Supersymmetry breaking Supersymmetry breaking as the origin of flavor as the origin of flavor

(from empirical formulas for the fermion spectra to (from empirical formulas for the fermion spectra to radiative fermion mass generation)radiative fermion mass generation)

Javier FerrandisJavier Ferrandis

IFICIFIC

Barcelona, January 12th 2005Barcelona, January 12th 2005

J.F ph/0406004 PRD70J.F ph/0406004 PRD70J.F. ph/0404068 PRD70J.F. ph/0404068 PRD70J.F & N. Haba ph/0404077 EPJCJ.F & N. Haba ph/0404077 EPJC

Outline

• I will argue that there is evidence for low energy empirical formulas that connect six dimensionless fermion mass ratios and the CKM elements

• There is a plausible reconstruction of the underlying SM Yukawa matrices that accounts for these empirical formulas

• I will present an effective SUSY GUT flavor model for the radiative generation of 1st and 2nd generation of fermion masses and mixing angles that can explain some of the features of the reconstructed Yukawa matrices

3

Some precision analysis of SUSY GUT Some precision analysis of SUSY GUT modelsmodels

YU =yt

0 ′bε 3 ′c ε 4

′bε 3 ε 2 ′a ε 2

′c ε 4 ′a ε 2 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

ht ,hb( ) ≈ mt,mb( )

Texture analysis Roberts-Romanino-Ross-Velasco hep-ph/0104088 -> H.D.Kim-Raby-Schradin ph/0401169Ross-Velasco hep-ph/0208208SO(10)/SU(3) Ross-Velasco-Vives hep-ph/0401064SO(10)/SU(2)xSU(2)xSU(4) Babu-Pati-Rastogi ph/0410200

10 input parameters

YD =yb

0 bε 3 cε 4

bε 3 ε 2 aε 2

cε 4 aε 2 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

YU =yt

0 cεε 0cεε βε 2 bε0 aε 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

YD =yb

0 ε 0ε αε ε0 t 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟asymmetric

symmetric

a,b,c,ε ,ε, ′b( ) c, ′b( )→ ceiφ, ′beiψ( )

12 inputparameters

ht ,hb( ) ≈ mt,mb( )

a,b,c,β,ε ,ε,α,t( ) φ1,φ2( )

CP-phases

CP-phases

A simple and predictive set A simple and predictive set of Yukawa matricesof Yukawa matrices

YD =yb

0 ϑλ2 e−iγϑλ2

ϑλ2 ϑλ 2ϑλeiγϑλ2 2ϑλ 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=yb

0 0 00 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+ϑ

0 λ2 e−iγλ2

λ2 λ 2λeiγλ2 2λ 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

YU =yt

ϑλ6 0 00 ϑλ2 00 0 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=yt

0 0 00 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+ϑ

λ6 0 00 λ2 00 0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

YL =yτ

0 ϑλ2 O(λ 3)ϑλ2 3ϑλ O(λ2 )

O(λ 3) O(λ2 ) 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=yτ

0 0 00 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+ϑ

0 λ2 O(λ2 )λ2 3λ O(λ)

O(λ2 ) O(λ) 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

yt , yb , yτ ,θ,λ,eiγ( )

6 parameters

Precision predictionsPrecision predictions(using quark data)(using quark data)

mD =m̂b

0 ϑλ2 e−iγϑλ2

ϑλ2 ϑλ 2ϑλeiγϑλ2 2ϑλ 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

β =Arg 2 − e−iγ( ) 1 + ϑ λ 1 − 2eiγ

( )⎡⎣ ⎤⎦⎡⎣

⎤⎦

md

ms

=λ2 1−ϑλ 4cγ −9⎡⎣ ⎤⎦( )ms

mb

=ϑλ 1−4ϑλ + λ2( )

Vus =λ −2 cγ −2( )ϑλ2

Vud =1−λ2

2+ 2 cγ −2( )ϑλ 3

Vub =ϑλ2 + 2cγϑ2λ 3

Vcb =2ϑλ 1+ϑλ( )

Vcs =1−λ2

21+ 4ϑ 2( ) + 2 cγ −2( )ϑλ 3 Vtb =1−2λ2ϑ 2 1+ 2λϑ( )

Vtd = 5−4cγ( )12 ϑλ2 1+ 4ϑλ( )

Vts =2ϑλ 1+ϑλ( ) + cγ −1( )ϑλ 3

Vus 0.2224(36)

mc / mt 3.7 ±0.4( )×10−3

γ 61o ±11o

λ 0.211(7)

ϑ 0.083(14)

sin 2β( ) 0.824(4)

Vud0.975(2)

Vub0.0037(9)

Vcs

0.9771(17)

Vcb0.035(7)

Vtd 0.007(2)

Vts 0.035(7)

Vtb0.9993(2)

mu 2.1±0.9 MeV

md 4.2 ±1.4 MeV

ms 84 ±19 MeV

mephys 0.49 ±0.13 MeV

mμphys 92 ±17 MeV

mU =m̂t

ϑλ6 0 00 ϑλ2 00 0 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

€

VLD

( ) mDVRD =

md 0 0

0 ms 0

0 0 mb

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

VCKM =VLD

running fermion massesrunning fermion massesmq μ( )MS

m̂q

=2β0αs μ( )

π⎛⎝⎜

⎞⎠⎟

γ0

β01+

αs μ( )π

⎧⎨⎩

γ1

β0

−γ0β1

β02

⎡

⎣⎢

⎤

⎦⎥+

+1

2

α s μ( )

π

⎛⎝⎜

⎞⎠⎟

2γ 1

β0

−γ 0β1

β02

⎡

⎣⎢

⎤

⎦⎥

2

+γ 2

β0

−γ 0β1

2

β03

−γ 1β1 + γ 0β2

β02

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎫⎬⎪

⎭⎪

n is the number of light quarksn is the number of light quarksγ0 = 1, γ 1 =202

3−

20n

9⎛⎝⎜

⎞⎠⎟

1

16

O.V. Tarasov, A.A.Vladimirov, A.Y.Zharkov PLB93(1980) O.V. Tarasov, A.A.Vladimirov, A.Y.Zharkov PLB93(1980)

γ2 = 1249 −2216

27+

160

3ζ 3

⎛⎝⎜

⎞⎠⎟

n −140n2

81

⎛

⎝⎜⎞

⎠⎟1

64

β2 = 2857 −5033

9n −

325n2

27

⎛

⎝⎜⎞

⎠⎟1

128

β1 = 51−19

3n

⎛⎝⎜

⎞⎠⎟

1

8β0 = 11−

2

3n

⎛⎝⎜

⎞⎠⎟

1

4

α s μ( ) =π

β0t1−

β1

β02

ln t( )

t+

β12

β04t 2

ln t( ) −1

2⎛⎝⎜

⎞⎠⎟

2

+β0β2

β12

−5

4

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

t =lnμ2

Λn2

⎛

⎝⎜⎞

⎠⎟

ml μ( ) =mlpole 1+ Δ l + ΔZ + ΔW( ) Δl =

α μ( )

π

3

2ln

ml μ( )

π

⎛⎝⎜

⎞⎠⎟

−1⎡

⎣⎢

⎤

⎦⎥ self-energy correctionself-energy correction

mq μ( )MS

mqpole =1+

αs μ( )π

L −43

⎡⎣⎢

⎤⎦⎥+

+α s μ( )

π

⎛⎝⎜

⎞⎠⎟

2

−3019

288+

71

144n +

445

72−

13

36n

⎛⎝⎜

⎞⎠⎟

L +n

12−

19

24⎛⎝⎜

⎞⎠⎟

L2 +ξ3

6− ξ2 2 +

2

3ln 2 −

n

3⎛⎝⎜

⎞⎠⎟

−π 2

6Δ

⎡

⎣⎢

⎤

⎦⎥

R.Tarrach NPB183 (1981)R.Tarrach NPB183 (1981)N.Gray, D.J.Broadhurst, W.Grafe, K.Schilcher Z.Phy s C48 (1990)N.Gray, D.J.Broadhurst, W.Grafe, K.Schilcher Z.Phy s C48 (1990)J.Fleischer, F.Jegerlehner, O.V.Tarasov, O.L.Veretin NPB539 (1999)J.Fleischer, F.Jegerlehner, O.V.Tarasov, O.L.Veretin NPB539 (1999)K.G.Chetyrkin, M.Steinhauser NPB573 (2000)K.G.Chetyrkin, M.Steinhauser NPB573 (2000)K. Melnikov, T.V.Ritbergen PLB482 (2000)K. Melnikov, T.V.Ritbergen PLB482 (2000)

Δ =mqi

mqpole

i≤n∑

Fermion masses and CKM Fermion masses and CKM elementselements

€

mt =174.3± 5.1 GeV

€

mb (mb )M S

= 4.2 ± 0.1 GeV

mc (mc )MS =1.28 ±0.09 GeV

€

ms (2 GeV )M S

=117 ±17 MeV

€

md (2 GeV )M S

= 5.2 ± 0.9 MeV

€

mu(2 GeV )M S

= 2.9 ± 0.6 MeV

€

mτ =1776.99 ± 0.30mμ =105.6583568(52)

me =0.510998902(21)

MeV

MeV

MeV

PDG 2003 off year partial updatePDG 2003 off year partial updateA.H.Hoang PRD61(2000), K.Melnikov & A.Yelkhovsky PRD59(99)A.H.Hoang PRD61(2000), K.Melnikov & A.Yelkhovsky PRD59(99)M.Eidemuller PRD67(2003), J.H.Kuhn & M.Steinhauser NPB619(2001)M.Eidemuller PRD67(2003), J.H.Kuhn & M.Steinhauser NPB619(2001)D.Beciveric, V.Lubicz & G.Martinelli. , PLB524 (2002)D.Beciveric, V.Lubicz & G.Martinelli. , PLB524 (2002)E.Gamiz, M.Jamin, A.Pich, J.Prades, F.Schwab. , JHEP0301 (2003)E.Gamiz, M.Jamin, A.Pich, J.Prades, F.Schwab. , JHEP0301 (2003)M.Jamin,J.A.Oller, A.Pich , EJPC24 (2002)M.Jamin,J.A.Oller, A.Pich , EJPC24 (2002)

sin 2β( ) 0.78(8)Vud 0.9739(5)

Vub 0.00357(31)

Vcs 0.9740(8)

Vcb 0.045(8)

Vtd ≤0.005

Vts 0.0405(35)

Vtb 0.99915(15)

Vus 0.2224(36)

γ 61o ±11o

PDG 2003 off year partial updatePDG 2003 off year partial update2002 CERN Workshop2002 CERN WorkshopCKM FitterCKM Fitter

3

Why should we expect correlations between Why should we expect correlations between dimensionless ratios of fermion masses ?dimensionless ratios of fermion masses ?

Y =y0 0 00 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+ f λ,θ,⋅⋅⋅⋅( )

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

• Third generation is much heavier than 1st and 2nd generations

• We expect the theory of flavor to provide a perturbative calculation of the fermion mass ratios and mixing angles

• are perturbative flavor breaking parameters, <0.22

• The same parameters describe the Yukawa matrices in the three sectors(λ,θ)

m1

m3

= fU ,D,L (λ,ϑ ,⋅⋅⋅) ≈cU ,D,Lλnθm

m2

m3

=gU ,D,L (λ,ϑ ,⋅⋅⋅) ≈dU ,D,Lλpθq

λ =λU ,D,L

m1

m3

,m2

m3

⎛

⎝⎜⎞

⎠⎟

ϑ =θU ,D,L

m1

m3

,m2

m3

⎛

⎝⎜⎞

⎠⎟

ma

mb

ma2

mbmc

ma3

mb2mc

Correlations between mass Correlations between mass ratiosratios

cab1[ ] =

ma

mb

cabc2[ ] =

ma2

mbmc

cabc3[ ] =

ma3

mb2mc

4.73711(7)×10−3

5.882(1)×10−2

1.6390(6)×10−5

1.3199(3)×10−6

12.417(2)

6.253(1)×10−9

9.640(5)×10−7

3.678(1)×10−10

4.567(2)×10−9

0.7304(2)

c121[ ]

c231[ ]

c1232[ ]

c3122[ ]

c2132[ ]

c1233[ ]

c1323[ ]

c3123[ ]

c2313[ ]

4.4 ±1.4( )×10−2

2.4 ±0.4( )×10−2

(2.5 ±1.0)×10−5

(4.7 ±2.5)×10−2

0.53±0.26(2.1±1.8)×10−6

(5.0 ±3.7)×10−8

(6.0 ±3.5)×10−7

(2.7 ±1.6)×10−8

(1.3±0.9)×10−2

2.6 ±0.8( )×10−3

3.7 ±0.6( )×10−3

3.65 ±1.5( )×10−8

2.52 ±1.4( )×10−8

1.45 ±0.71( )

6.5 ±5.8( )×10−11

2.44 ±2.0( )×10−13

c3213[ ] 1.4 ±0.8( )×10−10

3.5 ±2.4( )×10−13

5.4 ±3.5( )×10−3

Charged leptonsCharged leptons Down-type quarksDown-type quarks Up-type quarksUp-type quarks

First empirical formulaFirst empirical formula

md

ms

⎛

⎝⎜⎞

⎠⎟

12

=0.211±0.033

mu

mc

⎛

⎝⎜⎞

⎠⎟

14

=0.225 ±0.018

3me

mμ

⎛

⎝⎜

⎞

⎠⎟

12

=0.20648(2)

md

ms

⎛

⎝⎜⎞

⎠⎟

12

me

mμ

⎛

⎝⎜

⎞

⎠⎟

12

=3.06 ±0.48

mu

mc

⎛

⎝⎜⎞

⎠⎟

14

md

ms

⎛

⎝⎜⎞

⎠⎟

12

=1.06 ±0.25

€

{md

ms

⎛

⎝⎜⎞

⎠⎟

12

≈mu

mc

⎛

⎝⎜⎞

⎠⎟

14

≈3me

mμ

⎛

⎝⎜

⎞

⎠⎟

12

the exact relation isthe exact relation iscompatible with compatible with measurementsmeasurements

Scale evolution of fermion mass Scale evolution of fermion mass ratiosratios

md ,s

mb

⎛

⎝⎜⎞

⎠⎟μ

=md,s

mb

⎛

⎝⎜⎞

⎠⎟μ0

ξb

mu,c

mt

⎛

⎝⎜⎞

⎠⎟μ

=mu,c

mt

⎛

⎝⎜⎞

⎠⎟μ0

ξt

me,μ

mτ

⎛

⎝⎜⎞

⎠⎟μ

=me,μ

mτ

⎛

⎝⎜⎞

⎠⎟μ0

ξτ

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

ξ μ0 , μ( ) ≈ exp3

32π 2ln

μ

μ 0

⎛

⎝⎜⎞

⎠⎟1 −

mb

mt

⎛

⎝⎜⎞

⎠⎟

2⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

≈μ

μ 0

⎛

⎝⎜⎞

⎠⎟

3

32π 2

ξb( )SM

= ξ t−1( )

SM= ξ

SM

θb

θt

⎛

⎝⎜⎞

⎠⎟μ

≈θb

θt

⎛

⎝⎜⎞

⎠⎟μ0

ξ2

θb

θτ

⎛

⎝⎜⎞

⎠⎟μ

≈θb

θτ

⎛

⎝⎜⎞

⎠⎟μ0

ξ

⎧

⎨

⎪⎪

⎩

⎪⎪

ϑ b =ms

3

mb2md

⎛

⎝⎜⎞

⎠⎟

1

2

,ϑ t =mc

3

mt2mu

⎛

⎝⎜⎞

⎠⎟

1

2

,ϑ τ =me

3

mτ2mμ

⎛

⎝⎜

⎞

⎠⎟

1

2

ξ mZ , MG( ) ≈ 1.36

The second empirical relation gets spoiled when The second empirical relation gets spoiled when extrapolated at very high energy scalesextrapolated at very high energy scales

16π 2 ddt

m1,2

m3

⎛

⎝⎜⎞

⎠⎟=−

32

ayb2 +byt

2( ) PRD47 Babu-Shafi ph/9210251PRD47 Babu-Shafi ph/9210251

Second empirical formulaSecond empirical formula

θb =ms

3

mb2md

⎛

⎝⎜⎞

⎠⎟

1

2

= 0.114 ± 0.039

ms3

mb2md

⎛

⎝⎜⎞

⎠⎟

12

mc3

mt2mu

⎛

⎝⎜⎞

⎠⎟

12

=1.5 ±1.0

€

{ms

3

mb2md

⎛

⎝⎜⎞

⎠⎟

12

≈mc

3

mt2mu

⎛

⎝⎜⎞

⎠⎟

12

≈19

mμ3

mτ2me

⎛

⎝⎜⎞

⎠⎟

12

θt =mc

3

mt2mu

⎛

⎝⎜⎞

⎠⎟

1

2

= 0.073 ± 0.023

θτ =mμ

3

mτ2me

⎛

⎝⎜⎞

⎠⎟

1

2

= 0.8545(1)

mμ3

mτ2me

⎛

⎝⎜⎞

⎠⎟

12

ms3

mb2md

⎛

⎝⎜⎞

⎠⎟

12

=7.5 ±2.6

6, 7,8,9,10{ } 9 and 10 give the better fit9 and 10 give the better fit

the exact relation isthe exact relation iscompatible with compatible with measurementsmeasurements

γexp

Fermion mass ratios and CKM elementsFermion mass ratios and CKM elements

ϑ ≈ms

3

mb2md

⎛

⎝⎜⎞

⎠⎟

1

2

≈mc

3

mt2mu

⎛

⎝⎜⎞

⎠⎟

1

2

≈1

9

mμ3

mτ2me

⎛

⎝⎜⎞

⎠⎟

1

2

≈ 0.095

λ ≈md

ms

⎛

⎝⎜⎞

⎠⎟

1

2

≈mu

mc

⎛

⎝⎜⎞

⎠⎟

1

4

≈ 3me

mμ

⎛

⎝⎜

⎞

⎠⎟

1

2

≈ 0.21

€

{ md ,ms( ) ≈ ϑλ 3,ϑλ( )mb

mu ,mc( ) ≈ ϑλ6 ,ϑλ2( )mt

me, mμ( ) ≈13ϑλ 3, 3ϑλ⎛

⎝⎜⎞⎠⎟mτ

Vus =0.2224 ±0.0036

1

2

Vcb

Vus

=0.094 ±0.003

VCKM =

1−12λ2 −λ a

λ 1−b2 −2ϑλc 2ϑλ 1−2ϑ 2λ2

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

VCKM =

1−12λ2 −λ ϑλ2

λ 1−12λ2 −2ϑ 2λ2 −2ϑλ

ϑλ2 2ϑλ 1−2ϑ 2λ2

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

requiring unitarityrequiring unitarity

Vub =Vts =θλ2 ≈0.0041

Reconstructed Yukawa Reconstructed Yukawa matrices without CP-matrices without CP-

violation violation

YD =yb

0 ϑλ2 ϑλ2

ϑλ2 ϑλ 2ϑλϑλ2 2ϑλ 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=yb

0 0 00 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+ϑ

0 λ2 λ2

λ2 λ 2λλ2 2λ 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

YU =yt

ϑλ6 0 00 ϑλ2 00 0 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=yt

0 0 00 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+ϑ

λ6 0 00 λ2 00 0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

YL =yτ

0 ϑλ2 O(λ 3)ϑλ2 3ϑλ O(λ2 )

O(λ 3) O(λ2 ) 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=yτ

0 0 00 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+ϑ

0 λ2 O(λ2 )λ2 3λ O(λ)

O(λ2 ) O(λ) 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

yt , yb , yτ ,θ,λ( )

5 parameters

mD =m̂b

0 eiψ1ϑλ2 eiψ 2ϑλ2

e−iψ1ϑλ2 ϑλ eiψ 3 2ϑλe−iψ 2ϑλ2 e−iψ 3 2ϑλ 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Introducing CP-violationIntroducing CP-violation

requiring hermiticityrequiring hermiticity

mD =m̂b

0 ϑλ2 e−iγϑλ2

ϑλ2 ϑλ 2ϑλeiγϑλ2 2ϑλ 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

VCKM =

1−12λ2 λ −e−iγϑλ2

−λ 1−12λ2 −2ϑ 2λ2 −2ϑλ

eiγ −2( )ϑλ2 2ϑλ 1−2ϑ 2λ2

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

γ =Arg −VudVub

*

VcdVcb*

⎡

⎣⎢

⎤

⎦⎥

β =Arg −VcdVcb

*

VtdVtb*

⎡

⎣⎢

⎤

⎦⎥

β =Arg 2 − e−iγ⎡⎣ ⎤⎦

γ =−ψ 2 −ψ 1 −ψ 3( )

β =Arg 2 − e−iγ⎡⎣ ⎤⎦

βexp

γexp

β =Arg 2 − e−iγ( ) 1 + ϑ λ 1 − 2eiγ

( )⎡⎣ ⎤⎦⎡⎣

⎤⎦

Texture Parameters Predictions

Babu/Pati/Rastogi 13 0

Roberts/Romanino/Ross/Velasco

asymmetric

symmetric

12 (+1,2)

10 (+1,2)

1

2

J.F. 6 7

Characteristics of the Characteristics of the reconstructed SM Yukawa reconstructed SM Yukawa

matricesmatrices

YD =yb

0 0 00 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+ϑ

0 λ2 e−iγλ2

λ2 λ 2λeiγλ2 2λ 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

YU =yt

0 0 00 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+ϑ

λ6 0 00 λ2 00 0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

YL =yτ

0 0 00 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+ϑ

0 λ2 O(λ2 )λ2 3λ O(λ)

O(λ2 ) O(λ) 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

yt , yb , yτ ,θ,λ,eiγ( )

6 parameters

• They work at low or intermediate energy scales

• All the entries execpt (33) in the normalized Yukawa matrices are proportional to a factor

• The factor 3 between down-type quark and the charged lepton Yukawa matrices

θ ≈0.095

€

(AijD hd + μYij

D hu )

€

( ˜ d j )R

€

( ˜ d i)L

€

˜ g R

€

˜ g L

€

m ˜ g

€

(YDrad )ij =

2α s

3π(Aij

D − μ(YDtree )ij tβ )m ˜ g F(m

( ˜ d i )L,m

( ˜ d j )R,m ˜ g )

€

mD = YDtreevcβ + mD

rad = YDtree + YD

rad( )vcβ

€

(di)L

€

(d j )R

Radiative Yukawas in the MSSMRadiative Yukawas in the MSSM

€

MD2 =

MDL

2 + v 2cβ2 (YD

tree )YDtree + ΔD ADcβ − μYD

treesβ( )v

ADcβ − μ∗(YDtree )sβ( )v MDR

2 + v 2cβ2YD

tree (YDtree ) + Δ

D

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟6×6

†

†

€

(YUrad )ij =

2α s

3π(Aij

U − μ(YUtree )ij tβ

−1)m ˜ g F(m( ˜ u i )L,m( ˜ u j )R

,m ˜ g )

€

mU = YUtreevsβ + mU

rad = YUtree + YU

rad( )vsβ

€

m ˜ g

€

˜ g R

€

(u j )R

€

(ui)L

€

˜ g L€

( ˜ u i)L

€

( ˜ u j )R

€

(AijU hu − μYij

U hd )

†

†

W.Buchmuller & D.Wyler, PLB121 (Oct 82)W.Buchmuller & D.Wyler, PLB121 (Oct 82)A.Lahanas & D.Wyler, PLB122 (Nov 82)A.Lahanas & D.Wyler, PLB122 (Nov 82)L.Hall & Kostelecky & Raby NPB267 (Oct L.Hall & Kostelecky & Raby NPB267 (Oct 85)85)T.Banks, NPB303 (Sep 87)T.Banks, NPB303 (Sep 87)E.Ma PRD39 (Jul 88)E.Ma PRD39 (Jul 88)E.Ma & D.Ng PRD65 (May 90)E.Ma & D.Ng PRD65 (May 90)E.Ma & McIlhany MPLA6 (Dec 90)E.Ma & McIlhany MPLA6 (Dec 90)

Radiative mass matrix generationRadiative mass matrix generation

AD =Ab

0 λ2 λ2

λ2 λ 2λλ2 2λ 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟→ mD =vcβ yb

0 0 00 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+ ρDAb

0 λ2 λ2

λ2 λ λλ2 λ 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟−ρDμybtβ

0 0 00 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

• There is FV only in the (LR), i.e. trilinear, soft mass matricesThere is FV only in the (LR), i.e. trilinear, soft mass matrices• There is no FV in the (LL) and (RR) soft mass matricesThere is no FV in the (LL) and (RR) soft mass matrices• I will assume a particular one parameter soft trilinear textureI will assume a particular one parameter soft trilinear texture

tree leveltree level

mD =m̂b

0 ϑ bλ2 ϑ bλ

2

ϑ bλ2 ϑ bλ 2ϑ bλ

ϑ bλ2 2ϑ bλ 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

€

ϑ b =Abρ D

yb + Abρ D 1−μyb tβ

Ab

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

m ˜ d >m ˜ g ⏐ → ⏐ ⏐ 2α s

3πyb

Ab

m ˜ b

⎛

⎝ ⎜

⎞

⎠ ⎟m ˜ g

m ˜ b

⎛

⎝ ⎜

⎞

⎠ ⎟

€

ρD =2α s

3πm ˜ g F(K )

m ˜ d >m ˜ g ⏐ → ⏐ ⏐ 2α s

3π

m ˜ g

m ˜ q 2

Radiative down-type quark mass matrixRadiative down-type quark mass matrix

€

{

€

VLD

( ) mDVRD =

md 0 0

0 ms 0

0 0 mb

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

€

mb = vcβ yb + Abρ D 1−μtβ yb

Ab

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

md

ms

⎛

⎝ ⎜

⎞

⎠ ⎟= λ2 + O(θbλ3)

ms

mb

⎛

⎝ ⎜

⎞

⎠ ⎟= θbλ + O(θb

2λ2)

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

€

λ =md

ms

⎛

⎝ ⎜

⎞

⎠ ⎟comp

1

2

= 0.211± 0.033

θb =ms

3

mb2md

⎛

⎝ ⎜

⎞

⎠ ⎟exp

1

2

= 0.114 ± 0.039

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

€

m ˜ d

m ˜ b

⎛

⎝ ⎜

⎞

⎠ ⎟≤

2α s

3πybϑ b

Ab

m ˜ b

⎛

⎝ ⎜

⎞

⎠ ⎟m ˜ g

m ˜ d

⎛

⎝ ⎜

⎞

⎠ ⎟≈

14

tβ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟Ab

m ˜ b

⎛

⎝ ⎜

⎞

⎠ ⎟≤

40 ±14

tβ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

†

€

Ab ≤ 2.5m ˜ b

Non degenerateNon degeneratedown squarksdown squarks

€

ϑ b → ϑ b

m ˜ b

m ˜ d

⎛

⎝ ⎜

⎞

⎠ ⎟

2

⇒

€

m ˜ d ≠ m ˜ b

Flavor breaking F-termsFlavor breaking F-terms

F = F S + F F ϑ2 F

S=0

F F ≠0

€

{

F

MΨLΨRH( )

No flavor violation No flavor violation at tree level in the at tree level in the Yukawa couplingsYukawa couplings

U(2)+ SUSY breaking flavor modelU(2)+ SUSY breaking flavor model

€

Q3,D3,U3,L3,E3,Hu,Hd( )

ΨQ =Q1

Q2

⎛⎝⎜

⎞⎠⎟

ΨU =U1

U2

⎛⎝⎜

⎞⎠⎟

Ψ D =D1

D2

⎛⎝⎜

⎞⎠⎟

ΨL =L1

L2

⎛⎝⎜

⎞⎠⎟

Ψ E =E1

E2

⎛⎝⎜

⎞⎠⎟

flavor singletsflavor singlets

flavor flavor vectorsvectors

€

{

€

Sab, Aab , F a, (a,b =1,2)

S =vS 00 VS

⎛⎝⎜

⎞⎠⎟ϑ 2 , A =

0 vA

−vA 0⎛⎝⎜

⎞⎠⎟ϑ 2 , F =

vF

VF

⎛⎝⎜

⎞⎠⎟ϑ 2

€

{flavor flavor breaking breaking fields fields (F-(F-terms)terms)

vS ,vF ,vA ,VF ,VS( ) = 0,λ2 ,λ2 ,2λ,λ( )MF %m

R.Barbieri, G. Dvali & L.J. Hall PLB377 (96)R.Barbieri, G. Dvali & L.J. Hall PLB377 (96)

J.F & N.Haba ph/0404077J.F & N.Haba ph/0404077

Borzumati et al.,(May 98)Borzumati et al.,(May 98)

U(2)+ SUSY breaking flavor modelU(2)+ SUSY breaking flavor model

€

y tQ3U3Hu + ybQ3D3Hd + yτ L3E3Hd + μHuHd SuperpotentialSuperpotential(only third generation)(only third generation)

Soft Soft trilinearstrilinears

€

{

€

{Soft massesSoft masses

€

Q3L3D3 + L3Hd

€

1

Md2θ∫ ZHφLφR

€

1

MF

d2θ∫ Z abΨaLΨb

R Hα

Z = S,A

∑ + cc

€ €

1

MF

d2θ∫ φR F aΨaL + φLF aΨa

R( )Hα + cc

1

M F2

d 4θ∫ Zac† Zcb Ψa( )

†Ψb

Z=S,A∑ + Fa

†Fb Ψa( )†Ψb

⎛

⎝⎜⎞

⎠⎟

1

Md 2ϑ∫ G%λ%λ + cc G is a flavor singletG is a flavor singlet

m %λ

=GM

1

Md 2θ∫ κ %λGφ

LφRHα + cc

1

M 2d 4θ∫ κ %λ

' G†G Ψ†Ψ +φ†φ( )

1

MM F

d 4θ∫ κ %λ''G†F aφ†Ψa +h.c. }

1

M 2d 4θ∫ Z†Zφ†φ

J is a second flavor singletJ is a second flavor singlet

Boundary conditions for soft Boundary conditions for soft parametersparameters

A =A0 σλ2 σλ2

−σλ2 σλ 2σλσλ2 2σλ 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

M2 =m%f2

1+σ '2 λ 4 −σ '2 λ 3 σ '2 2λ 3

−σ '2 λ 3 1+ 5σ '2 λ2 σ '2 4λ2

σ '2 2λ 3 σ '2 4λ2 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

A =κ %λm%λ

σ =

%mA

m %f

2 =κ '%λ m%λ2

σ ' =

%m

m %f

Ytree =0 0 00 0 00 0 y

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

flavons =λnMF %mϑ 2

Tree level Yukawa matricesTree level Yukawa matrices

SU(5), lepton and up-type quark YukawasSU(5), lepton and up-type quark Yukawas

me

mμ

⎛

⎝⎜

⎞

⎠⎟ ≈

19λ

mμ

mτ

⎛

⎝⎜⎞

⎠⎟≈3θλ

⎧

⎨

⎪⎪

⎩

⎪⎪

λ =3me

mμ

⎛

⎝⎜

⎞

⎠⎟

exp

1

2

= 0.20648

θτ =mμ

3

mτ2me

⎛

⎝⎜⎞

⎠⎟exp

1

2

= 0.09495

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

mL =m̂τ

0 ϑ τλ2 ϑ τλ

2

ϑ τλ2 3ϑ τλ 2ϑ τλ

ϑ τλ2 2ϑ τλ 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Sab ∈75, Aab ∈1, F a ∈(1,24)

mU =m̂t

0 0 ϑ tλ2

0 0 ϑ tλϑ tλ

2 ϑ tλ 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

1

M F

SabHd10a5b

1

M F

AabHd10a5b

1

M F

FaHd10a53

Fadoes not mix with up-type sectordoes not mix with up-type sector (discrete symmetry)(discrete symmetry)

S 'ab ∈1 S(1) =vS(1) 00 VS(1)

⎛

⎝⎜⎞

⎠⎟ϑ 2

vS(1),VS(1)( ) = λ6 ,λ2( )MF %m

{€

mU = ˆ m t

ϑ bλ6 0 0

0 ϑ bλ2 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

charm quark mass too lightcharm quark mass too lightup quarm masslessup quarm massless

FCNCs suppression by radiative alignementFCNCs suppression by radiative alignement(degenerate squarks)(degenerate squarks)

€

d

€

d

€

s

€

s

€

˜ g

€

˜ g

€

AD

€

AD

non diagonal non diagonal gaugino vertexgaugino vertex

€

VLD

( )ADVRD

€

s€

d

€

d€

s

€

˜ g

€

˜ g

diagonal diagonal gaugino vertexgaugino vertex

€

VLD

( )ADVRD

superKMsuperKM basisbasis

†

†

AD =Ab

0 λ2 λ2

λ2 λ 2λλ2 2λ 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

ADSKM = VL

D( )ADVRD =Ab

λ 3 θbλ5 λ 4

θbλ5 λ 2 θb −1( )λ

λ 4 2 θb −1( )λ 1+ 2θbλ2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

†

€

˜ d

€

˜ s

Constraints from FCNCsConstraints from FCNCs(degenerate squarks)(degenerate squarks)

€

ADSKM = Ab

9 ×10−3 4 ×10−5 2 ×10−3

− 2.1×10−1 1.9 ×10−1

− − 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

€

ΔmK =2α s

2

648m ˜ q 2

mK fK2 δ12

d( )

LR

2 mK

ms + md

⎛

⎝ ⎜

⎞

⎠ ⎟

2

268xf x( ) +144g x( )( ) + 84g x( ) ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

δ12d

( )LR

=ζ D

SKM( )

12

m ˜ d Lm ˜ d R

€

x =m ˜ g

2

m ˜ q 2

€

m ˜ q

€

ΔmK theo< 5 ×10−16 MeV ,ΔmK exp

= (3.490 ± 0.006) ×10−12 MeV

€

δ12d

( )LR theo

= 4 ×10−5 Ab

m ˜ d

⎛

⎝ ⎜

⎞

⎠ ⎟

v

m ˜ d

⎛

⎝ ⎜

⎞

⎠ ⎟cβ ~ 7 ×10−6 1 TeV

m ˜ d

⎛

⎝ ⎜

⎞

⎠ ⎟1

tβ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

{

€

{

€

{€

δ13d

( )LR theo

= 2 ×10−3 Ab

m ˜ d

⎛

⎝ ⎜

⎞

⎠ ⎟

v

m ˜ d

⎛

⎝ ⎜

⎞

⎠ ⎟cβ ~ 3.5 ×10−4 1 TeV

m ˜ d

⎛

⎝ ⎜

⎞

⎠ ⎟

1

tβ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

ΔmB theo< 7 ×10−13 MeV ,

€

ΔmB exp= (3.22 ± 0.05) ×10−10 MeV

€

{

€

δ23d

( )LR theo

=1.9 ×10−1 Ab

m ˜ d

⎛

⎝ ⎜

⎞

⎠ ⎟

v

m ˜ d

⎛

⎝ ⎜

⎞

⎠ ⎟cβ ~ 3.7 ×10−2 1 TeV

m ˜ d

⎛

⎝ ⎜

⎞

⎠ ⎟

1

tβ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

B(b → sγ)theo

< 3.4 ×10−5,

€

B(b → sγ)exp

= (3.3± 0.4) ×10−4

€

{

average average squark masssquark mass

€

tβ > 5

Gabbiani et al., NPB477 (96)Gabbiani et al., NPB477 (96)Berolini et al. PLB192, 437 (87)Berolini et al. PLB192, 437 (87)

€

m ˜ q > 400 GeV

€

tβ > 40

Constraints from FCNCs on soft mass Constraints from FCNCs on soft mass matrices matrices (degenerate squarks)(degenerate squarks)

ΔmK =2α s

2

648m%q2

mK fK2 δ12

d( )LL

2 mK

ms + md

⎛

⎝⎜⎞

⎠⎟

2

384xf x( ) − 24g x( )( ) +120xf (x) +168g x( )⎡

⎣⎢⎢

⎤

⎦⎥⎥

δ12d( )

LL

2= σ '4 λ 6

€

x =m ˜ g

2

m ˜ q 2

€

m ˜ q

if σ =1→ ΔmK theo

LL < ΔmK expfor m%q > 400 GeV

€

{

€

{

€

{

1

6

mb

m%g

⎛

⎝⎜

⎞

⎠⎟δ13

d( )LL

δ13d( )

LR

≈3×10−3tβm%b

m%g

⎛

⎝⎜

⎞

⎠⎟

B(b → sγ)

average average squark masssquark mass

Gabbiani et al., NPB477 (96)Gabbiani et al., NPB477 (96)

M2 =m%f2

1+σ '2 λ 4 −σ '2 λ 3 σ '2 2λ 3

−σ '2 λ 3 1+ 5σ '2 λ2 σ '2 4λ2

2σ '2 λ 3 σ '2 4λ2 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

ΔmK exp= (3.490 ± 0.006) ×10−12 MeV

σ ' =

%m

m %f

if σ =1→ ΔmK theo

LL < 6 ×10−15 MeV for m%q > 9 TeV

if σ =0.5 → ΔmK theo

LL < 6 ×10−15 MeV for m%q > 2.5 TeV

LL contribution to LL contribution to suppressed compared with the LR contributionsuppressed compared with the LR contribution

if σ =1→ ΔmB theo

LL < 5 ×10−12 MeV for m%q > 600 GeVΔmB exp

= (3.22 ± 0.05) × 10−10 MeV

€

(AijL hd − μλ ij

L hu )

€

(˜ l j )R

€

(˜ l i)L

€

BR

€

BL

€

mB€

(li)L

€

(l j )R

Lepton flavor violationLepton flavor violation

W.Buchmuller & D.Wyler, PLB121 (Oct 82)W.Buchmuller & D.Wyler, PLB121 (Oct 82)E.Ma PRD39 (Jul 88)E.Ma PRD39 (Jul 88)

ϑ τ =

m%l >mB α

π

1

λ τ

Aτ

m%l

⎛

⎝⎜

⎞

⎠⎟

mB

m%l

⎛

⎝⎜

⎞

⎠⎟σ =

α

4π cW2

1

λ τ

%m

m%l

⎛

⎝⎜

⎞

⎠⎟

mB

m%l

⎛

⎝⎜

⎞

⎠⎟

ALSKM =Aτ

13λ 3 2

3θτλ

5 23λ2

− 3λ λ− − 1+ 2θbλ

2

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=Aτ

3×10−3 5.6 ×10−4 3×10−2

− 0.648 0.216− − 1.007

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

mL =m̂τ

0 ϑ τλ2 ϑ τλ

2

ϑ τλ2 3ϑ τλ ϑ τλ

ϑ τλ2 ϑ τλ 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

m%e

m%τ

⎛

⎝⎜⎞

⎠⎟≤

απyτϑ τ

%mm%τ

⎛

⎝⎜⎞

⎠⎟mB

m%e

⎛

⎝⎜⎞

⎠⎟≈

2.6tβ

⎛

⎝⎜

⎞

⎠⎟

%mm%τ

⎛

⎝⎜⎞

⎠⎟⇒ m%e ≤

2.6tβ

⎛

⎝⎜

⎞

⎠⎟ %mIf sleptons are If sleptons are

non degeneratenon degenerate

me

mμ

⎛

⎝⎜

⎞

⎠⎟ ≈

19λ

mμ

mτ

⎛

⎝⎜⎞

⎠⎟≈3θλ

⎧

⎨

⎪⎪

⎩

⎪⎪

λ =3me

mμ

⎛

⎝⎜

⎞

⎠⎟

exp

1

2

= 0.20648

θτ =mμ

3

mτ2me

⎛

⎝⎜⎞

⎠⎟exp

1

2

= 0.09495

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

Γμ→ eγ < 8 × 10−12

€

{ if m%l

>1 TeV,tβ > 50,m%l > mB → Γτ→ μγ < 4 ×10−10

Γτ→ eγ < 8 ×10−11

Γτ→ eγ < 2.7 × 10−6

Γμ→ eγ < 1.2 ×10−11

Γτ→ μγ <1.1 × 10−6

′Aτ Hd*LE → Aτ + ′Aτ tβ → m%e ≤ 2.6 %mnon-holomorphic soft trilinearnon-holomorphic soft trilinear

Borzumati et al.,(May 98)Borzumati et al.,(May 98)

Proton decay suppresionProton decay suppresion

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WSU (5) =1

4λ ij

Uψ10ψ10H5 + 2λ ijDψ10ψ 5

H5

+L

€

Wdim 5 Op. =1

2M H c

λ ijU λ kl

D QiQ j( ) QkLl( ) +L

SU(5) superpotential

dimension 5 operators

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λD,L =

0 0 0

0 0 0

0 0 λ b,τ

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

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λU =

0 0 0

0 0 0

0 0 λ t

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Tree level cancellationof dimension fiveoperators

u

d s

νt

b

n-loop generated, n>1