Fitting fermion masses and mixings in F-Theory GUTs

Transcription

Fitting fermion masses and mixings in F-Theory
GUTs
Federico Carta.
IFT-UAM/CSIC
24th of June 2016
Federico Carta. (IFT-UAM/CSIC)
24th of June 2016
1 / 16
Based on...
F.C., Marchesano, Zoccarato. 2015.
(hep-th 1512.04846)
Related work:
Marchesano, Regalado, Zoccarato. 2015
Font, Ibañez, Marchesano, Regalado. 2012
Font, Marchesano, Regalado, Zoccarato. 2013
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Yukawa couplings in MSSM & SU (5) GUT.
SM has 19 free parameters.1
Yukawa couplings must be placed by hand in SM Lagrangian.
Can String Theory address this problem?
u
d
e
WM SSM ⊃ yij
Hu Qi uj + yij
Hd Qi dj + yij
Hd Li ej
d/e
WSU (5) ⊃ Yiju 10M · 10M · 5U + Yij 10M · 5̄M · 5̄D
E6 enhancement for 10M · 10M · 5U
−→
SO(12) enhancement for 10M · 5̄M · 5̄D
1
NO in (pert.) IIB
−→
OK in (pert.) IIB
Excluding masses, mixing, CP phases for neutrinos.
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F-Theory generalities
D7 located at singularity
locus.
Matter located at
codimension 2.
Yukawas located at
codimension 3.
Non perturbative Type IIB.
Geometrize the axio-dilaton
τ = C0 + ie−ϕ
To generate 10M · 10M · 5U a
T-brane is needed.
[Φ, Φ† ] 6= 0
Elliptic fibration.
24th of June 2016
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Adding fluxes and non-perturbative effects.
At tree level, T-brane implies
fluxes threading the base.
(see S. Schwieger’s talk)
Non perturbative effects.
R
Wnp =
θ0 Tr(F ∧ F )
2 S
Euclidean D3.
Gaugino condensate.
Taken from Camara, Ibañez and Valenzuela, ’14
Needed to have a right
hierarchy.
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E7 models.
Generate Up & Down at the same point.
(
G = E7
G ⊃ E6 and SO(12) =⇒
G = E8
Heckman et al. ’09
Our choice
Marchesano et al. ’15
The commutant of SU (5) inside of E7 is SU (3) × U (1)
Adjoint Higgs is a 3 × 3 matrix
Different possibilities are:
3
Unique model.
Chiou et al. 2011
Bad hierarchy for masses.
2+1
Two models
Good hierarchy for masses.
1+1+1
No mass for the top quark.
Not a T-brane background
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Higgs vev. A and B models
The adjoint higgs vev has the form2
0
m
su(2) :=
u(1)1 := λ1
m2 x 0
Matter fields
A model.
10M := 21,0
B model.
10M := 21,0
5U := 1−2,0
5U := 1−2,0
5̄M := 20,1
5̄M := 1−1,−1
5̄D := 1−1,−1
Both give good hierarchies.
Only A model is physical.
2
u(1)2 := λ2
5̄D
2
(O( ), O(), O(1)).
:= 20,1
In holomorphic gauge
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Matter curves and Yukawa point.
Matter curves.
Σ10M := {λ21 − m3 x = 0}
Σ5̄M := {λ22 − m3 x = 0}
Σ5̄D := {λ1 + λ2 = 0}
Σ5U := {λ1 = 0}
Yukawa point.
The points can be separated.
pup := {λ1 = 0 ∩ x = 0}
pdown := {λ2 = 0 ∩ x = 0}
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Yukawa Couplings in 8d SYM.
Twisted d = 8 SYM theory lives on D7 worldvolume.
Beasley et al. ’08
Yukawa couplings can be computed by dimensional reduction of
the superpotential3
Z
Z
R
¯ ∧Φ+
W =
F (0,2) ∧ Φ =
∂A
SA∧A∧Φ
S
S
Rank 1 Yukawa matrix −→ only 3rd generation is massive.
3
One should correct this with Wnp
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E7 model. Holomorphic Yukawa
Wavefunctions localize at Yukawa point.
Compute by residue integral.
Cecotti et al. ’09
Up type quarks
YU =
2
π 2 γU γ10,3
2ρm ρµ

0






0
˜
γ10,1
2ρµ γ10,3
0
˜
˜
2
γ10,2
2
2ρµ γ12,3
0
γ10,1 
2ρµ γ10,3 


0



1
Down type quarks and leptons

YD/L
0


π 2 γ5,3 γ10,3 γD 
2γ
5,1 γ10,1
 κ̃˜
=−
(d + 1)ρµ ρm 
(d + 1)2 ρ2µ γ5,3 γ10,2


γ5,1
˜
(d + 1)ρµ γ5,3
2γ5,2 γ10,1
(d + 1)2 ρ2µ γ5,3 γ10,3
γ5,2 γ10,2
−˜
(d + 1)ρµ γ5,3 γ10,3
κ̃˜
0

1







2κ̃2
− ˜
2
(d + 1) ρµ
γ10,2
−κ̃
(d + 1)ρµ γ10,3
γ10,1
(d + 1)ρµ γ10,3
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10 / 16
Wavefunction normalization
To have physical couplings we need to compute kinetic terms
Z
Kij̄ =
dvolS Tr(ψi ψ̄j̄ )
S
Imposing canonical normalization gives physical couplings
Physical couplings depend on flux densities.
Particularly important for hypercharge flux
Approximate S as C2 . (Valid only for very peaked wavefunction)
Determine explicitly γ5i , γ5̄i , γ10i as functions of flux quanta and
other parameters defining the matter curves.
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Fermion Masses at GUT scale
Ross, Serna. ’07
tan β
mu /mc
mc /mt
md /ms
ms /mb
me /mµ
mµ /mτ
Yτ
Yb
Yt
10
2.7 ± 0.6 × 10−3
2.5 ± 0.2 × 10−3
5.1 ± 0.7 × 10−2
1.9 ± 0.2 × 10−2
4.8 ± 0.2 × 10−3
5.9 ± 0.2 × 10−2
0.070 ± 0.003
0.051 ± 0.002
0.48 ± 0.02
38
2.7 ± 0.6 × 10−3
2.4 ± 0.2 × 10−3
5.1 ± 0.7 × 10−2
1.7 ± 0.2 × 10−2
4.8 ± 0.2 × 10−3
5.4 ± 0.2 × 10−2
0.32 ± 0.02
0.23 ± 0.01
0.49 ± 0.02
50
2.7 ± 0.6 × 10−3
2.3 ± 0.2 × 10−3
5.1 ± 0.7 × 10−2
1.6 ± 0.2 × 10−2
4.8 ± 0.2 × 10−3
5.0 ± 0.2 × 10−2
0.51 ± 0.04
0.37 ± 0.02
0.51 ± 0.04
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Fitting the empirical value
Fix first second generation masses
Look at the following ratios:
mµ /mτ
ms /mb
= 3.3 ± 1
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24th of June 2016
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mµ /mτ
ms /mb
�
mc /mt
ms /mb
= 3.3 ± 1
��
0.55
��
0.50
��
�
= 0.13 ± 0.03
0.45

d  0.015

d  0.035

d  0.065
0.40
��
0.35
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1
2
3
4
5
6
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24th of June 2016
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mµ /mτ
ms /mb
mc /mt
ms /mb
= 3.3 ± 1
��
0.55
��
0.50
��
0.45
mc /mt = 2.5 · 10−3
= 0.13 ± 0.03
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0.40
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0.35
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
d  0.015

d  0.035

d  0.065
 ��
ϵ � �
μ��
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24th of June 2016
13 / 16
Look at third generation masses.
Yτ
Yb
= 1.37 ± 0.3
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24th of June 2016
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Yτ
Yb
= 1.37 ± 0.3
Typical value fo Yb
��
0.037
0.13
0.036
��
0.12
0.035
�

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0.11
0.034
0.10
��
0.033
0.032
0.2
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0.09
0.08
0.3
0.4
0.5
0.6
0.7
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24th of June 2016
14 / 16
Yτ
Yb
= 1.37 ± 0.3
Confront with
Typical value fo Yb
hep-th/1503.02683
��
0.037
0.036
0.13
0.036
��
0.12
0.034
0.035
�

��
��
0.11
0.034
0.10

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0.032
a=b=1
a=-0.4, b=-0.6
��
0.033
0.09
0.030
0.032
0.2
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0.08
0.3
0.4
0.5
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0.6
0.7
0.028
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
�
24th of June 2016
14 / 16
Separating the points
Induce a separation between
the up and down points.
Change of base for the
wavefunctions.
Effect on CKM.
Randall, Simmons-Duffin ’09

VCKM

Vud Vus Vub
=  Vcd Vcs Vcb 
Vtd Vts Vtb
|Vtb | ∼ 1 −→ κ̃ ∼ 10−5
24th of June 2016
15 / 16
Conclusions
Not all E7 models accomodate a good hierarchy.
We analize two local models with good hierarchy.
(Only one is compatible with empirical data)
Find a wider good region of parameter space, by considering
more general matter curves.
Find some preferred values for MSSM parameters
(tan β ' 10 − 20)
Achieve b − τ non-unification.
A small splitting of the points generates a CKM matrix.
Thank you for your attention.
24th of June 2016
16 / 16

Fitting fermion masses and mixings in F-Theory GUTs

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