Neutrino Physics III (Additional Material)

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Neutrino Physics III (Additional Material)

Neutrino mass models

Arcadi Santamaria

IFIC/Univ. València

TAE 2014, Benasque, September 19, 2014

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Neutrino Physics III Outline

1 Introduction

Effective Lagrangian Approach

2 Tree-level Majorana Masses See-saw Types I,II,III

Variations (Inverse See-saw)

3 One-loop Majorana Masses The Zee Model

The Inert Doublet (Inert)

4 Two-loop Majorana Neutrino Masses Double W Exchange contributions The Zee-Babu Model

5 Masses not Described by the Weinberg Operator

6 Masses in extensions of the SM Grand Unification (GUT) Supersymmetry

Extra Dimensions

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Neutrino Physics III Outline

1 Introduction

Extra Dimensions

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Neutrino Physics III Outline

1 Introduction

Extra Dimensions

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Neutrino Physics III Outline

1 Introduction

Extra Dimensions

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Neutrino Physics III Outline

1 Introduction

Extra Dimensions

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Neutrino Physics III Outline

1 Introduction

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1 Introduction

2 Tree-level Majorana Masses

3 One-loop Majorana Masses

4 Two-loop Majorana Neutrino Masses

6 Masses in extensions of the SM

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Masses of Neutrinos in Models BSM

In the SM the neutrinos do not have mass by several reasons:

(a)There are notν_R

(b)It only contains a doublet of scalars

(c)Renormalizable theory (interactions with dim≤4) (a)=⇒Neutrinos can not have Dirac masses

(b)+(c)=⇒LandBconserved exactly (perturbativelly)

=⇒no Majorana masses m_ν 6=0 requires to abandon(a),(b)or(c)

We will begin by(c), the most radical but the most general.

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Effective Lagrangian Approach

If the new particles are much heavier thanm_W their effect can be parametrized at low energies by an effective Lagrangian

L =LSM+

∞

∑

n=5

∑

i

C_i⁽ⁿ⁾

Λⁿ⁻⁴O_i⁽ⁿ⁾+h.c.

!

withC_i⁽ⁿ⁾coupling constants andΛthe new physics scale IfE,mΛ, effects ofO_i⁽ⁿ⁾ suppressed by powers ofE/Λor m/Λ

(They decouple and we recover the SM).

Effects of new physics dominated by thelowest dimension operators.

Arcadi Santamaria Neutrino Physics III (Additional Material) TAE 2014, Benasque, September 19, 2014 2/33

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Weinberg Operator

Only a dimension 5 operator (Weinberg operator) which does not conserve LN

O⁽⁵⁾= (˜L_LΦ)( ˜Φ^†L_L) withL˜_L=iτ₂L^c_LandΦ =˜ iτ₂Φ^∗

(transform well under Lorentz and SU(2)) After SSB (hΦ⁽⁰⁾i=v/√

2) in the unitary gauge O⁽⁵⁾→ −1

2v²ν_L^cνL

For three generationsC⁽⁵⁾

α β has family indices (M_ν)_{α β} =C⁽⁵⁾

α β

v² Λ

To obtainm_ν <1 eVone needsΛ>10¹⁴GeVifC⁽⁵⁾∼1

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Most neutrino Majorana masses can be paremetrized by the Weinberg operator, but

Not always!

Not, if the models contain particles lighter than the electroweak scale

Light Dirac neutrinos Light sterile fermions Majorons, Axions,. . . Additional light scalars

Not, if the scalar discovered is not exactly the SM Higgs.

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Most neutrino Majorana masses can be paremetrized by the Weinberg operator, but

Not always!

Not, if the models contain particles lighter than the electroweak scale

Light Dirac neutrinos Light sterile fermions Majorons, Axions,. . . Additional light scalars

Not, if the scalar discovered is not exactly the SM Higgs.

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1 Introduction

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Opening the Weinberg Operator at Tree Level

If we want to generate the Weinberg operator at tree level with renormalizable interactions, need to open the effective vertex:

Fermionic lines do not branch Only Yukawa interactions allowed

Only 2 topologies (Notation:Ψ^(Y)_T ,Ψfermion andΦscalar)

ℓL

Φ⁽¹⁾1

φ ℓL

φ φ

Ψ⁽⁰⁾1,0

ℓL

φ

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L_LΦ→Ψ⁽⁰⁾_1,0

L_LL_L→Φ⁽¹⁾_1,0,Φ⁽¹⁾₀ does not have neutral component and does not give mass at tree level. Besides there is no Φ⁽¹⁾₀ ΦΦcoupling with only one doublet

L_LΦ^∗→Ψ⁽¹⁾_1,0, has hypercharge, cannot have Majorana mass

L_L¯L_L→Φ⁽⁰⁾_1,0by chirality does not couple to scalar. Besides it does not carry LN

Only three possibilitiesΨ⁽⁰⁾₀ ,Φ⁽¹⁾₁ ,Ψ⁽⁰⁾₁ Ψ⁽⁰⁾₀ Fermion singlet withY =0 Φ⁽¹⁾₁ Scalar triplet withY =1 Ψ⁽⁰⁾₁ Fermion triplet withY =0

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See-saw Type I

As discussed already we consider the caseν_R= Ψ⁽⁰⁾₀ LY =−L¯_LYeΦe_R−¯L_LY_νΦν˜ _R−1

2ν_R^cMν_R+h.c.

We will considernνR fields and only three families ofL_L. Then Misn×nsymmetric matrix andY_ν a 3×nmatrix. After (SSB)

Lνmass=−1 2

νL ν_R^c





0 M_D M_D^T M







 ν_L^c νR



+H.c. ,

WithM_D=Yv/√

2. IfMM_D, diagonalized by blocks M_ν^†' −M_DM⁻¹M_D^T

M_N'M

andM_ν will be small.M_ν has, in general, rank min(3,n)

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Comparing with the Weinberg operator (identifyingΛwith the smaller eigenvalue ofM,M₁= Λ)

C^(5)†=−1 2Y_νM₁

M Y_ν^T

Small neutrino masses withM very large and/orY_ν very small In any case the effects of theν_R extremelly suppressed:

Difficult to check!

ThecaseM.m_Z cannot be described by the Weinberg operator,in particular

M=0 Dirac neutrinos

0<Mm_Z sterile neutrinos (new mass scales: effects in oscillations)

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See-saw Type II

Consider the caseχ= Φ⁽¹⁾₁ (triplet scalar withY =1)

χ=



 χ⁺/√

2 χ⁺⁺

χ0 −χ⁺/√ 2





with Lagrangian

Lχ=−

L˜_LY_χχL_L+h.c.

−V(φ,χ)

V(Φ,χ) =m²_χTr{χ χ^†}+

µΦ˜^†χ^†Φ+H.c.

+. . . Y_χ is a symmetric matrix. Allow to assign LN−2 to theχ µ breaks LN explicitly

Ifµ=0 andm²_χ<0,LN spontaneously broken (Goldstone theorem===⇒⇒⇒Majoron)

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Ifµ6=0 a VEV for the triplet is induced even ifm_χ>0

χ⁽⁰⁾

hφ⁽⁰⁾i hφ⁽⁰⁾i In the limitm_χv

hχi ≡v_χ' −µv² 2m²_χ And therefore

M_ν =−2Y_χv_χ=Y_χµv² m²_χ Comparing with the expression obtained with the Weinberg operator (withm_χ= Λ)

C^(5)†=Y_χ µ m_χ

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Triplet scalar VEV’s contribute to theρ parameter ρ= m_W²

m²_Zcos²θ_W =hΦi²+2hχi²

hΦi²+4hχi²≈1−2hχi²

hΦi² ≈1−2µ²hΦi² m⁴_χ ρ=1.0008^+0.0017₋_0.0010 andhχi.3GeV.

Can be achieved easily by takingµ m_χ and/orm_χ hΦi. Possible to adjust the masses of the neutrinos with values of m_χ acccessible at the LHC andY_χ not too small

(m_ν∝Y_χ(µ/m_χ)(hΦi/m_χ)three-factor suppression)

In this case the model produces a very rich phenomenology tied to the masses of the neutrinos:

Production at the LHC. Doubly charged scalar easy.

Processes with LNVin the charged sector

µ→eγ,τ→µ γ,µ →3e,τ→3e,µ2e,2eµ,3µ,µ ↔ein nuclei,. . .

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See-saw Type III

Takezero hypercharge fermion tripletsΣ = Ψ⁽⁰⁾₁

(Will need at least 2 to have 2 light massive lneutrinos) In cartesian components

~Σ = (Σ₁,Σ₂,Σ₃)

Σhas family indices. Under LorentzΣtransform like two component right-handed fields. The Lagrangian is

LΣ=i~ΣD/~Σ− 1

2~Σ^cM~Σ + ¯`Y(~Σ·~τ) ˜φ+h.c.

!

and after SSB Lνmass=−1

2

ν_L Σ^c₃





0 M_D M_D^T M







 ν_L^c Σ₃



+h.c.

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Same results for the mass as in see-saw type I M_ν^†' −M_DM⁻¹M_D^T.

Charged components are combined into a Dirac field E⁻= 1

√2(Σ₁+iΣ₂+ Σ^c₁+iΣ^c₂) with massM(present limits giveM&100GeV)

TheE mix with charged leptons giving rise to tree-level FCNC (proportional toM_D/M1, therefore supressed)

~Σhave gauge interactions: much easily produced thanν_R . Masses of neutrinos like see-saw type I

New charged particles, more restricted (M>100GeV) Gauge Interactions

Much richer phenomenology

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Variations (Inverse See-saw)

See-saw I very general: adding more sterile fermions one obtains qualitativelly different pictures.

A peculiarity of see-saw I,III: active-sterile mixing is order M_D/M while mass of active isM_D²/M

=⇒

==⇒⇒Small masses require small mixings!

Interesting variation: 3ν_L, 3ν_R And 3s_Lwith mass matrix (in the basisν_L,ν_R^c,s_L)







0 M_D 0

M_D^T 0 M

0 M^T µ







M_D=Y_νhΦiandY_ν standard Yukawas

Mandµ mass matrices of singlets (µ symmetric)

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νR νR

νL νL

sL

Φ Φ

Ifµ=0 LN conserved

Spectrum contains 3 masslesν’s and 3 heavy Dirac.

IfµM_DM: the 3 heavy split into 2×3 pseudo-Dirac the 3 light obtain a mass

M_ν^†≈M_DM⁻¹µ(M^T)⁻¹M_D^T while the active-sterile mixing remains∼M_D/M:

masses and active-sterile mixing decoupled!

The active neutrino masses can be small and the active-sterile mixing big!

Interesting phenomenology

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1 Introduction

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One-loop Majorana Masses

To have small masses from the Weinberg operator one needs Λ hΦiorC⁽⁵⁾1:

IfΛ hΦidifficult to distinguish the different models, the only effect is the generation of the Weinberg operator (higher dimension operators are supressed)

IfC⁽⁵⁾1only in the Weinberg operator (which does not conserve LN),

=⇒

=⇒New accessible physics not tied to neutrino masses.

But, what is thereason forC⁽⁵⁾1?

If there are no fields with the quantum numbers of the see-saw Ψ⁽⁰⁾₀ ,Φ⁽¹⁾₁ ,Ψ⁽⁰⁾₁ masses cannot be generated at tree level But, if LN is not conserved the Weinberg operator will be generatedat one loop (or more).

This gives a natural justification forC⁽⁵⁾1

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Opening the Weinberg at One Loop

Topologies 1PI

Topologies that are not 1PI can be reduced to variations of the see-saw I,II,III.

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The Zee Model

Adds to the SM a scalar singleth⁺and a new doubletΦ⁰ h⁺∼(0,1), Φ⁰∼(1

2,1 2) L =LSM+LZee.The relevant terms terms are

LZee= ˜L_Lf L_Lh⁺+µh⁺Φ^†Φ˜⁰+···

f_ab antisymmetric 3×3 µ can be taken real

Theµ coupling only exists for two different doublets (iτ₂is antisymmetric).

Variationsdepending on the Yukawa couplings of the new doubletΦ⁰

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φ⁺

e_R eL

h⁺

φ^′

A finiteM_ν is generated at one loop.

Simplest versions (with onlyΦcoupling to leptons) give M_ν ∼hΦihΦ⁰iµ

(4π)²m²_h

f Y Y^†+Y^∗Y^Tf

Y charged lepton Yukawas (can be taken diagonal)m_`=Y_``v.

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(M_ν)_{α β} ∝f_{α β}(m²_α−m²_β)

Zero entries in the diagonalbecause thef_{α β} antisymmetry!

This structure predicts maximal mixing for the solar angle (in fact too maximal).

Thesimplest version “too predictive”and cannot explain the spectrum of masses and mixings

More complicated versions of the model can fit all the data Smallm_ν by takingµ small and/orf small and/orm_hlarge Ifµ small (natural):

hcould be seen at the LHC

Simplest version has no 0ν β β ((M_ν)_ee=0)

0ν β β linked the departure of maximal mixing in the solar angle.

Rich LFV phenomenology (µ→eγ,τ→eγ, . . . )

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The Inert Doublet

Add a fermion singlet (like theν_R)s_R, and scalar doubletη s_R∼(0,0) η∼(1

2,1 2) with (in addition to the SM couplings ofΦ)

LY =−¯L_LY_sη˜s_R−1

2s^c_RMs_R−λ5 Φ^†η2

+···+h.c.

Like see-saw type I ifη↔Φands_R↔ν_R, but

η does not take VEV (m²_η >0), cannot generate masses.

Discrete symmetryη→ −η ands_R→ −s_Rforbids the couplingss_R-Φ

LNassigned as (L(η) =−1,L(s_R) =0)brokenexplicitly in the potential by theλ5term

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ΦWeinberg op. cannot be generated at tree level W. op. generated for theη, buthηi=0

=⇒

==⇒⇒no tree-levelm_ν

LN is broken (byλ5): Majorana masses will be generated

η sR

η η

M_ν^†∼λ5hΦi²

(4π)² Y_sM⁻¹Y_s^T, m_η M Ifm_ηM (changeM⁻¹→M/m²_η) M_ν like in see-saw but with a supressionλ₅/(4π)²

ΦWeinberg op. induced by theη Weinberg op. at one loop!

Charged particles could be seen in theLHC

The discrete symmetry makes the lightest ofs_R andη’s stable: it could be a gooddark mattercandidate

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1 Introduction

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Double W Exchange Contributions

If one of the neutrinos is Majorana and mixes with the others it will necessarily generate two-loop masses for the others

νi

νj

eα

W

W eβ

ν4, ν¯4

Ifν₄mass is generated by see-saw I (M_ν)⁽²⁾_ij =− g⁴

m⁴_Wm_Rm²_D

∑

α

V_αiV_α4m²_α

∑

β

V_βjV_β4m²_βI_{α β},

I_{α β} two-loop integralm_α,β masses charged leptons.

If neutrino masses are Majoranam_lightest cannot be zero!

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The Zee-Babu Model

Minimal model: add only two scalar singlets to the SM h⁺∼(0,1), k^±±∼(0,2)

L =LSM+LZB.The relevant terms are

LZB= ˜L_Lf L_Lh⁺+e^c_Rg e_Rk⁺⁺+µ(h⁻)²k⁺⁺+···

f_ab Antisymmetric 3×3 g_ab Symmetric 3×3 µ Can take real

f coupling allows to assign LN 2 to theh⁻ gcoupling allows to assign LN 2 to thek⁻⁻

µ couplingbreaksLN in 2 units===⇒⇒⇒Majorana neutrino masses

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h⁻

eR k⁻⁻

νL eR

eL eL

νL h⁻

hHi hHi

Mass arises attwo loops, calculable andFinite!

M_ν= hΦi²µ 48π²M²

˜I f Y g^†Y^Tf^T, M≡max(m_h,mk), ˜I≈1 Y charged lepton Yukawa coupling.m_`=Y_``v.

det(Mν) =0:one of theν’s is massless The corresponding eigenvector is fixed:

feτ/f_{µ τ},feµ/f_{µ τ} fixed in terms of mixings

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Main features

k⁺⁺easily producedby Drell-Yan

Decays mainly to leptons (ifm_k>2m_halso to 2h⁺)

γ^∗, Z^∗

q q

k⁻⁻

k⁺⁺

`⁻_a →`⁺_b`⁻_c`⁻_d: limits ong_ab

τ⁻

µ⁺

µ⁻

µ⁻ k⁻⁻

`a→`_bνν: limits on¯ f_ab

µ⁻

¯ ν

e⁻ h⁻

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`⁻_a →`⁻_bγ : constraintsg_ab andf_ab

h⁺, k⁺⁺

µ⁻ ν, e⁺ e⁻

γ

Testable Model!

Predictions onν masses and mixings LHC (nowm_k>200–400GeVat 95% CL)

Many LFV process which should be large ifm_k,hare discovered

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1 Introduction

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Models with Light Sterile Neutrinos

The Weinberg Op. parametrizes a large class of models but!

Does not parametrizeDirac neutrinos!

Does not parametrizelight sterile neutrinos Minimum model 3ν_Land 2ν_R:

If∆L=0 (noν_R Majorana mass):

1 masslesνL

2 massive Dirac neutrinos if∆L6=0 (ν_Rwith Majorana masses)

1 masslessν_L(will get a two-loop Majorana mass) 4 massive Majorana neutrinos ( 4 mass differences) Scenario not excluded at present, could explain LNSD-MiniB, reactor and Gallium anomalies andNsin cosmology!

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Majoron Models

In models with Majorana masses there is always a dimensionful parameter associated with the breaking of LN

I can be promoted to a complex field:

LN spontaneously broken .Example:

ν_R^cMν_R→hσ ν_R^cν_R σ→ 1

√2(v_σ+ρ)e^iθ/v^σ θGoldstoneboson “Majoron”. It survives at low energies Majoron-neutrino couplings (Goldstone theorem)

m_ν v_σ ν γ5ν θ Possibility of decays and annihilations

ν→ν⁰θ, ν ν→θ θ

Could avoid cosmological bounds(modification of primordial abundances, CMB, LSS, BBN)

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1 Introduction

Extra Dimensions

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Extensions of the SM

Apart from the neutrino mass problem, the SM has a series of unsatisfactory aspects:

Only 3 generations? Why?

Hierarchy problem

Several gauge coupling constants

Flavour problem (hierarchies of masses and mixings) It does not includes gravity

It can not explain the baryon asymmetry of the universe It does not have a good dark matter candidate

Several theories have been proposed to solve these problems.

Theyhave to incorporate neutrino masses!

The mechanisms will be generalizations of the mechanisms discussed above (after all these theories should behave like the SM at low energies)

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Grand Unification (GUT)

Unification based in SU(5) is like the SM (ν_R singlet under SU(5))

Unification based in SO(10) very interesting:

ν_Rmember of the 16 representation which fits a complete family of the SM

Nc

z}|{3







Q_L

z}|{2 +

d_R

z}|{1 +

u_R

z}|{1





+

L_L

z}|{2 +

e_R

z}|{1 +

νR

z}|{1 =16

Automatic cancellation of anomalies

B−Lis a generatorof the group, necessarily broken spontaneously (Majorana neutrino masses)

GUT relations between the Yukawas of the top and of the neutrino

Neutrinos in SO(10) were the inspiration of the see-saw!

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Supersymmetry

In the Minimal Supersymmetric Standard Model (MSSM)LN conservation imposed“by hand” when imposing the

conservation of the parityR,(−)^3B+L+2j.

Many terms in the superpotential could break explicitlyL Le^cΦ1, LLe^c ,LΦ2 ,LQd^c

Masses of neutrinos even withoutν_R!

SUSY containsB,˜ W˜ which just have the quantum numbers of the see-saw I,III and and also Majorana masses!

hν˜i

B,˜ W˜3

νL

h˜νi

B,˜ W˜3

˜

˜ ν ν νL

hΦi hΦi

νL

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Extra Dimensions

In extra dims couplings can be small because we only see

“The tip of the iceberg”. In 5 dims Yukawa couplings are not dimensionless [Y_d=5] =M⁻^1/2. (M Natural scale of interactions in 5 dims). Then, size of Yukawas in 4 dims

Y_d=4∼p

McY_d=5∼ rMc

M 1 In 5 dims γ₅ is γ₅, (no chirality). Neutrinos are naturally Dirac with small masses.