javier ferrandis ific barcelona, january 12th 2005
DESCRIPTION
Supersymmetry breaking as the origin of flavor (from empirical formulas for the fermion spectra to radiative fermion mass generation). J.F ph/0406004 PRD70 J.F. ph/0404068 PRD70 J.F & N. Haba ph/0404077 EPJC. Javier Ferrandis IFIC Barcelona, January 12th 2005. - PowerPoint PPT PresentationTRANSCRIPT
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
€
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
€
λD,L =
0 0 0
0 0 0
0 0 λ b,τ
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
€
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