)

essentially presupposes the single quantum production wavefunction, e.g. in 3 momentum space in a given fixed frame,propagating from a production time t P to a specific detectionspace-time point x D and characterized accounting for all quantummechanical uncertainties by the distance d . In this framework 〈 p 〉stands for the so evaluated single quantum expectation valu e a .

aThis was the content of my notes in ref. [ 16 ] (1966) . h.o. →

– p. 26

27

This was implicit in refs. (e.g.) [ 5 ] − [ 7 ], and became obvious indiscussing matter effects, specifically for neutrino oscil lations (e.g.in the sun) .

Coherence and decoherence in neutrino oscillations (h.o.)

The ensuing is an incomplete attempt of a historical overview, goingzigzag in time, starting with ref. [ 17 ] a.Just mention is due to two papers : Shalom Eliezer and Arthur S wift[ 18 ] b and Samoil Bilenky and Bruno Pontecorvo [ 19 ] c ,

where the phase argument ∆m 2 d / ( 2 〈 p 〉 ) appears correctly.

a [ 17 ] Carlo Giunti, ”Theory of neutrino oscillations”, hep-ph/0 409230, in itself a h.o.b [ 18 ] Shalom Eliezer and Arthur Swift, ”Experimental consequenc es of ν e − ν µ mixing in

neutrino beams”, Nucl. Phys. B105 (1976) 45, submitted 28. July 1975.c

[ 19 ] Samoil Bilenky and Bruno Pontecorvo, ”The lepton-quark ana logy and muonic charge”, Yad.

Fiz. 24 (1976) 603, submitted 1. January 1976.

– p. 27

28

Also in 1976 a contribution by Shmuel Nussinov [ 20 ] appeared a .

Matter effects - MSW for neutrinos [ 21 ] b , [ 22 ]c

The general remark hereto is”Every conceivable coherent or incoherent phenomenon invo lvingphotons, is bound to happen (and more) with neutrinos.”

→ refraction, double refraction, Cerenkov radiation, · · · [ 23 ] d .

The forward scattering amplitude and refractive index rela tion is(a semiclassical one) →

a[ 20 ] Shmuel Nussinov, ”Solar neutrinos and neutrino mixing”, Ph ys.Lett.B63 (1976) 201, submitted

10. May 1976.b

[ 21 ] Lincoln Wolfenstein, ”Neutrino oscillations in matter”, P hys.Rev.D17 (1978) 2369.c [ 22 ] Stanislav Mikheyev and Alexei Smirnov, Sov. J. Nucl. Phys. 4 2 (1985) 913.d [ 23 ] see e.g. Arnold Sommerfeld, ”Optik”, ”Elektrodynamik”, ”A tombau und Spektrallinien”,

Akademische Verlagsgesellschaft, Geest und Ko., Leipzig 1 959.

– p. 28

29

plane wave distortion in the z-direction

f 0 ≡ f labforward = ( 8π m target )

−1 T forward

Im f 0 = ( k lab / (4π ) ) σ tot ; k lab → k

e i k z → e i n k z = e i k z e i ( 2π / k ) N f 0 z

= e i k z e i [ ( 2π / k 2 ) N f 0 ] k z

n = 1 + ( 2π / k 2 ) N f 0 ; 〈 v 〉 mat. ∼ 1 / ( Re n )<> ? 1

N = mean number density of (target-) mattera

T = invariantly normalized (elastic-) scattering amplitude

(22)a

key question → which is the fully quantum mechanical description ?

– p. 29

30

for neutrinos [ 24 ] a at low energy :

H ν ∼ 2√2 G F

ν α γ µL ν β ℓ β γ µ L ℓ α

+ ν α γ µL ν α j µ n ( ℓ , q )

+ 14 ν α γ µ

L ν α ν β γ µL ν β

α , β = I, II, III for family ; : e.w. neutral current parameter

(23)

The second and third b terms on the r.h.s. of eq. (23) – in matterconsisting of hadrons and electrons – do not distinguish

a [ 24 ] Hans Bethe, ”A possible explanation of the solar neutrino pu zzle”, Phys.Rev.Lett.56 (1986)

1305, indeed, tribute – to many who ’really did it’ – and to a pioneer of solar physics and beyond (no h.o.) .b

the latter induces – tiny – matter distortions on relic neutr inos , maybe it is worth while to work

them out ?

– p. 30

31

between neutrino flavors. So the relative distortion of ν e –by electrons at rest – is

∆ e H ν →√2 G F ν β ∗

e ν βe 〈 e ∗ e 〉 e

=(√

2 G F n e

)ν β ∗e ν β

e

K e =√2 G F n e

; (K e → −K e for e − → e + )

(24)

The spinor field equation in the above semiclassical approximationin (chiral) basis becomes – suppressing all indices – and all owingfor an ~x dependent electron density

– p. 31

32

( i ∂ t − κ ) ν = M † ν

( i ∂ t + κ ) ν = M ν; i ∂ t → E

κ = K e P e − 1i ~σ

~∇ ¶ ; ν = ν βf ; ν = ε α β

(ν βf

) ∗

f = 1, · · · , 6 ; P e = δ ef δ ef ′ ; ¶ = δ ff ′ ; M = M ff ′

( ν , ν ) ( t , ~x ) : fields ; ∗ : hermitian conjugation

K e = K e ( ~x ) =√2 G F n e

( ~x )

(25)

Here we substitute fields by wave functions →

– p. 32

33

( ν , ν ) → e −iEt ( ν , ν ) ( E , ~x )∣∣ a

→ ( E − κ ) ν = M † ν ; ( E + κ ) ν = M ν(26)

From eq. (26) we obtain the ’squared’ form

(E 2 − κ 2 − M †M

)ν =

[κ , M † ] ν

(E 2 − κ 2 − MM † ) ν = [M , κ ] ν

[M , κ ] = K e [M , P e ] ;[κ , M † ] = [M , κ ] †

∣∣∣ b

κ 2 = −∆ − 2 P e K e1i ~σ

~∇ − P e ~σ ( 1i~∇K e ) + P e K

2e

(27)

aThe quantities ν (E , ~x ) are not related to complex conjugate entries for ν (E , ~x ) .

bThe purple quantities in eq. (27) give (e.g. in the sun) negli gible effects for the light flavors → 0 .

– p. 33

34

In eqs. (26-27) the mainly neutrinos have negative helicity, whichcan be specified precisely if the purple quantities are ignored,whereas the mainly antineutrinos carry positive helicity, in theultrarelativistic limit p = | ~p | ≫ |m | .

κ 2 → p 2 ± 2 p K e P e :

+ for neutrinos

− for antineutrinos(28)

The mass diagonalization yields correspondingly for the mi xing invacuo approximatively

M †M → u m 2diag u

−1 ; MM † → u m 2diag u

−1(29)

In eq. (29) the red quantities refer to the three light flavors a .a

So for real (i.e. orthogonal) u , neutrinos distorted by elec trons react identically to antineutrinos

relative to positrons .

– p. 34

35

a

aFrom Hans Bethe, ref. [ 24 ], with apologies to Mikheyev and Smirnov and many. →

– p. 35

36

(Further) references to the ⊙ LMA solution a

[ 25 ] Alexei Smirnov, ”The MSW effect and matter effects in neutri no oscillations”, hep-ph/0412391.

∆m 2⊙ = 7.9 . 10 −5 ev 2 ; tan 2 ϑ ⊙ = 0.40 ; ϑ ∼ 32.3 (30)

[ 26 ] Serguey Petcov, ”Towards complete neutrino mixing matrix a nd CP-violation”, hep-ph/0412410.

∆m 2⊙ =

(7.9 +0.5−0.6

). 10 −5 ev 2 ; tan 2 ϑ ⊙ = 0.40 +0.09

−0.07 Fig. →(31)

[ 27 ] John Bahcall and Carlos Pena-Garay, ”Global analyses as a ro ad map to solar neutrino fluxes and

oscillation parameters”, JHEP 0311 (2003) 004, hep-ph/030 5159.

[ 28 ] Gian Luigi Fogli, Eligio Lisi, Antonio Marrone, Daniele Mon tanino, Antonio Palazzo and A.M.

Rotunno, ”Neutrino oscillations: a global analysis”, hep- ph/0310012.

[ 28 ] P. Aliani, Vito Antonelli, Marco Picariello and Emilio Torr ente- Lujan, ”The neutrino mass matrix

after Kamland and SNO salt enhanced results”, hep-ph/03091 56.

[ 29 ] Samoil Bilenky, Silvia Pascoli and Serguey Petcov, ”Majora na neutrinos, neutrino mass spectrum,

CP violation and neutrinoless double beta decay. 1. The thre e neutrino mixing case”, Phys.Rev.D64

(2001) 053010, hep-ph/0102265.

aSee also many refs. cited therein, (no h.o.) .

– p. 36

37

– p. 37

38

Experimental references to the ⊙ LMA solution a

[ 30 ] Bruce Cleveland, Timothy Daily, Raymond Davis, James Diste l, Kenneth Lande, Choon-kyu Lee,

Paul Wildenhain and Jack Ullman, ”Measurement of the solar e lectron neutrino flux with the

Home-stake chlorine detector”, Astrophys. J. 496 (1998) 505.

reaction : ν e ⊙ + 37Cl → 37Ar + e −

[ 31 ] V. Gavrin, ”Results from the Russian American gallium exper iment”, for the SAGE collaboration,

J.N. Abdurashitov et al., J. Exp. Theor. Phys. 95 (2002) 181, astro-ph/0204245.

J.N. Abdurashitov, V.N. Gavrin, S.V. Girin, V.V. Gorbachev , P.P. Gurkina, T.V. Ibragimova,

A.V. Kalikhov, N.G. Khairnasov, T.V. Knodel, I.N. Mirnov, A .A. Shikhin, E.P. Veretenkin, V.M. Vermul,

V.E. Yants, G.T. Zatsepin, Moscow, INR

T.J. Bowles, W.A. Teasdale Los Alamos

J.S. Nico, NIST, Wash., D.C.

B.T. Cleveland, S.R. Elliott, J.F. Wilkerson, Washington U ., Seattle.

reaction : ν e ⊙ + 71Ga → 71Ge + e − →

Nobel Prizes : 2002 to Raymond Davis jr. , Masatoshi Koshiba a nd Riccardo Giacconi

and 2015 to Takaaki Kajita and Arthur B. McDonald.

aSee also many refs. cited therein, (no h.o.) .

– p. 38

39

[ 31 ] W. Hampel et al., GALLEX collaboration, ”GALLEX solar neutrino observations: resu lts for

GALLEX IV”, Phys. Lett. B 447 (1999) 127.

W. Hampel, J. Handt, G. Heusser, J. Kiko, T. Kirsten, M. Laube nstein, E. Pernicka, W. Rau, M. Wojcik,

Y. Zakharov, Heidelberg, Max Planck Inst.

R. von Ammon, K.H. Ebert, T. Fritsch, D. Heidt, E. Henrich, L. Stieglitz, F. Weirich, Karlsruhe U., EKP

M. Balata, M. Sann, F.X. Hartmann, Gran Sasso

E. Bellotti, C. Cattadori, O. Cremonesi, N. Ferrari, E. Fior ini, L. Zanotti, Milan U. and INFN, Milan

M. Altmann, F. von Feilitzsch, R. M ossbauer, S. Wanninger, Munich, Tech. U.

G. Berthomieu, E. Schatzman, Cote d’Azur Observ., Nice

I. Carmi, I. Dostrovsky, Weizmann Inst.

C. Bacci, P. Belli, R. Bernabei, S. d’Angelo, L. Paoluzi, Rom e U.,Tor Vergata and INFN, Rome

M. Cribier, J. Rich, M. Spiro, C. Tao, D. Vignaud, DAPNIA, Sac lay

J. Boger, R.L. Hahn, J.K. Rowley, R.W. Stoenner, J. Weneser, Brookhaven.

reaction : ν e ⊙ + 71Ga → 71Ge + e − like SAGE . →

– p. 39

40

[ 32 ] Y. Fukuda et al., Kamiokande collaboration ”Solar neutrino data covering solar cycle 22 ”, Phys.

Rev. Lett. 77 (1996) 1683.

Y. Fukuda, T. Hayakawa, K. Inoue, K. Ishihara, H. Ishino, S. J oukou, T. Kajita, S. Kasuga, Y. Koshio,

T. Kumita, K. Matsumoto, M. Nakahata, K. Nakamura, K. Okumur a, A. Sakai, M. Shiozawa, J. Suzuki,

Y. Suzuki, T. Tomoeda, Y. Totsuka, Tokyo U., ICRR

K.S. Hirata, K. Kihara, Y. Oyama, KEK, Tsukuba

Masatoshi Koshiba, Tokai U., Shibuya

K. Nishijima, T. Horiuchi, Tokai U., Hiratsuka

K. Fujita, S. Hatakeyama, M. Koga, T. Maruyama, A. Suzuki, To hoku U.

M. Mori, Miyagi U. of Education

T. Kajimura, T. Suda, A.T. Suzuki, Kobe U.

T. Ishizuka, K. Miyano, H. Okazawa, Niigata U.

T. Hara, Y. Nagashima, M. Takita, T. Yamaguchi, Osaka U.

Y. Hayato, K. Kaneyuki, T. Suzuki, Y. Takeuchi, T. Tanimori, Tokyo Inst. Tech.

S. Tasaka, Gifu U.E. Ichihara, S. Miyamoto, K. Nishikawa, INS, Tokyo.

reaction : ν e ⊙ + e − → ν e + e − →

– p. 40

41

[ 33 ] Y.Ashie et al., Super-Kamiokande collaboration, ”Evidence for an oscillatory signature in

atmospheric neutrino oscillation”, Phys.Rev.Lett.93 (20 04) 101801, hep-ex/0404034 and

Y. Suzuki, ”Super-Kamiokande: present and future”, Nucl.P hys.Proc.Suppl.137 (2004) 5.

M. Goldhaber, Masatoshi Koshiba, J.G. Learned, S. Matsuno, R. Nambu, L.R. Sulak, Y. Suzuki,

R. Svoboda, Y. Totsuka, R.J. Wilkes · · · a

reactions : ν e ⊙ + e − → ν e + e −

ν e,µ

ν e,µ

+ H 2 O → e ∓ , µ ∓ +X

8B solar neutrino flux :

5.05(1 +0.20−0.16

). 10 −6 / cm 2 / s for BP2000

5.82 ( 1 ± 0.23 ) . 10 −6 / cm 2 / s for BP2004→

a ”Wer z ahlt die Seelen, nennt die Namen,

die gastlich hier zusammenkamen.”

”Who counts the souls, relates the names,

who met in piece here for the games.”

Friedrich Schiller

– p. 41

42

– p. 42

43

[ 34 ] D. Sinclair for the SNO collaboration, ”Recent results from SNO”, Nucl.Phys.Proc .Suppl.137 (2004)

150.

Art McDonald, A. Hamer, J.J. Simpson, D. Sinclair, David War k · · · ← a

∆m 2⊙ =

(7.1 +1.0−0.6

). 10 −5 ev 2 ; ϑ ⊙ =

(32.5 +2.4

−2.3

) (32)

reactions : ν e ⊙ + d → p + p + e − (CC)

ν x ⊙ + d → p + n + ν x (NC)

ν x ⊙ + e − → e − + ν x (ES)

– p. 43

44

Reference(s) to ν e ↔ ν x oscillations

[ 35 ] A. Suzuki for the Kamland collaboration, ”Results from Kamland”, Nucl.Phys.Proc.S uppl.137 (2004)

21.

T. Araki, K. Eguchi, P.W. Gorham, J.G. Learned, S. Matsuno, H . Murayama, Sandip Pakvasa,

A. Suzuki, R. Svoboda, P. Vogel · · · ← a

reaction : ν e ⊙ + p → e − + n from ∼ 20 reactors →

leading to the present best estimates (ref. [ 26 ] ) for 3 flavor oscillations :

∆m 2⊙ =

(7.9 +0.5−0.6

). 10 −5 ev 2 ; tan 2 ϑ ⊙ = 0.40 +0.09

−0.07

∣∣∆m 223

∣∣ =(2.1 +2.1−0.8

). 10 −3 eV 2 ; sin 2 2 ϑ 23 = 1.0 −0.15

sin 2 ϑ 13 ≤ 0.05 at 99.73 % C.L.

(33)

This shall conclude my – necessarily partial – historical overview, from the time of the yeae 2005 [ -1].

What follows must be cut short as well .

– p. 44

45

– p. 45

46

2 Mass from mixing → the subtle things

– p. 46

47

How is the mass matrix of the form in eq. (9) diagonalized exactly ?

M =

0 µ T

µ M

= UM diag U

T = KM bl.diag KT

M diag =M diag (m 1 , m 2 , m 3 ; M 1 , M 2 , M 3 )

M bl.diag = U 0M diag UT0 ; U 0 =

u 0 0

0 v 0

U = K U 0 =

u 11 u 12

u 21 u 22

→

(34)

– p. 47

48

The at least perturbatively appropriate notion of mass in theassumed fundamental presence of gravity, has to beselfconsistently justified. It can then only have perturbat ive validitythrough a limited range of energies of multopartile lepton s catteringstates.Leptonic heavy gauge bosons, necessarily shall thus have ma sseslow enough, to allow their decay amplitudes, i.e. their widt hs, to becalculated as if gravity effects are negligible, i.e. whenc e Poincar einvariance, again approximately, holds.We shall come back to this point in (a) subsequent section(s) ,leading to the formulation of the treaceanomaly in ref. [ -5] .Here a seminal paper by Mijhail Shifman, Arkady Vainshtein a ndValentin Zakharov showed the way, in ref. [ -7] , after quite s ometime of derivation and checking.

– p. 48

49

U = KU 0 ; K−1MK −1 T = M bl.diag. =

M 1 0

0 M 2

U =

(1 + t t †

)−1/2u 0

(1 + t t †

)−1/2t v 0

−t †(1 + t t †

)−1/2u 0

(1 + t † t

)−1/2v 0

↓ · · · →

M 1 = − tM 2 tT ;

M 1 = u 0 m diag uT0 : light 3

M 2 = v 0 M diag vT0 : heavy 3

(35)

– p. 49

50

M 1 = − tM 2 tT ;

M 1 = u 0 m diag uT0 : light 3

M 2 = v 0 M diag vT0 : heavy 3

(36)In eq. (36) all matrices are 3 × 3 , t describes light - heavy mixing

generating mass by mixing . a

u 0 (unitary) accounts for light-light mixingand v 0 (unitary) for heavy-heavy (re)mixing .

t is ’driven’ by µ ( in M ← ) and determined →

aThis is documented in [ 36 ] Clemens Heusch and Peter Minkowski, ”Lepton flavor violatio n in-

duced by heavy Majorana neutrinos” , Nucl.Phys.B416 (1994) 3 . Of course 6 by 6 matrices have been

diagonalised before .

– p. 50

51

from the quadratic equation

t = µ T M −1 − t µ t M −1 ; to be solved by iteration →

t n+1 = µ T M −1 − t n µ t n M −1 ; t 0 = 0

t 1 = µ T M −1 , t 2 = t 1 − µ T M −1 µ µ †M−1

M −1

· · · ; lim n → ∞ t n = t a , b

(37)

The subtleties inherent to the (iterative) solution of eq. 3 7 haverecently been correctly tied to the breaking of supersymmet ryon a l l scales in ref. [ -8] . −→

aThis is essentially different from the mixing of identical SU2 L × U1 Y representations, i.e.

charged base fermions.b

[ 37 ] Ziro Maki, Masami Nakagawa and Shoichi Sakata, Prog. Theor. Phys. 28 (1962) 870.

→ the PMNS-matrix : The authors assumed (in 1962) only light-light mixing .

– p. 51

52

Finally lets turn to the mixing matrix u 11 ( eqs. 34-35 )

u 11 =(1 + t t †

)−1/2u 0 ∼ u 0 − 1

2 t t † u 0

t t † = O(m /M

)∼ 10 meV / 10 10 GeV = 10 −21

(38)

The estimate in eq. (38) is very uncertain and assumes amongother things m 1 ∼ 1 meV .Nevertheless it follows on the same grounds as the smallness oflight neutrino masses, that the deviation of u 11 from u 0 is tiny .

”Much ado about nothing” , William Sakespeare

For details in the derivation of the approximations display ed in eq.37 I refer to ref. [ -9] .

– p. 52

53

I should like - in commemoration of my former collaboratorBernhard Scheuner - to show a photograph of Heimenschwand, B E,the community where he was born in the southern Emmental .

– p. 53

54

3. Mass and Mixing of 3 + 3 light and heavy neutrino flavors

The details of the complex symmetric 6 by 6 matrix M wasdisplayed in eqs. 9 and 35 - 37 .In the framework of three families of minimal neutrino massextended fermions, within a unified or enveloping charge lik e gaugegroup [ -5] , [ (-6)]

G env = SO10(39)

It may be useful, to define the mass in a renormalization groupcovariant way, in which case it does not describe the physica l massof neutrino’s , neither light nor heavy ones. Then it is neces sary touse a loop expansion in the SO10 associated couplings, aprocedure denoted loop expansion of physical neutrino mass .For a recent discussion of these issues let me refer to ref. [ - 10] . →

– p. 54

55

It must here be ’remembered’ that the notion of physical neut rinomass in a unifying theory including gravity is not unconditionallydefined , except after establishing conditions of (here) lep tonicscattering amplitudes , for which local Poincar e covariance is agood approximation. It can not be an exact one, as follows fro m thespontaneous scale breaking induced by the trace anomaly, asexposed in refs. [ -5] and here at ICNFP 2014 [ (-6)] . →

– p. 55

56

Given the premises of simultaneous

a) charge like gauging as in SO10

b) gauging of orientation as in gravity,

but in d + 1 space time dimensions with ∞ > d ≥ 3

c) a finite number of defining constants, as required for

substrate fields, compatible with strict locality,

implyinging causality

it turns out that the conditions a) , b) , c) can not be fulfilled a . →a

This statement is at present a conjecture, in the process of b eeing checked .

– p. 56

57

Two conseuences shall be singled out below

1 Chargelike gauge fields have to be generated as composite on esfrom structures in space time

2 any realization of supersymmetry, global and/or local, co ntaininga super Yang Mills structure in accord with our SO10unification hypothesis, is broken on all scales as aconsequence of the trace anomaly.

→

– p. 57

58

In ref. [ -11] Keith Olive considers the unification of the 3 ch argelike coupling constants , with the field content underlying S O10unification , but without supersymmetry . First let me quote h isintroductory remarks”1. IntroductionFor well over 30 years, supersymmetry and the minimalsupersymmetric standard model (MSSM) have been primeexamples of extensions beyond the standard model (SM) of str ongand electroweak interactions. Apart from its algebraic bea uty, thereare many motivations for supersymmetry. These include thetheory’s ability to provide gauge coupling unification and a ddressthe gauge hierarchy problem. →

– p. 58

59

It is well known that the additional fields predicted in the MS SM, ifpresent at low energy, alter the running of the gauge couplin gs asshown on Fig. 1 ( → Fig. -2 ) . At one-loop, the running of the gaugecouplings is given by

Q

d

Q

α i =

b i

2 π

α 2i(40)

where the coefficients of the SM beta functions areb i = 41/10, -19/6, -7 for the U1 Y , SU2 L , and SU3 c gaugecouplings respectively.The scale evolution is shown in Fig. -2 below →

– p. 59

60

Fig. -2 :

The left panel shows the SO10 evolution of the

coupling constants, while the right panel shows the same but→

– p. 60

61

but with supersymmetric doubling of the 16-representation

of fermions modifying the evolution at a scale of energy with in

∼ 1 TeV of the electroweak scale .I further quote from ref. [ -11] :”It is also well known that the quartic coupling, λ, of the Higgs

potential, (denoting the physical Higgs scalar field by h)

V = 14 λ h 4 + 1

2 m 2 h 2(41)

runs negative at a scale Q ∼ 10 10 GeV · · · ”Indeed the 1 scalar doublet realization of electroweak glob alsymmetry breaking is a nontrivial source of estimate of the m assscale , where new physics must appear. This svale beeing of th eorder of 10 10 GeV is an independent derivation from the standardmodel →

– p. 61

62

as referred to in eq. 41 above and the estimate , derived in eq. 38repeated below :

u 11 =(1 + t t †

)−1/2u 0 ∼ u 0 − 1

2 t t † u 0

t t † = O(m /M

)∼ 10 meV / 10 10 GeV = 10 −21

(42)

In order to get the numerical relation in the last line of eq. 4 2normal hierarchy for the three light masses and 1 meV for the m assof the lightest neutrino flavor have to be assumed.

→

– p. 62

63

3.1 Rare decay processes associated with B - L breaking , show ingdeviations from oppositely charged lepton pair decays, wit h

assessed significant deviations in their e - mu ratiofrom the standard model at the LHC

In ref. [ -12] the four authors, Andreas Crivellin, Dario Mu ller, AdrianSigner and Yannick Ulrich, consider the constituent transi tions

b → s ℓ + ℓ − ; ℓ − = e − , µ − ,(τ −

)(43)

Upon hadronization the directly leptonnumber violating pr ocessesB s → µ e ; B → K µ e appear, which are within the reach ofLHCb and Belle II .

→

– p. 63

64

I quote from ref. [ -12] :”Taking this approach and including the new LHCb result forR(K) =

(B → K µ +µ −

)/(B → K e + e −

), measuring

lepton flavour universality (LFU) violation, the global sig nificancefor NP increased above the 5 σ level [ -16] . In addition, thecombination of the ratiosR (D() ) = ( B → D () τ ν) / ( B → D () ℓ ν ) also differs by3.9 σ from its SM prediction. All together, this strongly motivat es toexamine LFU violation in semileptonic B decays in the contex t ofNP. Since b → s ℓ + ℓ − processes are semileptonic, leptoquarks(LQ) provide a natural explanation for these anomalies. The y givetree-level contributions to these processes but contribut e forexample to ∆ F = 2 processes only at the loop level, thereforerespecting the bounds from other flavour observables.

→– p. 64

65

Furthermore, since in R (D ∗ ) an O(10%) effect compared to thetree-level SM is needed, also a NP tree-level effect is requi red. HereLQ are probably even the most promising solution. In fact ... amodel for a simultaneous explanation of b → s ℓ + ℓ − data

together with R (D () ) has been proposed which is compatible

with the bounds from B K () − ν ν , electroweak precision data ...and direct LHC searches ... . Interestingly, LQ also provide a naturalsolution to the anomaly in the magnetic moment of the muon due tothe possible enhancement by m t / m µ through an internalchirality flipping ... .The dots in the text quoted above and below stand for referenc esquoted in ref. [ -12], that are not essential for our discussi on here.The model independent fit to R (K ) and R (K ∗ ) allows for NPcontributions to electrons or muons separately, but also to bothsimultaneously ... . ” →

– p. 65

66

”Once the other data on b → s µ + µ − is included, NP in muonsis required but is only optional for electrons. However, the best fitvalue suggests a simultaneous NP contribution to electrons as well... . It is well known that once LQ couple to muons and electron ssimultaneously, they give rise to lepton flavour violating B decaysand to µ → e γ ... .

Fig. 1 ( → Fig. -3 )

” →– p. 66

67

” Both µ → e γ and lepton flavour violating B decays with µ efinal states are experimentally very interesting and precis e upperlimits for these processes already exist. For µ → e γ the currentexperimental bound, obtained by the MEG collaboration ... , is

Br [ µ → e γ ] ≤ 4.2 . 10 −13(44)

and MEG II ... at the Paul Scherrer Institute (PSI) will signi ficantlyimprove on this bound in the future. ”Here let me recall my estimate in ref. [7] ( µ → e γ at a rate of 1 outof 10 9 muon decays ? )

Br [ µ → e γ ] ∼ 3.9 10−56(45)

Between the estimates in eqs. 44 and 45 there are 43 orders ofmagnitude .

Thank you for your attention

– p. 67

ref1

Additional references to those in the text

References[ -1] Peter Minkowski (Bern U.), May 2005, 22 pp., ’Neutrino o scillations: A Historical overview and its

projection’, Contributed to Contribution to XI Internatio nal Workshop ”Neutrino Telescopes in

Venice”, 22.-25. February 2005, Venice, Italy, published i n the proceedings, e-Print: hep-ph/0505049,

URL: http://www.mink.itp.unibe.ch/ lectures.html in the file venice3.pdf .

[ -2] James Chadwick, ”Possible Existence of a Neutron”, Nat ure, 129 (1932) 312 .

[ -3] Johann Wolfgang Goethe, ”Der Zauberlehrling”,

/www.youtube.com/results?search-query=der+zauberleh rling+youtube+ .

[ -4] E.C.G. Sudarshan, R.E. Marshak (Rochester U.), 1994, ” The nature of the four-fermion interaction”,

in *Sudarshan, E.C.G. (ed.): A gift of prophecy 508-515* .

[ -5] Peter Minkowski (Bern U.), Feb 2012, 39 pp., ”Canonical (anti-)commutation rules in QCD and

unbroken gauge invariance, QCD - the two central anomalies a nd canonical structure”, Conference:

C12-01-09.1, e-Print: arXiv:1202.2488 [hep-th] — PDF .→

– p. 68

ref2

References[ (-6)] Peter Minkowski, (Albert Einstein Center for Fundam ental Physics - ITP, University of Bern,

Switzerland), ”Avenues of cognition of nongravitational l ocal gauge field theories”, published in

EPJ Web of Conferences 95, 03025 (2015), DOI: 10.1051/epjco nf/20159503025 .

[ -7] Mikhail A. Shifman, A.I. Vainshtein, Valentin I. Zakha rov (Moscow, ITEP), ”QCD and Resonance

Physics: Applications”, 1978, 71 pp., Published in Nucl.Ph ys. B147 (1979) 448-518 ITEP-94-1978,

ITEP-81-1978 DOI: 10.1016/0550-3213(79)90023-3 .

[ -8] Nicola Ambrosetti, Daniel Arnold, Jean-Pierre Derend inger (Bern U.), Jelle Hartong (Brussels U.,

PTM & Intl. Solvay Inst., Brussels), Jul 28, 2016. 50 pp., ”Ga uge coupling field, currents, anomalies

and N=1 super-YangMills effective actions”, Published in N ucl.Phys. B915 (2017) 285-334,

DOI: 10.1016/j.nuclphysb.2016.12.011, e-Print: arXiv:1 607.08646 [hep-th] — PDF .

[ -9] Peter Minkowski (Bern U.), 2009, 31 pp., ”The origin of n eutrino mass: Stations along the path of

cognition”, Published in J.Phys.Conf.Ser. 171 (2009) 0120 16, DOI: 10.1088/1742-6596/171/1/012016;

Conference: C08-12-11 Proceedings, Link to Fulltext .

[ -10] Pavel Fileviez Prez, Clara Murgui, Aug 7, 2017, 7 pp., ” Sterile Neutrinos and B-L Symmetry”,

e-Print: arXiv:1708.02247 [hep-ph] — PDF .→

– p. 69

ref3

References[ -11] Keith A. Olive (Minnesota U., Theor. Phys. Inst.), 201 7, 20 pp., ”Supersymmetric versus SO(10)

models of dark matter”, Published in Int.J.Mod.Phys. A32 (2 017) no.13, 1730010,

DOI: 10.1142/S0217751X17300101 ,

for a review of searches for related signals by the ATLAS cola boration see:

URL: https://arxiv.org/pdf/1707.03144.pdf .

[ -12] A. Crivellin, (Paul Scherrer Institut, CH5232 Villig en PSI, Switzerland), Dario Muller, Adrian Signer

and Yannick Ulrich (Paul Scherrer Institut, CH5232 Villige n PSI, Switzerland and Physik-Institut,

Universit at Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Swi tzerland), PSI-PR-17-11, ZU-TH

16/17, ”Correlating Lepton Flavour (Universality) Violat ion in B Decays with µ → e γ using

leptoquarks”, arXiv:1706.08511v1 [hep-ph], 26 Jun 2017 .

[ -13] W. Altmannshofer and D. M. Straub (2015), 1503.06199.

[ -14] S. Descotes-Genon, L. Hofer, J. Matias, and J. Virto, J HEP 06, 092 (2016), 1510.04239.

[ -15] S. Descotes-Genon, L. Hofer, J. Matias, and J. Virto, J HEP 06, 092 (2016), 1510.04239.

[ -16] B. Capdevila, A. Crivellin, S. Descotes-Genon, J. Mat ias, and J. Virto (2017), 1704.05340.→

– p. 70

ref4

References[ -17] Y. Amhis et al. (2016), 1612.07233.

→

– p. 71

61’

Additional Material

– p. 72