minimal flavour violation: from quarks to leptons · minimal flavour violation: from quarks to...

Minimal Flavour Violation: from quarks to leptons

Gino Isidori [INFN-Frascati]

Introduction

The flavour sector of the SM & the flavour problem

The EFT approach to Minimal Flavour Violation

MFV in the quark sector

The MFV hypothesis in the MSSM

MLFV: MFV in the lepton sector

CP violation and Leptogenesis

MFV within Grand Unified Theories

Conclusions

Ringberg Workshop Ringberg, Oct. 2006

Minimal Flavour Violation: from quarks to leptons

Gino Isidori [INFN-Frascati]

Introduction




The MFV hypothesis in the MSSM

MLFV: MFV in the lepton sector

CP violation and Leptogenesis

MFV within Grand Unified Theories

Conclusions

Ringberg Workshop Ringberg, Oct. 2006

1st lecture

Introduction

Given the great phenomenological success of the SM up to LEP energies and the limitations / unsatisfactory-aspects of the model above the e.w. scale [ v = ⟨ φ ⟩ ≈ 250 GeV] ⇒ natural to consider the SM as an effective theory

or the low-energy limit of a more fundamental theory with new degrees of freedom appearing above some energy threshold Λ :

The stability of the Higgs sector

naturally points toward Λ ∼ TeV

ℒeff = ℒgauge(Ak, ψi) + ℒHiggs(φ, Ak, ψi ; Y, v ) + ....

G. Isidori – MFV: from quarks to leptons Ringberg – Oct. 2006

⇒ High-energy experiments are the key tool to determine the energy scale of thenew d.o.f. (or the value of Λ) via their direct production

⇒ Low-energy experiments are a fundamental ingredient to determine the symmetry properties of the new d.o.f. via indirect effects in precision observables

Precision measurements of rare flavour-changing processes, both in the quark and in the lepton sector, are a key ingredients

to study the flavour symmetries of physics beyond the SM

ℒeff = ℒgauge(Ak, ψi) + ℒHiggs(φ, Ak, ψi ; Y, v) + Σ Ond (φ, Ak, ψi)

cn

Λd-4d≥5

most general parameterization of the new (heavy) d.o.f. as long as we perform low-energy experiments

ℒSM = renormalizable part of ℒeff [= all possible operators with d ≤ 4

compatible with the gauge symmetry]


3 identical replica of the basic fermion family [ ψi = QL , uR, dR, LL, eR ] ⇒ huge flavour-degeneracy [ U(3)5 group ]

ℒSM = ℒgauge(Ak, ψi) + ℒHiggs(φ, Ak, ψi ; v)



ψi → Uikψk = eiα0 × [eiαΤ] ikψk

|det(U) |= 1 U(1) × SU(3)


Within the SM the flavour-degeneracy is broken only by the Yukawa interaction

QLi YD

ik dRk φ → QL

i MD ik dR

k

QLi YU

ik uRk φc → QL

i MU ik uR

k

Quark sector:

_

_

_

_

_

_





Within the SM the flavour-degeneracy is broken only by the Yukawa interaction

QLi YD

ik dRk φ → QL

i MD ik dR

k

QLi YU

ik uRk φc → QL

i MU ik uR

k

Quark sector:

_

_

MD = diag(md ,ms ,mb)

MU = VCKM × diag(mu ,mc ,mt)

Thanks to the residual flavour symm.: Nowadays we have a rathergood knowledge of all the

10 observables entries[6 masses + 4 CKM angles]

_

_

_

_




No neutrino masses (& mixing angles) unless we extend the model

Lepton sector: LLi YL

ik eRk φ → LL

i ML ik eR

k _ _

ML = diag(me ,mµ ,mτ)





dim.-5 Majorana mass term

right-handedneutrinos

LLi YL

ik eRk φ → LL

i ML ik eR

k



completely equivalentat low energies if we assume

MνR ≫ ⟨φ⟩ [see-saw]

νL νR νR νL

⟨φ⟩ ⟨φ⟩

Lepton sector:_ _




dim.-5 Majorana mass term

completely equivalentat low energies if we assume

MνR ≫ ⟨φ⟩ [see-saw]

Lepton sector:

(LLT)i gLL

ik LL

k φT φ → (LL

T)i Mνik LL

k

1ΛLN

Mν = U* × diag(m1 ,m2 ,m3) × U

+ Pontecorvo-Maki-Nakagawa-Sakata matrix

+ Σ Ond (φ, Ak, ψi)

cn

Λd-4d ≥5

right-handedneutrinos

LLi YL

ik eRk φ → LL

i ML ik eR

k _ _


νL νR νR νL

⟨φ⟩ ⟨φ⟩


ℒeff = ℒgauge(Ak, ψi) + ℒHiggs(φ, Ak, ψi ; v)

The flavour sector beyond the SM

(LLT)i gLL

ik LL

k φT φ 1

ΛLN


cn

Λd-4

The dim-5 operator responsible for neutrino masses is quite special since it violates lepton number

d ≥5

If ΛLN ≫ v some of the gLLik

couplings can

be O(1) ⇒ natural effective theory [no fine-tuning for the smallness of neutrino masses]



(LLT)i gLL

ik LL

k φT φ 1

ΛLN


cn

Λd-4

The dim-5 operator responsible for neutrino masses is quite special since it violates lepton number

d ≥5

If ΛLN ≫ v some of the gLLik

couplings can

be O(1) ⇒ natural effective theory [no fine-tuning for the smallness of neutrino masses]

However, ΛLN ≫ v cannot be the only scale

of NP if we want to avoid the fine-tuning problem of the Higgs sector



The stabilization of the Higgs sector ⇒ Λ ~ TeV [e.g. MSSM]

Λeff ~ 1-10 TeV should be the natutal suppression factor of high. dim. operators

which preserves SM symmetries

flavour problem

if cn ~ O(1) rare processes

already imply bounds on the effective scale of new physics

well above 1 TeV

Most remarkable example of flavour problem in the quark sector:neutral-meson mixing [ ∆F =2 ]

b (s)

s (d)

s (d)

q

u = u,c,t

No SM tree-level contribution

Additional suppression within the SM because of CKM hierarchy

If cnew~ cSM ~1, in order to maintain the agreement between data and SM predictions:

Λ > 104 TeV for O (6) ∼ (s d)

2 Λ > 103 TeV for O (6) ∼ (b d)

2

b (s)

(yt Vtb*Vtd)2

16 π2 ΜW2

E.g.: M(Bd-Bd) ~ cSM + cnew

1

Λ2

_

[ K0-K0 mixing ] [ B0-B0 mixing ]

_ _

_ _


A very similar (numerically even more serious) problem is present also for FCNC processes in the lepton sector:

c e

2eL

R F

BR(µ→eγ) = 9×105 |cµe|21 TeV

Λ

4

ℒeff ⊂

BR(µ→eγ)exp < 1.2×10-11 Λ > 105 TeV × (cµe)1/2

_


Two possible solutions:

pessimistic: ΛFCNC

≫ v Higgs fine-tuning problem ignored [no new d.o.f.

carrying flavour quantum numbers close to the e.w. scale - e.g. split SUSY - ]

natural: ΛFCNC

~ 1 TeV + flavor-mixing protected by additional symmetries

⇒ still a lot to learn from flavour physics and particularly rare decays[we need to determine the underlying flavour symmetry in a model-independent way]


Two possible solutions:

pessimistic: ΛFCNC

≫ v Higgs fine-tuning problem ignored [no new d.o.f.

carrying flavour quantum numbers close to the e.w. scale - e.g. split SUSY - ]

natural: ΛFCNC

~ 1 TeV + flavor-mixing protected by additional symmetries

⇒ still a lot to learn from flavour physics and particularly rare decays[we need to determine the underlying flavour symmetry in a model-independent way]


Minimal Flavour Violation (MFV) hypothesis:

The breaking of the flavour symmetry occurs at very high scales and is mediated at low energies only by terms proportional to SM Yukawa coupl.

Natural implementation in many consistent scenarios [SUSY, technicolour, extra dimensions,...]

Possible to build a predictive low-energy EFT model-independent approach

most restrictive possibility [strongly suggested at least in the quark secor]

The maximal group of unitary field transf. allowed by ℒgauge is: SM

GF = U(3)5 = SU(3)2× SU(3)3×U(1)PQ×U(1)eR×U(1)B×U(1)L×U(1)Yl q

U(1)PQ: glob. phase of DR & eR

U(1)ER : glob. phase of eR SU(3)3 = SU(3)Q

L×SU(3)U

R×SU(3)D

Rq

groups relevant in multi-Higgs models

[overall Yukawa normalization]

subgroup responsible for quark mixing[Yukawa structure, CKM matrix]

SU(3)2 = SU(3)LL×SU(3)e

Rl

SMsubgroup broken by ℒYukawa

ℒYukawa = QL YD DR φ + QL YUUR φc + LL YL eR φ + h.c._ _ _


I. Definition of the Flavour Group


Since GF is already broken within the SM, it is not consistent to impose it as anexact symmetry beyond the SM.

However, we can (formally) promote GF to be an exact symmetry, assuming the Yukawa matrices are the vacuum expectation values of appropriate auxiliary fields:

_ _E.g.: YD ~ (3,1,3) & YU ~ (3,3,1) under SU(3)Q

L×SU(3)U

R×SU(3)D

R

(1,1,3) (3,1,1) _

(3,1,3) _

(1,1,1)

II. Definition of the symmetry-breaking terms

ℒYukawa = QL YD DR φ + QL YUUR φc + LL YL eR φ + h.c._ _ _


unknownflavour−blind

dynamics

⟨Y ⟩ ⟨Y ⟩

ΛF

breaking of GF

by means of ⟨Y ⟩

Λ (~ TeV)

flavour-blind dynamics [non-SM degrees of freedom

stabilizing the Higgs potential]

SM degreesof freedom

natural cut-off scale of the EFT


unknownflavour−blind

dynamics

⟨Y ⟩ ⟨Y ⟩

ΛF

breaking of GF

by means of ⟨Y ⟩

Λ (~ TeV)

flavour-blind dynamics [non-SM degrees of freedom

stabilizing the Higgs potential]

SM degreesof freedom

natural cut-off scale of the EFT

D'Ambrosio,Giudice, G.I., Strumia '02

III. The rules of the game

A low-energy EFT (including only SM fields) satisfies the criterion of MFV if all higher-dimensional operators, constructed from SM and Y fields, are (formally) invariant under GF


YD = diag(yd ,ys ,yb) YU = V+ × diag(yu ,yc ,yt)

CKM matrix (= the only source of quark mix.)

Typical FCNC dim.-6 operator: QLi (YUYU

+)ij QLj × LL LL

_ _

We can always choose a quark basis where:



(3,3,1) _

(1,1,1) SU(3)Q

L×SU(3)U

R×SU(3)D

R

(3,3,1) _


YD = diag(yd ,ys ,yb) YU = V+ × diag(yu ,yc ,yt)

CKM matrix (= the only source of quark mix.)

We can always choose a quark basis where:




same CKM - Yukawa structure of the SM contribution !

(YU YU+)ij ≈ yt

2 V3iV3j *

V+ × diag( yu

2, yc2, yt

2 ) × V

≈ V+ × diag(0, 0, yt

2) × V

Typical FCNC dim.-6 operator: QLi (YUYU

+)ij QLj × LL LL

_ _

N.B.: the CKM as only source of flavour mixing is a consequence of the

MFV hypothesis



λFC is the effective coupl. ruling all FCNCs with external d-type quarks

[as within the SM for s.d. dominated amplitudes]

All FCNC amplitudes have the same CKM structure as in the SM [e.g.: A(bsγ) ∝VbtVts, A(sdγ) ∝VstVtd, ...]

Only the flavour−independent magnitude of FCNC amplitudes can be modified by (non−standard) dimension−six operators

Phase measurements [e.g.: a(BψKS), a(BφKS), ∆MBd∆MBs ]

are completely unaffected by new physics

[ (YU YU+)n ]ij ≈ (YU YU

+)ij ≡ ( λFC )ij ≈ yt2 V3iV3j

*


Comparison with other definitions/approaches to MFV:

Early attempts based on (what I call) the pragmatic MFV definition [Buras et al. '01]:- no other FCNC operators beyond the SM ones ; - NP modifies only the Ci and not the CKM factors of FCNC operators;

Equivalent, in practice, to our approach [D'Ambrosio et al. '02] under the additional assumption of a single light Higgs doublet.

However...

Not based on a symmetry principle ⇒ not clear how to extend this definition to cases with additional degrees of freedom (2HDM, SUSY, etc...)

Focus on the CKM matrix rather than on the Yukawa couplings is very misleading (and not even consistent when subleading Yukawa entries are taken into account).

The idea that Yukawa couplings are the only relevant sources of flavour symmetry breaking has been implemented in specific NP frameworks, such as technicolor [Chivukula & Georgi, '87] or supersymmetry [Hall & Randall, '90], since a long time.

What is more recent is the attempt to give a general (model-independent) definition/description of this class of models.

⇒ Ι. The MFV hypothesis is very powerful (restrictive) !

within this framework the flavour problem is basically solved

Recent update M. Bona et al. '05

2.6 4.9

6.0 6.1

2.3 3.8

4.2 3.8

4.5 4.0

2.5 2.5

Bounds on the scale of New Physics within MFVmodels:

The bounds are typically weaker (or at most as stringent as) those obtained from flavour -conserving e.w. dynamics [~ 5-10 TeV]

⇒ II. We are slowly entering in the interesting region...

e φ

φ+ iDµφ

g φ


0.9

1.6

2.8


⇒ ΙΙΙ. The MFV hypothesis leads to (testable) predictions on FCNCs

A violation of these prediction would be an unambiguous signal of new (non-Yukawa) sources of flavour symmetry breaking

link between b s , b d , s d

Given we have not observed any deviation from the SM, so far these predictions are only upper bounds on possible deviations from the SM:


Possible extensions/generalizations of this approach:

I) Inclusion of extra Higgs doublets ⇔ breaking of the U(1)'s

The breaking of the SU(3) groups is not necessarily related to the breaking of the U(1)'s

With two Higgs doublets we can assume the Yukawa interaction is invariant under U(1)PQ

⇒ the CKM matrix remains the only source of quark mixing, but we are free to modify the overall normalization of the Y

(YDYD+)ij ≈ yb

2 δ3iδ3j potential new non-negligible

source of GF-breaking if tanβ = ⟨HU⟩⟨HD⟩ ≫ 1

II) Inclusion of SUSY degrees of freedoms ⇒ EFT valid to higher scales

⋮

ℒYukawa = QL YD DR HD + QL YUUR HU + LL YL ER HD + h.c._ _ _


MFV within the MSSM

Basic principles of Supersymmetry:

Q∣fermion⟩ = ∣boson⟩Q∣boson⟩ = ∣fermion⟩

{Q, Q+} = Pµ

{Q, Q} = {Q+, Q+} = 0[Q, Pµ] = [Q+, Pµ] = 0

Main "phenomenological virtue":

"−" "="

no power dependence from the cut-off

in the Higgs potentialH

boson fermion

∆v2 ~ [ Mb2 - Mf

2 ] g2

16π2SUSY breaking must occur not far from the e.w. scale

The existence of a large energy gap [the "big hierarchy"] between the electroweak [& SUSY-breaking] scale

and the Planck [or GUT, or Flavor-breaking, ....] scale is not a technical problem

⇒


The Minimal Supersymmetric extension of the SM includes:

scalar partners of the ordinary quarks and leptons [QL, uR , ...]

spin-1/2 partners of the ordinary gauge bosons [gauginos]

Two Higgs doublets [HU , HD] with their corresponding spin-1/2 partners

~ ~

N.B.: the presence of (at least) two Higgs doublets is mandatory:cancellation of triangular gauge anomalies induced by the higgsinos

SUSY invariance forbids non-analytic Yukawa couplings of the type QL Y UR (H)*

_


The Minimal Supersymmetric extension of the SM includes:

scalar partners of the ordinary quarks and leptons [QL, uR , ...]

spin-1/2 partners of the ordinary gauge bosons [gauginos]

Two Higgs doublets [HU , HD] with their corresponding spin-1/2 partners

~ ~

N.B.: the presence of (at least) two Higgs doublets is mandatory:cancellation of triangular gauge anomalies induced by the higgsinos

SUSY invariance forbids non-analytic Yukawa couplings of the type QL Y UR (H)*

_

The SUSY version of ℒgauge is completely determ. by its symmetry properties

The SUSY version of ℒYukawa is also strongly constrained:

ℒYMSSM = QL YD DR HD + QL YUUR HU

+ QL YD DR HD + QL YUUR HU + ...

_ _

~+ ~ ~+ ~

HDQL

DR

HDQL

DR

~ ~+

HDQL

DR

~+~

+ 4 bosoncouplings


All the "difficulties" of the theory (e.g. a large number of free parameters) are hidden in the so-called soft-breaking sector:

potential new sources offlavor-symmetry breaking

ℒsoft = (Mf)ij χi χj + (Ms

2)ij φi φj + Aijk φiφiφk

gaugino/higgsino masses

squark/slepton masses

trilinear scalarcouplings



potential new sources offlavor-symmetry breaking

N.B.: while for the SM quarks [Dirac fermions] only LR mass terms are allowed, in the case of the s-quarks [scalars] all possibilities LL, LR and RR are allowed ⇒ 6×6 mass matrices.

M 2LL M 2LR

(M 2LR)+ M 2RR













General MFV prescription: (M2)LL ∝ Σ an (YU YU+)n

~ a0 I + a1YU YU+

The MFV hypothesis provides a strong restriction on the favour structure of these terms

E.g.: (M2)LL QL QL ~+ ~

coefficients linked by RGEfully determined only if the UV completion of the theory is known

Hall & Randall, '90D'Ambrosio et al. '02








(M2)LL ∝ I [universality] (M2)LL ∝ diag [a,b,c]

compatible with the MFV hypothesis(sufficient condition)not RGE invariant

not compatible with the MFV hypothesis (new source of GF breaking)not RGE invariant

General MFV prescription: (M2)LL ∝ Σ an (YU YU+)n

~ a0 I + a1YU YU+

To be compared with more specific ansatz:

The MFV hypothesis provides a strong restriction on the favour structure of these terms

E.g.: (M2)LL QL QL ~+ ~


Which is the effective scale Λ in the (MFV) MSSM?

The largest effects are expected in gluino-mediated amplitudes:

qLi

qLj

qLi

qLj

αS2(MS

2)

MS2~

~

[ vs. Λ0 = Mw αw(Mw) ≈ 2.4 TeV ]

ΛS ≈ 10 MS ~

If the SUSY-breaking scale coincides with the e.w. scale,

we should expect ΛS ~ 1 TeV – 10 TeV

q q~ ~

g ~

g ~


Which is the effective scale Λ in the (MFV) MSSM?

The largest effects are expected in gluino-mediated amplitudes:

qLi

qLj

qLi

qLj

αS2(MS

2)

MS2~

~

[ vs. Λ0 = Mw αw(Mw) ≈ 2.4 TeV ]

ΛS ≈ 10 MS ~


we should expect ΛS ~ 1 TeV – 10 TeV

q q~ ~

g ~

g ~

MS

MW vanishing mass in the limit of unbroken SU(2)L

tZ

⋮⋮However,

since we have not seen any SUSY partner yet, the situation maybe more complicated...

"little hierarchy"


2.6 4.9

6.0 6.1

2.3 3.8

4.2 3.8

4.5 4.0

2.5 2.5

e φ

φ+ iDµφ

g φ


we should expect ΛS ~ 1 TeV – 10 TeV...

⇒ We are slowly entering in the interesting region for MFV-SUSY models...

minimal flavour violation: from quarks to leptons · minimal flavour violation: from quarks to...

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