p h ysics of neu trin o m as s - fermilabp hysics of neu trino ma ss p h ysics of neu trin o m as s...

Physics of Neutrino Mass


R. N. Mohapatra

University of Maryland,College Park.

FERMILAB/KEK Neutrino Summer Institute, 2007

c! Rabi Mohapatra, 2007University of Maryland

1/138 fermilab.tex


Main theme of the talk

! Outline

" HOW WELL DO WE UNDERSTAND THE NEUTRINOOBSERVATIONS ?

" WHAT HAVE WE LEARNED ABOUT NEW PHYSICSFROM THEM ?

" HOW CAN WE TEST THESE NEW IDEAS ?

" HOW DO THEY FIT INTO THE BIG PICTURE THATWE MAY HAVE FOR OTHER PHYSICS e.g. GRANDUNIFICATION, COSMOLOGY etc ?


2/138 fermilab.tex


Outline of the three elctures

!

1. Understanding small neutrino masses and mixings:• Seesaw Mechanism and some implications;

• Mass matrices and Family Symmetries.

2. Origin of seesaw scale and implications for unification

• Local B-L symmetry and left-right symmetric weakinteractions;

• Neutrino mass and grand unification:

(i) Is SU(5) enough ?

(ii) SO(10), a more suitable group and its predictions.

3. Neutrino mass related new physics

• possible TeV scale WR and Z !

• light doubly charged Higgs;

• Neutron-anti-neutron oscillation in reactors.


3/138 fermilab.tex


Summary of what we now know

(A. de Gouvea and B. Kayser’s talks for details)

! Solar + Atmospheric + KamLand + K2K

" MIXINGS Defined as:!

"""""#

!e

!µ

!!

$

%%%%%&= U"i

!

"""""#

!1

!2

!3

$

%%%%%&

" where UPMNS =!

"""""#

c12c13 s12c13 s13e"i#

!s12c23 ! c12s23s13ei# c12c23 ! s12s23s13ei# s23c13

s12s23 ! c12c23s13ei# c12s23 ! c12c23s13ei# c23c13

$

%%%%%&K

" K = diag(1, ei$1, ei$2)

" SOLAR: sin2"12 " 0.31 ± 0.02

" ATMOS: tan2"23 " 0.89+0.31"0.21

" REACTOR: "13 # 0.23


4/138 fermilab.tex


MASSES

! We only know mass two di!erence squares

" ATMOS: |!m213| = (2.6 ± 0.2) $ 10"3 eV2; (1#)

" SOLAR: !m221 = (7.9 ± 0.3) $ 10"5 eV2 ; (1#)

" MASS PATTERN STILL UNKNOWN

! Possibilities

1. ::NORMAL::% m1 & m2 & m3

% !m231 > 0; m3 " 0.05 eV; m2 " 0.009 eV

In particular, in this case m3m2

' 6;What is m1 ?

2. ::INVERTED::% m1 " m2 ( m3

% !m231 < 0; m1 " m2 " 0.05 eV

3. ::DEGENERATE:: m1 " m2 " m3 % !m231 > or < 0


5/138 fermilab.tex


INVERTEDNORMAL

MA

SS

3!

!2

!1

!2!1

!3

!e

µ!

!"

Figure 1: Two possible pattern of masses; Compare with quarks for which mu,d # mc,s #mt,b

! Compare with quarks for which mu,d & mc,s & mt,b

and "q13 " .004; "q

23 " 0.04 and "q12 " 0.22


6/138 fermilab.tex


! Another fundamental property of the neutrinothat we stll do not know is:

Is neutrino its own antiparticle ? i.e. is ! = !

If ! = !, it is Majorana; otherwise Dirac

For Majorana neutrino, we have

2N % 2P + 2e"

i.e. neutrinoless double beta decay; amplitudedirectly proportional to neutrino Majorana mass


7/138 fermilab.tex


Overall mass scale

! We need to know the lightest mass in Case (i)(normal) and (ii)(inverted) and absolute mass incase(iii)

: ! Experimental Information

1. 3H Decay end point: 'i m2

i |Uei|2 # 2.2 eV2 (KATRINexpected to improve it to 0.2 eV)

2. Cosmology: ' mi # 0.4 eV (WMAP, SDSS: will beimproved by Planck)

3. If neutrino Majorana i.e. ! = !, $$o% results imply:'

i U 2eimi # 0.3 ! 0.5 eV (Expected improvement to 0.03

eV)


8/138 fermilab.tex


How many neutrinos?

!

" Z-width measurement at LEP and SLC implies three !’scoupling to Z (active neutrinos !e,µ,!)

" Most persuasive argument for sterile neutrinos is theoscillation results from LSND: negative results byMiniBooNe can still be made consistent with LSND, ifthere are two sterile neutrinos with specific masses:!m2

41 = 0.66 eV2 and |Ue4Uµ4| = 0.044; e.g.!m251 = 1.44

eV2 and |Ue5Uµ5| = 0.022 (Maltoni and Schwetz (07)).

" Severe constraints on the number of !s from BBN (# 0.3;

" Also severe constraint on the number from structureformation; (# 2.5 at 68% c.l.) (Hannestad review, 2006)


9/138 fermilab.tex


WHERE WE ARE HEADED ?

! (i) Dirac vrs Majorana in $$0% decay search:

Expts: MAJORANA (Ge76), EXO(Xe), CUORE(Te)...

(ii) Normal vrs Inverted hierarchy:

Long baseline neutrino experiments- JPARC, NoVa,

(iii) Determining "13:

Reactor experiments: Double CHOOZ, Daya Bay"13 # 0.05 will reveal important new symmetries:


10/138 fermilab.tex


Testing inverted hierarchy in !!0! decaysearches:

! < m&& >" 'i U 2

eimi: leads in the case of invertedhierarchy to a lower bound if neutrino is Majorana.

10-4 10-3 10-2 10-1 100

m1 [eV]

10-4

10-3

10-2

10-1

100

|mee

| [e

V]

3# best fit

10-4 10-3 10-2 10-1 100

m3 [eV]

10-3

10-2

10-1

100

|mee

| [e

V]

3# best fit

Heidelberg-Moscow (76Ge)

CUORE (130Te)

EXO (136Xe, 10t)

MOON (100Mo)

GENIUS

Invertedhierarchy

Almostdegenerate

Heidelberg-Moscow

CUORE

MOON

EXO (10 t)

GENIUS (76Ge)

Almostdegenerate

normalhierarchy


11/138 fermilab.tex


! CAUTION! even with normal hierarchy, therecould be “large” e!ects from heavy particles such assparticles, doubly charged Higgs bosons or RHMajorana neutrinos.

d

d

u

u

WR

WR

H!!

e

e

vR


12/138 fermilab.tex


0 0.005 0.01 0.015 0.02 0.025 0.03!1

2

20

40

60

80

100

meff!meV

"

Figure 2: Scatter plot of |meff | "< mee > in the case of inverted hierarchy in the presence of a sterile neutrino withmass # 1 eV as a function of !2

1 . Note that the minimum value of |meff | depends on the best fit value for the LSNDmixing parameter in a 3+1 scheme. The scatter of points is due to the variation of the phases " and # between 0 to $.

Similarly presence of sterile neutrinos couldmask the inverse hierarchy e!ects!


13/138 fermilab.tex


Prospects for discriminating between Diracand Majorana neutrino

! Sign of !m2, $$0% and KATRIN result can tell usa lot:

$$0% !m232 KATRIN Conclusion

yes > 0 yes Degenerate, Majoranayes > 0 No Degenerate, Majorana

or normal or heavy exchangeyes < 0 no Inverted, Majoranayes < 0 yes Degenerate, Majoranano > 0 no Normal, Dirac or Majoranano < 0 no Diracno < 0 yes Diracno > 0 yes Dirac


14/138 fermilab.tex


Neutrino Mass and new physics

! Challenges for theory:

" Why m% & mu,d,e ?

" Why are neutrino mixings so much larger than quarkmixings ?

" Why is !m2!

!m2A& 1 but (

(mµm!

)2?

" What do neutrinos tell us about physics beyond thestandard model e.g. new symmetries; any new forces ?

" How do neutrinos fit into the big picture of grandunification,cosmology etc.


15/138 fermilab.tex


A Primer on Fermion masses and mixings

! Look for bilinears of the form %L%R in theLagrangian

If there are more fermions of the same kind, thenLmass = Mab%a,L%b,R

! Masses and mixings from the Lagrangian

" Mab= Mass matrix

" Diagonalize the mass matrixU †MV = diag(m1,m2, ·, ·)

" U, V gives the mixings between di!erent (L,R) fermions,%a and mi are the actual masses e.g. for quarks, Uab

contains the CKM mixings (e.g. UCKM = U †uUd,

whereU and V denote the rotations in the up and thedown sector)


16/138 fermilab.tex


Oscillations

!

" The “flavor” eigenstate :|a > = 'i Uai|i > and as it

evolves with time, we have oscillations.

" Key to understanding masses and mixings is clearly theform of the mass matrix


17/138 fermilab.tex


! DIRAC vrs MAJORANA MASS

" Lorentz Invariance allows two kinds of mass terms forfermions: %L%R or %T

LC"1%L (or L ) R)

" Note under a symmetry transformation % % ei"%, thefirst mass is invariant whereas the second term is not;

" Fermions only with the first kind of mass are called Diracfermions and those with both kinds are called Majoranafermions

" Dirac fermion unlike Majorana requires an extrasymmetry: e.g. for e, µ, q.., extra symmetry is U(1)em;since Q(!) = 0, no such symmetry is there for !

" Hence for neutrinos, Majorana-ness is more natural; alsosmall mass is easier for Majorana neutrino.

" For Majorana mass, ! = !


18/138 fermilab.tex


Neutrino Majorana mass matrices

! If there are no RH or SM singlet neutrinos, theMajorana mass matrix is symmetric and has 6 realentries and three phases- 9 parameters;

Physical observables: mi; "ij; &D; 'M1,2;

Measured observables: 4;Can have mass matrices with three real parametersdescribing observations- but entries at suitableplaces;

e.g. M% =

!

"""""#

0 a ba 0 cb c 0

$

%%%%%&is already ruled out by solar

observations;

An example of a minimal mass matrix that is allowedis:

M% =*

!m2A

!

"""""#

( b( b(b( 1 + ( b( ! 1b( b( ! 1 1 + (

$

%%%%%&


19/138 fermilab.tex


! A four parameter example: M% =

!

"""""#

d a ba 0 cb c 0

$

%%%%%&;

leads to inverted hierarchy and observableneutrino-less double beta decay.


20/138 fermilab.tex


Standard model

! Details

" Gauge group SU(3)c $ SU(2)L $ U(1)Y

" Matter: Doublets: Q *!

"#uL

dL

$

%&; %L *!

"#!L

eL

$

%&;

Singlets: uR; dR; eR

Higgs: H *!

"#H0

H"

$

%&

" LY = huQLHuR + hdQLHdR + he%LHeR + h.c.


21/138 fermilab.tex


Fermion masses

! Masses arise from symmetry breaking < H0 >= vwk

" Lm = ua,LMuabub,R + da,LMd

abdb,R + ea,LM eabeb,R;

" gives masses to quarks and charged leptons only

" Therefore m% = 0 in the standard model.

" No e!ect of the standard model can give mass to theneutrino since B-L is a good symmetry of the standardmodel.

! Could there be some source of B-L breakinghidden in the standard model that would give m% += 0?


22/138 fermilab.tex


Could it be gravity ?

! Standard model + gravity

" global symmetries could be broken by nonperturbativegravitational e!ects such as black holes or worm holes etc.

" if so, they could induce B-L breaking operators intostandard model e.g. (%LH)2/MP ';

" They lead to m% " v2wk

MP "' 10"5 eV- clearly too small to

explain atmospheric neutrino deficit. The Answer is: NO!


23/138 fermilab.tex


! : NEUTRINO MASS AND NEW PHYSICS:

MECHANISMS AND MODELS


24/138 fermilab.tex


Nu-standard model

! Simplest possibility: Add !R to the standard model

" New term in the LY : h†%%LH!R + h.c.

Could it be that this is the source of neutrino mass whensymmetry breaks ?

" In principle it could but this is very unnatural because ofthe following reasons:(i) since m% # 0.1 eV, this will mean h% ' 10"12 ? why isit so small ?

(ii) This Yukawa coupling then is likely to be very similarto the quarks in which case it will be hard to understandwhy(a) the neutrino mixings are so di!erent from quarkmixings and(b) why m2

m3' 1

6 for neutrinos whereas for up quarks thesame number is 1/100 and for down quarks it is 1/40-50 ?


25/138 fermilab.tex


Seesaw mechanism

!

" RH neutrinos are di!erent from the rest of the standardmodel particles- since they are SM singlets unlike allothers.

" Because of this gauge invariance allows a new term thatcan be added to the SM i.e. 1

2MR!TRC"1!R + h.c.

Important point: MR breaks B-L symmetry

" This makes neutrino masses di!erent from the masses ofother SM fermions:


26/138 fermilab.tex


Seesaw diagram

Figure 3: Seesaw diagram

! Leads to Majorana neutrinos.% Mass matrix for (!L, !R) system:!

"#0 h%v

hT% v MR

$

%& MR ( h%v, mass eigenvalues: heavy :

%: MR and light: M% " !v2wkh

T% M"1

R h%;

Roughly m% " !h2#v2

MR. Since MR ( vwk, this explains

why m%i & mu,d,e.... (Type I seesaw)

Minkowski (77); Gell-Mann, Ramond, Slansky; Yanagida; Glashow; R. N. M., Senjanovic (1979)


27/138 fermilab.tex


Detailed derivation of seesaw formula

! L%,mass = h%,ijvwk!R!L + 12!

TRMR,ij!R + h.c.;

Write ! =!

"#)

i#2*$

$

%&;

),* two-component objects;

+ matrix convention: +i =!

"#0 #i

!#i 0

$

%&; +0 =!

"#0 II 0

$

%&;

+5 =!

"#I 00 !I

$

%&;

!L =!

"#)0

$

%& and !R =!

"#0

i#2*$

$

%&;

L%,mass = ! i ( )T *T )!

"#0 h%vwk

hT% vwk MR

$

%&

!

"#)*

$

%& + h.c.

(Same as in the previous page.)


28/138 fermilab.tex


Higgs triplets and Type II seesaw

! Instead of adding the !R, add a Higgs triplets, !L

with Y = 2

i.e. !L =

!

"""""#

!++

!+

!0

$

%%%%%&;

L =!

"#!L

eL

$

%&.

The new contribution to Yukawa coupling:

L!Y = fTLTC"1,- · ,!LL + h.c.;

Gives a term: !TLC"1!0!L;

If < !0L >' eV, then we understand small neutrino

mass.

The big question is:why < !0L >& vwk ?

: TYPE II SEESAW.


29/138 fermilab.tex


Other variations

!

" Fermionic seesaw: Add a triplet fermion ,F which thenallows a new Yukawa of the form L,-H · ,F ;

" Give a large mass to ,F ;

" TYPE III SEESAW;Diagram similar to type I seesaw.

Figure 4: Seesaw diagram

E. Ma


30/138 fermilab.tex


Alternatives to seesaw and ultralightness ofDirac "’s

! Two classes of alternatives to seesaw:

(i) Loop models- populate weak scale with many newparticles e.g. singly and doubly charged bosons- notmotivated by any other physics; then one cangenerate two loop Majorana masses that are small.

(ii) Extra dimensions: Brane-bulk models forfundamental forces and forces;in such models, SM particles are in the brane andRH neutrinos in the bulk. The overlap of the wavefunction is given by M/MP where M is the multi-TeVstring scale- this is a new mechanism forunderstanding the smallness of neutrinos:

Neutrinos are Dirac particles and are accompanied byan infinite tower of RH neutrinos.Such models are highly constrained by cosmologyand astrophysics.


31/138 fermilab.tex


Derivation of neutrino mixings

! Lm(!, e) = 12!

TLC"1M%!L + eLM eeR + h.c.

Diagonalize:

UT% M%U% =

!

"""""#

m1 0 00 m2 00 0 m3

$

%%%%%&;U †

' MeV' =

!

"""""#

me 0 00 mµ 00 0 m!

$

%%%%%&

Neutrino mixing matrix UPMNS = U †' U%

Note that observed mixings encoded in UPMNS

depends on both the charged lepton as well asneutrino mass matrices !

This fact is important while trying to understandneutrino mixings.

Is there any way to separately measure U' ?Not if weak,em and gravity are the only interactionsof leptons.Could be possible for example if there arelepto-quark type interactions or even supersymmetryunder certain circumstances !


32/138 fermilab.tex


! : IMPLICATIONS OF HIGH SCALE TYPE ISEESAW


33/138 fermilab.tex


How many right handed neutrinos forseesaw ?

! One RH neutrino seesaw does not work- it leadsto two massless light neutrinos!!

Either two or three RH neutrinos can give realisticmodels !!

! Advantages and disadvantages of 2 vs 3 case

" 3 !R’s: (i) # of parameters =18 (too many since possibleinputs from nu-mass physics=9)(ii) Complete quark-lepton symmetry- theoreticallyappealing.

" 2 !R’s: (i) # of param. = 12;(ii) however, theoretically less appealing;(iii) Predicts one massless neutrino.


34/138 fermilab.tex


What is the Seesaw Scale ?

! Three theory motivated benchmark values (for 3RH nus):

•: In GUT theories,Mu ' MD and hence mD,33 ' mt;*

!m2A ' m2

tMR

gives MR ' 1014 GeV- near GUT scale.Could m% be the first indication of grand unification ?

If we give up the GUT hypothesis, mD is free andother possibilities arise !!

•: MR " 1011 GeV if mD,33 ' m! ;

Also emerges in theories as MR ",

MWMP

•: MR " TeV if Dirac mass suppressed due tosymmetries:E!ects observable at LHC (see later).


35/138 fermilab.tex


High Scale Seesaw

! Most well motivated due to the way it fits intomany other discussed schemes of beyond thestandard model physics e.g.

•: Simple Grand Unification (see Lecture II);

•: Cosmology e.g. origin of matter, inflation etc.


36/138 fermilab.tex


Testing High Scale type I Seesaw

! Testing seesaw with Lepton Flavor ViolationIn Non-SUSY !R extended standard model (!-SM)with small m% due to high scale seesaw, rare leptondecays e.g. µ % e + + are highly suppressed.

gamma

mu e

W

nu

Figure 5: µ $ e + % in non-SUSY nu-SM

! A(µ % e + +) " eGFmµmem2#

(2m2W

µB

leads to B(µ % e + +) ' 10"50.(Same with other such LFV processes)


37/138 fermilab.tex


Present experimental situation

! B(µ % e + +) # 1.4 $ 10"11 and MEG expt to pushit by at least three orders of magnitude;

B(- % µ + +) # 4.5 $ 10"8: Belle;B(- % e + +) # 1.2 $ 10"8.


38/138 fermilab.tex


LFV in high scale SUSY seesaw models

review and references: Masiero, Vives and Vempati (2004)

! Large neutrino mixings can induce significant- % µ + + and µ % e + +.

With Seesaw + supersymmetry %, superpartnersremember high scale e!ects; large lepton mixings,radiatively inducing large 23 and 12 slepton mixing,and hence significant B(- % µ + +) and B(µ % e + +).


39/138 fermilab.tex


Predictions for LFV

! Formula:B(µ % e + +) = "3

G2FM8

S|(3+a2

0)m20

8(2 |2| ' h†%,µkln

(MUMk

)

h%,ke|2

Typical SUSY models: M 8S " 0.5m2

0m21/2(m

20 + 0.6m2

1/2)2


40/138 fermilab.tex


Predictions for LFV

! Predictions depend on supersymmetryparameters e.g. m1/2,m0, A, tan$ and µ as well as onseesaw scale MR and the Dirac coupling h%.GUT models can fix MR and h% leading to definitepredictions for LFV e!ects(see later). Typicalexpectations are between 10"11 ! 10"14 forB(µ % e + +) and upto 10"7 for B(- % µ + +)

10-18

10-17

10-16

10-15

10-14

10-13

10-12

10-11

10-10 10-9

B(µ

$e%

)

YB

tan&=5m0=m1/2=250GeVA0=-100GeV

'(&M=0 '(&M=)

Figure 6: Figures from Petcov, Rodejohann,Takanishi, Shindou; demonstrates LFV-leptogenesis connection

10-19

10-18

10-17

10-16

10-15

10-14

10-13

10-12

10-10 10-9

B("

$e%

)

YB

tan&=5m0=m1/2=250GeVA0=-100GeV

'(&M=0 '(&M=)


41/138 fermilab.tex


Seesaw, Leptogenesis and origin of matter

See lectures by Plumacher

! why is nB"nBn$

" 10"10?

" If there is CP violation in the lepton sector (in particularin RH neutrino couplings), then

" "(NR % . + H) ! "(NR % . + H) += 0 % leptonasymmetry;

" baryon violation at the weak scale converts the leptonasymmetry into baryon asymmetry.

" Same physics used for generating small neutrino masses.

Fukugita, Yanagida, 1984

! Seesaw cosmologically appealing


42/138 fermilab.tex


Dependence of Leptogenesis on Seesawparameters

! Formula for baryon asymmetry: YB:

YB = ! 10"2/(';/: wash-out factor;(' is lepton asymmetry and is dictated by seesawtheory.

(' " ! 38(

1

(h#h†#)11Im

+(h%h†

%)221

,M1M2

! Key point to notice is that YB depends on h%h†%

whereas LFV depend on h†%h%.


43/138 fermilab.tex


Parameterizing Seesaw

! Seesaw formula: M% = ! v2wkh

T% M"1

R h%:

This formula can be inverted:h%(MR) = 1

vwk

*

MdRR

*

Md% U †

PMNS

Casas, Ibarra (2001)

! where RTR = 1 i.e. a complex symmetricorthogonal matrix:Parameter counting:6(inR) + 6(masses) + 6(anglesandphases) = 18;

Parameterizing R: R = R12R23R31 where

R12 =

!

"""""#

cosz1 sinz1 0!sinz1 cosz1 0

0 0 1

$

%%%%%&and similarly for others.

: A consequence of this is: lepton asymmetry isindependent of PMNS phases observable at lowenergies whereas LFV depends on that.

In actual models however, they are linked !


44/138 fermilab.tex


Lepton edms also as tests of Seesaw,leptogenesis

! In leptogenesis models, CP violation will “sip”down from high scale to the neutrino mixings via theseesaw mechanism and can give rise to e!ects suchelectron and muon edm.


45/138 fermilab.tex


EDM predictions in generic seesaw modelsfor leptogenesis

10-37

10-36

10-35

10-34

10-33

10-32

10-31

10-30

10-29

10-28

300 400 500 600 700 800m1/2[GeV]

EDM

(e c

m)

de

dµA0=0 GeV

300 GeV

800 GeVtan&=10, µ>0

Figure 7: Typical values of electron and muon edm in seesaw models that explain theorigin of matter: (Dutta and RNM, 2003)

! Present expt limits and future possibilities

de : Now: # 7 $ 10"28 ecm; Future: 10"32-10"35 ecm;

dµ : Now: # 3.7 $ 10"19 ecm; Future: 10"24 ! 10"26

ecm;


46/138 fermilab.tex


Radiative corrections from high scaleSeesaw

! • Seesaw mechanism is a high scale e!ect- itspredictions are therefore high scale predictions!Measurements are made at the weak scale; so howto calculate the weak scale values ?

Babu, Leung, Pantaleone; Chankowski, Plucciniek (93); Antusch, Kersten, Lindner and Ratz

! dM#dt = Clh

†lhlM% + M%Clh

†lhl + 'M%

M%(µ) = IC(µ)R(µ)M%(mZ)R(µ).

R - diag(1, 1, Z!(µ)), Z! ! 1 = Clh2

!16(2 log µ

mZ*

m2! tan2 &

8(2v2wk

log µmZ

,

SM: µ = 1010 GeV, (Z! ! 1)SM " 10"5.

MSSM: (Z! ! 1)MSSM - (Z! ! 1)SM tan2 $ Fortan $ = 50, Z! ! 1 ' 0.03 (significant!)

.


47/138 fermilab.tex


Typical e!ect on mixings

! e.g on "13:

"i3(MZ) ""13(MR) + (Z! ! 1)1

4sin2"12sin2"23

-m2+m3m2"m3

+ m1+m3m1"m3

.

(i) Normal hierarchy: e!ect very small i.e. &"13 # 0.01!!

(ii) For inverted or degenerate case, e!ect can belarge !


48/138 fermilab.tex


An interesting application

Parida, Rajasekaran, RNM (2003)

! It could be that just like there is grand unificationof forces, there could be unification of quarks withleptons, possibly implying that at the seesaw scale(or GUT scale),

"qij = "%

ij.What then are the values of neutrino mixings at theweak scale ?

This time look at "12:"12 - !A sin 2"12 sin2 "23

|m1+m2e%12|2

!m221

"13 case given earlier:

For normal hierarchy, this is a small e!ect. But fordegenerate neutrinos, m1,2 ( !m2

12; hence there is abig enhancement and large lepton mixings can arisefrom small lepton mixings at seesaw scale.


49/138 fermilab.tex


An actual computation

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25 30

sin*

ij

t=lnµ

sin*23sin*12sin*13

sin*q23sin*q12sin*q13

Figure 8: Large lepton mixings from small high scale mixings


50/138 fermilab.tex


Can we ever determine all the seesawparameters ?

! In principle, the answer is yes !

Case (i): 3 $ 3 seesaw:

:Measure the following:(a): three m%i, three mixings and three phases [ 9inputs];

(b) leptogenesis: [1 input];

(c) B(µ % e + +); B(- % (e, µ) + +): [3 inputs]

(d) CP violating parameters: µ, e, - edms; [3 inputs]

(e): CP violating asymmetries in µ % e + +;- % (e, µ) + + decays: [3 inputs]

One more than needed.


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Case of 3 ! 2 seesaw

! A more realistic possibility(a): Two m%i, three mixings and one phase [ 6inputs];

(b) leptogenesis: [1 input];

(c) B(µ % e + +); B(- % (e, µ) + +): [3 inputs]

(d) CP violating parameters: µ, e edms; [2 inputs]


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! : UNDERSTANDING LARGE MIXINGS


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How to understand large mixings?

! Mixings are intergenerational and thereforereflect the underlying structure of the mass matrices;

Start with a look at the quark sector. Mass beinghierarchical and small mixings uniquely determinethe structure to have the form:

Mu,d = mu,d

!

"""""#

d(4 a(3 b(3

a(3 (2 (2

b(3 (2 1

$

%%%%%&

where ( ' "C & 1.

Hierarchy makes it easier to think in terms ofsymmetries:


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A toy model for quark mixings with twogenerations

! Think of two generations (Q2, Q3)Land imagine a U(1) family symmetry with q = 0 for3rd and q = ±1 for (L,R) chiralities of the 2nd family;

Let there be a field * that has charge +1 under this;

LY = Q3L0Q3R + )M Q3L0Q2R + )"

M Q3R0†Q2L +)2

M2Q2L0Q2R + h.c.;

If <)>M * ( & 1, the mass matrix becomes:

Mq =!

"#' (2 ' (' ( ' 1

$

%&

the desired hierarchy and small mixings emerge.


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Understanding leptonic mixings is harder !!

! Leptonic case is very di!erent due to largemixings —many ideas but no consensus !!

! Why large leptonic mixings are so di"cult tounderstand–

" Remember: UPMNS = U †' U%;

When net mixings are small, take two hierarchical massmatrices and each with small mixings can combine to givea small mixing !! Case closed !

" When the net mixing is large however, one does not knowif one of them is large or both ?

" when both are large due to a symmetry, the net mixingarises from a cancellation and net mixing is small, contraryto what is desired !

" Since the lepton doublet contains both (!e, e)L,symmetries tend to act the same way on both !e and eL

and may naively tend to cancel. Seesaw can come torescue here.


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An example of why large mixings are sohard to understand:

! Start with a generic symmetric mass matrix fortwo generations:

M =!

"#a bb c

$

%&;

Maximal mixing (e.g. atmospheric case), needs a = c;

Use symmetry 2 ) 3; % " = (4 .

First problem: how to understand m2 & m3 sincethat requires a = b + ( with ( & 1.Not easy to find a symmetry that will do this !

Even if one is willing to accept that ( & 1, SMSU(2)L symmetry would say that most likely chargedlepton mass matrix also has a similar form- butseesaw can do things to the neutrinos despite thisand % large mixings.


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What does one do ?

! Di!erent approaches

" Symmetry approach:Must use seesaw cleverly to make U% di!erent from U'.Also note: Type II seesaw is not related to charged leptonssince it arises from LT!LL and typical Yukawa LH!R.

Keep Charged leptons and neutrino Dirac masses diagonalbut get all neutrino mixings from type II seesaw:

" Dynamical approach:May be the atmospheric mixing is not maximal butdynamically enhanced:• Ex. 1: RGE e!ects enhancing mixing as in the examplegiven;

• Ex. 2: An SO(10) example coming up later !!

May be one should not over emphasize the maximality of"23; recall: central value of tan*

23 " 0.89 and not 1.0.

" Anarchy -!!


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Dynamical underestanding of large mixing

! Since d*ijdt . 1

mi"mj, for deg neutrinos, small angles

at seesaw scale get enhanced;

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25 30

sin*

ij

t=lnµ

sin*23sin*12sin*13

sin*q23sin*q12sin*q13

Figure 9: Large lepton mixings from small high scale mixings

Examples of SO(10) models of dynamical scenariosin Lec. II


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Large mixings and neutrino mass matrix:The symmetry approach

! Model building Strategy:

•: Take a leptonic symmetry;

•: Add additional symmetries to keep chargedleptons diagonal;

•: Use a combination of Type I and II seesaw togenerate large neutrino mixings.

•: Make predictions so that the model can be tested.


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µ " # symmetry

! Two generation example: atmospheric angle

" Suppose µ, - are mass eigenstates

" M% =!

"#a bb a

$

%& % U% = 1%2

!

"#1 1!1 1

$

%&; Maximal

mixing and !m2& & !m2

A % (a ! b) & (a + b)

" Mass matrix for atmosph. angle and solar mass is:

M% =*

!m2A

!

"#1 + ( 1

1 1 + (

$

%&; ( =/0001

!m2!

!m2A

" Symmetric under the interchange of !µ ! !! . Largeatmospheric mixing strongly suggestive of !µ ) !!

symmetry.


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µ " # SYMMETRY for 3 gen.

! M% =*

!m2A

!

"""""#

d(n a( a(a( 1 + ( 1a( 1 1 + (

$

%%%%%&; n / 1

Predicts "23 = 1/4 and "13 = 0;

"12 ' a i.e. large not suppressed by the small number(.

Because of this feature, µ ! - symmetry has receivedconsiderable attention and may indeed have somereality to it.

A clear test of approximate µ ! - symmetry is acorrelation between "13 and "23 ! 1/4-

"13 " (2 " !m2!

!m2A" 0.04 whereas

"23 ! 1/4 ' (


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$13 correlation with $23 " %/4:

! Implications µ ! - symmetry

-0.6 -0.4 -0.2 0 0.2cos2!A

-0.01

-0.005

0

0.005

0.01

0.015!13

Figure 10: Departure from µ % & symmetry and correlation between '13 and 'A

! Magnitude of "13 ia a measure of the extent ofµ ) - symmetry in the neutrino mass matrix.


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$13 in Astrophysics

! Neutrino telescope tests of approximate µ ! -symmetryWhen neutrinos leave an astrophysical source faraway (D / 106 Km away or so), their relative fluxesare determined by neutrino mixings !How to calculate ?Calculate transition probability the usual way-

For large distances when !m2LE ( 1, cosine terms

average to zero.

Remember the initial flux ratio is:00

e : 00µ : 00

! = 1 : 2 : 0.

Observed fluxes on Earth:

#" = 00ePe" + 00

µPµ"


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Flux di!erence and neutrino mass models

! Using the formula given:• Result 1: When "13 = 0 and "23 = 1/4,

0e : 0µ : 0! = 1 : 1 : 1 (Exercise)Independent of the solar angle!!

• Result 2: If "13 += 0 due to breaking of µ ! -symmetry;

0e : 0µ : 0! = (1 ! 2!) : (1 + !) : (1 + !).

• Result 3: If "13 = 0 and "23 = (4 ! (,

0e : 0µ : 0! = (1 + 4!) : (1 ! 2!) : (1 ! 2!).


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Some references for understanding largemixings

!• Exact symmetry: Fukuyama, Nishiura (98); RNM,Nussinov (98); Lam (01);

• Broken µ ! - sym: RNM (04); RNM, Rodejohann(05); de Gouvea (04); Joshipura, Grimus, Lavoura...;Kitabayashi and Yasue (05); Review Rodejohann,Venice 2007 talk.

Astrophysics refs. Xing (06); Rodejohann (06)

• SU(5) embedding of µ ! - : RNM, Nasri and Yu,2006.

• Anarchy approach: Hall, Murayama, Weiner; deGouvea, Murayama; Altarelli, Ferruglio, MasinaRH neutrino dominance: S. F. King, King andSingh;..General analysis in seesaw: Akhmedov, Branco,Rebelo and ABR w/Joaquim


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• Tri-bi-maximal mixing and Highersymmetries

! Remarkable feature of neutrino mixing:

UPMNS =

!

"""""""""#

%2%3

1%3

0

!%

1%6

%1%3

%1%2%

1%36

!%

1%3

%1%2

$

%%%%%%%%%&

! An indication of higher symmetry ?Mass matrix has only 3 parameters

M% =*

!m2A

!

"""""#

( b( b(b( 1 + ( b( ! 1b( b( ! 1 1 + (

$

%%%%%&M0 + &M

where M0 =*

!m2A

!

"""""#

( b( b(b( ( b(b( b( (

$

%%%%%&and

&M =*

!m2A

!

"""""#

0 0 00 1 !10 !1 1

$

%%%%%&

Note M0 has an S3 permutation symmetry and isdiagonalized by UT

tbmM0Utbm but with two degenerateeigenvalues


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and &M which has a lower Z2 symmetry i.e. µ ! -symmetry removes that degeneracy !!

This would suggest that tri-bi-maximal mixing is anindication of broken S3 symmetry i.e.M% = M0(S3) + &M(Z2,µ"!)

One model building strategy: Get the first term fromtype II seesaw and second from type I.


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References for tri-bi-maximal case

Wolfenstein (1978) Harrison, Perkins, Scott (2002); Xing (2002); He andZee (2002)

A4 models: Ma; Babu, He; Altarelli, Ferruglio; Adhikary, et al; Volkas,Low,He, Keum; King, Ross, M-Varzeilas; Grimus, Lavoura, Rodejohann,

Pfletinger

S3 model: Nasri, Yu, RNM (06))


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Inverted hierarchy and Implications

! Inverted hierarchy: m1 " m2 ( m3

Unlike normal hierarchy, inverted hierarchy is moreclosely linked to leptonic symmetries:

To see this, one can write M% " M0 + &M, with

M0 =

!

"""""#

0 m1 m2

m1 0 0m2 0 0

$

%%%%%&.

M0% leads to two opposite parity degenerate states

with masses ±*m2

1 + m22 and tan"1 m2

m1= "A.

This begins to mimic inverted hierarchy. The abovemass matrix has Le ! Lµ ! L! symmetry.


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Le " Lµ " L- symmetry must be broken !

! In the Le ! Lµ ! L! symmetry limit, !m2& = 0 and

"& = (2 .

&M must therefore be very important. Typically, &Melements are large.

&M11 ' 0.5m1,2.

However fits to data exist with such breaking.

One key prediction is lower bound on < m&& >/ 30meV.


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Importance of measuring $13

!"13 "23 ! (

4 preferred model

“large” (/ 0.1) small QLC

small (# 0.05) ',

"13 µ ! - symmetry

"13 = 0 large Scaling

very small very small TBM+ NH

large (/ 0.07) large Minimal SO(10) with 126(see later)

Clearly the importance of "13 for understanding thetrue nature of new physics associated with neutrinoscannot be overstated !!


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p h ysics of neu trin o m as s - fermilabp hysics of neu trino ma ss p h ysics of neu trin o m as s...

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