Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,

Sven KrippendorfDAMTP, University of Cambridge

based on arXiv:1002.1790, with: M. Dolan, A. Maharana, F. Quevedo

Cambridge, 22nd April 2010

Phenomenology of D-branesat Toric Singularities

Disclaimer: This talk is somewhat independent of the LHC.

... string theorists could have talked about this 20 years ago

for more than 25 years...


Heterotic String Theory


Heterotic String Theory Type II brane models



e.g. intersecting branes in IIA,magnetised branes in IIB

Type II brane models

F-theory





0502005, 0702094

F-theory




0502005, 0702094




0502005, 0702094

moduli stabilisation




0502005, 0702094

moduli stabilisation

under control in Type IIB(KKLT, LVS)

Perspective: Local Brane Models (within IIB)

... bottom-up model building

e.g.: aldazabal, ibanez, quevedo, uranga (10 years ago),verlinde, wijnholt (5 years ago)

Modular String Model Building (in LARGE volume)

Standard (like) Modelswith fractional (D3/D7) branes

at singularities

and what can we get?

e.g.: aldazabal, ibanez, quevedo, uranga (10 years ago),verlinde, wijnholt (5 years ago)

E

Brane models: how good are they to date?

chiral matter, adjoint and (bi-)fundamental matter

U(n), SO(n), Sp(n) gauge groups (exceptional gauge groups in F-theory)

models with the correct matter content

BUT structure of yukawa couplings, e.g.: hierarchy of masses, ckm matrix

e.g. madrid model, 0105155

... this applies also to F-theory models

ContentWhy branes at singularities, what gauge theories do we get? ...background and history

Studying gauge theories with dimers

What general properties do gauge theories of the infinite class of toric singularities obey?

How good are the models regarding flavour physics, i.e. can we get the CKM-matrix?





Upper bound of#families (≤3)





Mass structure(0, m, M)






Mass structure(0, m, M)


yes, we can...there is enough structure aroundto build conrete models with the right ckm.

Motivation for branes at singularities

Local models -> a lot of information without addressing moduli stabilisation

Effective field theory (although distances below string scale) well under control

Gauge coupling unification (in principle)

Powerful (dimer) techniques for toric singularities

Gauge theories highly restricted (unlike intersecting branes in IIA)

E

Motivation for branes at singularities

Local models -> a lot of information without addressing moduli stabilisation

Effective field theory (although distances below string scale) well under control

Gauge coupling unification (in principle)

Powerful (dimer) techniques for toric singularities

Gauge theories highly restricted (unlike intersecting branes in IIA)

E

Hanany et al.

Classic Example:

ni D3-branes: U(n1)xU(n2)xU(n3)

mi D7-branes: U(m1)xU(m2)xU(m3)

Arrows: bi-fundamental matter

Anomaly cancellation

Hypercharge:

. C3/Z3

3[(n1, n̄2, 1), (1, n2, n̄3), (n̄1, 1, n3)]

D3 matter content:m2 = 3(n3 ! n1) + m1,

m3 = 3(n3 ! n2) + m1

Qanomaly!free =!

i

Qi

ni

aiqu: 0005067

let’s build models!

Standard-Model

let’s build models!Left-Right

Standard-Model


Pati-Salam

Left-Right

Standard-Model


Pati-Salam Trinification

Left-Right

Standard-Model

e.g. left-right model

3 families SU(3)xSU(2)LxSU(2)RxU(1)B-L

B-L normalisation 32/3:

+ 3 Higgses -> unification at 1012 GeV

Proton stability from global U(1) B-L

embedded in compact CY

sin2 ! =314

= 0.214

e.g. left-right model

3 families SU(3)xSU(2)LxSU(2)RxU(1)B-L

B-L normalisation 32/3:

+ 3 Higgses -> unification at 1012 GeV

Proton stability from global U(1) B-L

embedded in compact CY

sin2 ! =314

= 0.214

details in Aldazabal, Ibanez, Quevedo, Uranga hep-th/0005067

What’s bad about the model?

Problem: Yukawa couplings

W = !ijk QiLHj

uukR =

!

"Q1

LQ2

LQ3

L

#

$

!

"0 Z12 !Y12

!Z12 0 X12

Y12 !X12 0

#

$

!

"u1

Ru2

Ru3

R

#

$ .

Masses: (0, M, M)M2 = |X12|2 + |Y12|2 + |Z12|2

Resolution: Embed into singularities with more structure (in this case del Pezzo 1)

Problem: Yukawa couplings

W = !ijk QiLHj

uukR =

!

"Q1

LQ2

LQ3

L

#

$

!

"0 Z12 !Y12

!Z12 0 X12

Y12 !X12 0

#

$

!

"u1

Ru2

Ru3

R

#

$ .

Masses: (0, M, M)M2 = |X12|2 + |Y12|2 + |Z12|2

Resolution: Embed into singularities with more structure (in this case del Pezzo 1)

cmq: 0810.5660.C3/Z3 = dP0

What types of singularities are there?

Orbifold Singularities

del Pezzo singularities (P2 blown-up), Conifold -> TORIC SINGULARITIES

non-toric singularities





few suitable for model building





infinite class, techniques






infinite class, techniques

limited techniques


Gauge theories probing toric singularities

Toric CY-Cone:represented as T3 fibrationover rational polyhedral cone

D3 branes (at the tip of the cone)

D7 branes (wrapping 4-cyclespassing through singularity)





D3 branes





toric diagram





Gauge theories of toric singularitiesare always quiver gauge theories!






uR

QL

Hu

Hd dR

1

32

m

6!m

3!m






uR

QL

Hu

Hd dR

1

32

m

6!m

3!mbi-fundamental matter






uR

QL

Hu

Hd dR

1

32

m

6!m

3!mbi-fundamental matter

change rank byaddition of fractional branes






uR

QL

Hu

Hd dR

1

32

m

6!m

3!m

WARNINGonly matter content,

dimer for superpotential

Question:

How to determine the gauge theory associated to a given singularity (inverse problem)? -> most efficient solution via dimers

Aside: Dimer techniques useful in understanding gauge theories of M2 branes at singularities (no inverse algorithm)

0503149, 0504110, 0505211, 0511063, 0511287, 0604136, 0706.1660, 0803.4474Hanany, Kennaway, Franco, Vegh, Wecht, Martelli, Sparks, Feng, He, Vafa, Yamazaki

The Dimer LanguageDimers visualise the gauge theory of toric singularities.

Geometry: Toric Diagram Gauge Theory: Dimerinverse slopes in toric diagram

winding numbers of zigzag paths







(1,1)




(1,-1)




(0,-1)




(-2, 1)




Dimer Language IIReading off the gauge theory

Faces = gauge groups

Intersection of zigzag paths = bi-fundamental matter

Vertices (faces orbited by zigzag paths) = superpotential terms









W = X13X32X21 !X14X43X32X21

merge zigzag paths according to cutting of toric diagram

caveat: additional crossings, concrete prescription to be avoided by precise operations

Dimer Language III: How do I get a dimer?

embed toric singularity in orbifold of conifold whose dimer is known (chess-board).

collaps cycles in singularity(= cutting toric diagram)

Gulotta 0807.3012






Gulotta 0807.3012






Gulotta 0807.3012

1 2 3 1

4 5 6

7 8 9 7

1 2 3 1

10 11 12

! ! !

" " " "

! ! !

" " " "

! ! !






Gulotta 0807.3012

1 2 3 1

4 5 6

7 8 9 7

1 2 3 1

10 11 12

! ! !

" " " "

! ! !

" " " "

! ! !






Gulotta 0807.3012

1 2 3 1

4 5 6

7 8 9 7

1 2 3 1

10 11 12

! ! !

" " " "

! ! !

" " " "

! ! !






Gulotta 0807.3012

1 2 3 1

4 5 6

7 8 9 7

1 2 3 1

10 11 12

! ! !

" " " "

! ! !

" " " "

! ! !






Gulotta 0807.3012

1 2 3 1

4 5 6

7 8 9 7

1 2 3 1

10 11 12

! ! !

" " " "

! ! !

" " " "

! ! !

Operations on the dimer

(1,0) + (0,1) -> (1,1)Operation 1:


(1,0) + (0,1) -> (1,1)Operation 1:


(1,0) + (0,1) -> (1,1)Operation 1:


(1,0) + (0,1) -> (1,1)

(1,0) + (0,1) & (-1,0) + (0,-1)

I(1,1),(!1,!1) = 0Iab = namb !manb

Operation 1:

Operation 2:

-> (1,1) + (-1,-1)


(1,0) + (0,1) -> (1,1)

(1,0) + (0,1) & (-1,0) + (0,-1)

I(1,1),(!1,!1) = 0Iab = namb !manb

Operation 1:

Operation 2:

-> (1,1) + (-1,-1)


(1,0) + (0,1) -> (1,1)

(1,0) + (0,1) & (-1,0) + (0,-1)

I(1,1),(!1,!1) = 0Iab = namb !manb

Operation 1:

Operation 2:

-> (1,1) + (-1,-1)

e.g. del Pezzo 31 2 1

3 4

5 6 5

1 2 1

2 6

! !

" " "

! !

" "

! !

WdP3 = !X12Y31Z23 !X45Y64Z56 + X45Y31Z14!53 + X12Y25Z56!61

+X36Y64Z23"42 !X36Y25Z14!53!61"42

=

!

"X45

Y23

Z25

#

$

!

"0 Z14!53 !Y64

!Z14!53!61"42 0 X12!61

Y64"42 !X12 0

#

$

!

"X36

Y31

Z56

#

$ .

1 2 1

3 4

5 6 5

1 2 1

7 8

! !

" " "

! !

" " "

! !

e.g. del Pezzo 31 2 1

3 4

5 6 5

1 2 1

2 6

! !

" " "

! !

" "

! !

WdP3 = !X12Y31Z23 !X45Y64Z56 + X45Y31Z14!53 + X12Y25Z56!61

+X36Y64Z23"42 !X36Y25Z14!53!61"42

=

!

"X45

Y23

Z25

#

$

!

"0 Z14!53 !Y64

!Z14!53!61"42 0 X12!61

Y64"42 !X12 0

#

$

!

"X36

Y31

Z56

#

$ .

1 2 1

3 4

5 6 5

1 2 1

7 8

! !

" " "

! !

" " "

! !

1 2 1

3 4

5 6 5

1 2 1

2 6

! !

" " "

! !

" "

! !

1 2 1

3 2

3 6 3

1 2 1

2 6

! !

" " "

! !

" "

! !

Application of Gulotta’s algorithmto toric del-Pezzo surfaces

1 2 1

3 2

3 1 3

1 2 1

2 1

! !

" " "

! !

" "

! !

1 2 1

3 4

3 6 3

1 2 1

2 6

! !

" " "

! !

" "

! !

Gulotta’s dimers = Traditional dimers

Philosophy: Use this algorithmic view of gauge theories to find general features for gauge theories probing toric singularities!

Are there properties revealed that are not apparent from looking at the superpotential

and are they useful for model building?

Restricting the # of familiesThe dimers obtained with the operations of Gulotta are highly restricted (otherwise: inconsistent dimers)

example: additional crossing (-> mass term)

without add. crossings 3 (left),with add. crossings 4 but unique (right)

A B




A B

e.g.1 2

3 4

4 6

7 8

! !

" "

! !

" "




A B

e.g.1 2

3 4

4 6

7 8

! !

" "

! !

" "




A B

e.g.1 2

3 4

4 6

7 8

! !

" "

! !

" "




A B

e.g.1 2

3 4

4 6

7 8

! !

" "

! !

" "




A B

e.g.1 2

3 4

4 6

7 8

! !

" "

! !

" "

maximum of3 fields in/out forany gauge group

-> 3 families




A B

e.g.1 2

3 4

4 6

7 8

! !

" "

! !

" "

maximum of 4 fields(no add. branches)

between 2 gauge groups

-> 4 families




A B

e.g.1 2

3 4

4 6

7 8

! !

" "

! !

" "



-> 4 families




A B

e.g.1 2

3 4

4 6

7 8

! !

" "

! !

" "



-> 4 families(unique F0)

Upper bound on the number of families given by the physically

observed number with one exception (4 families in F0).



Can you break this bound?



Can you break this bound?

requires to go to non-toric phases/singularities. there are examples with more families.

Mass Hierarchies

Known result: dP0 (0, M, M); dP1 (0, m, M)

the zero eigenvalue is present in all toric dPs

appearing due to vanishing determinant of Yukawa matrix

Is the structure (0, m, M) generic in models at toric singularities?

CMQ: 0810.5660

m = |Y12|2 +|!16|2

!2(|X62|2 + |Z62|2)

M = |Y12|2 + |X62|2 + |Z62|2

Mass Hierarchies

Known result: dP0 (0, M, M); dP1 (0, m, M)

the zero eigenvalue is present in all toric dPs

appearing due to vanishing determinant of Yukawa matrix

Is the structure (0, m, M) generic in models at toric singularities?

uR

QL

Hu

Hd dR

1

32

m

6!m

3!mCMQ: 0810.5660

m = |Y12|2 +|!16|2

!2(|X62|2 + |Z62|2)

M = |Y12|2 + |X62|2 + |Z62|2

Mass hierarchies II

How do we choose “quarks”?one left & right handed quark in every coupling with quarks.-> every superpotential term has two quarks-> quarks aligned in closed lines

Connected or disconnected lines?connected to be able to higgs to common gauge group

Maximal or non-maximal number of quarks?after Higgsing the same result of vanishing determinant

1 2 1

3 4

5 6 5

1 2 1

7 8

! !

" " "

! !

" " "

! !

Mass hierarchies II




1 2 1

3 4

5 6 5

1 2 1

7 8

! !

" " "

! !

" " "

! !

1 2 1

3 4

5 6 5

1 2 1

7 8

! !

" " "

! !

" " "

! !

Mass hierarchies II




1 2 1

3 4

5 6 5

1 2 1

7 8

! !

" " "

! !

" " "

! !

1 2 1

3 4

5 6 5

1 2 1

7 8

! !

" " "

! !

" " "

! !

general structure (0, m, M)


non-vanishing mass?


non-vanishing mass?

corrections to Kähler potential, deformation of singularity (non-toric)

... back to model building

Flavour mixing: CKMAim: construct models with the correct flavour mixing among quarks

2 types of models: a) up & down from D3D3 states

... we have a non-trivial superpotential (Yukawa

structure) in these singularity models.

What does this imply for model building?

VCKM =

!

"1 ! !3

! 1 !2

!3 !2 1

#

$ .

b) up from D3D3 states & down from D3D7 states

2R2 2R

32L

E

Flavour mixing: CKMAim: construct models with the correct flavour mixing among quarks

2 types of models: a) up & down from D3D3 states

... we have a non-trivial superpotential (Yukawa

structure) in these singularity models.

What does this imply for model building?

VCKM =

!

"1 ! !3

! 1 !2

!3 !2 1

#

$ .

b) up from D3D3 states & down from D3D7 states

uR

QL

Hu

Hd dR

1

32

m

6!m

3!m2R2 2R

32L

E

A left-right modelwith the right CKM

2R2 2R

32L

W =

!

"XL

23

Y L23

ZL23

#

$

!

"0 Z12 !Y62

!Z12!61" 0 X12

!61"

Y62 !X12 0

#

$

!

"X36

Y31

Z36

#

$ .

E


!61

Y31R

B

X36R ,Z36

R

A

X23L , Y23

L , Z23L

X12, Z12

Y62

B"

A"

2R 2R2

32L

6

2

1

4

1 2

W =

!

"XL

23

Y L23

ZL23

#

$

!

"0 Z12 !Y62

!Z12!61" 0 X12

!61"

Y62 !X12 0

#

$

!

"X36

Y31

Z36

#

$ .

E


!61

Y31R

B

X36R ,Z36

R

A

X23L , Y23

L , Z23L

X12, Z12

Y62

B"

A"

2R 2R2

32L

6

2

1

4

1 2

W =

!

"XL

23

Y L23

ZL23

#

$

!

"0 Z12 !Y62

!Z12!61" 0 X12

!61"

Y62 !X12 0

#

$

!

"X36

Y31

Z36

#

$ .

After breaking of U(2)R

W =

!

"XL

23

Y L23

ZL23

#

$

!

"0 Zu

12 !Y u62

!Zu12

!! 0 Xu

12!!

Y u62 !Xu

12 0

#

$

!

"Xu

36

Y u31

Zu36

#

$ +

!

"Xd

23

Y d23

Zd23

#

$

!

"0 Zd

12 !Y d62

!Zd12

vd! 0 0

Y d62 0 0

#

$

!

"Xd

36

Y d31

Zd36

#

$

E


The CKM is given in terms of ratios of Higgs vevs.

!61

Y31R

B

X36R ,Z36

R

A

X23L , Y23

L , Z23L

X12, Z12

Y62

B"

A"

2R 2R2

32L

6

2

1

4

1 2

W =

!

"XL

23

Y L23

ZL23

#

$

!

"0 Z12 !Y62

!Z12!61" 0 X12

!61"

Y62 !X12 0

#

$

!

"X36

Y31

Z36

#

$ .


W =

!

"XL

23

Y L23

ZL23

#

$

!

"0 Zu

12 !Y u62

!Zu12

!! 0 Xu

12!!

Y u62 !Xu

12 0

#

$

!

"Xu

36

Y u31

Zu36

#

$ +

!

"Xd

23

Y d23

Zd23

#

$

!

"0 Zd

12 !Y d62

!Zd12

vd! 0 0

Y d62 0 0

#

$

!

"Xd

36

Y d31

Zd36

#

$

dP1:

E



!61

Y31R

B

X36R ,Z36

R

A

X23L , Y23

L , Z23L

X12, Z12

Y62

B"

A"

2R 2R2

32L

6

2

1

4

1 2

W =

!

"XL

23

Y L23

ZL23

#

$

!

"0 Z12 !Y62

!Z12!61" 0 X12

!61"

Y62 !X12 0

#

$

!

"X36

Y31

Z36

#

$ .


W =

!

"XL

23

Y L23

ZL23

#

$

!

"0 Zu

12 !Y u62

!Zu12

!! 0 Xu

12!!

Y u62 !Xu

12 0

#

$

!

"Xu

36

Y u31

Zu36

#

$ +

!

"Xd

23

Y d23

Zd23

#

$

!

"0 Zd

12 !Y d62

!Zd12

vd! 0 0

Y d62 0 0

#

$

!

"Xd

36

Y d31

Zd36

#

$

VCKM = V †u Vd =

!

""#

Xu12

Zu12

!Y u12

Zu12"61

1!Y u

12Xu12

(Zu12)

2"61

1 ! !Y u12

Zu12"61

1 0 !Xu12

Zu12

$

%%& .

!

""#

0 0 1a !Y d

12Zd

12"61a 0

a !a !Y d12

Zd12"61

0

$

%%& .

normalisation

=

!

"1 ! !3

! 1 !2

!3 !4 1

#

$ .

dP1:

E



!61

Y31R

B

X36R ,Z36

R

A

X23L , Y23

L , Z23L

X12, Z12

Y62

B"

A"

2R 2R2

32L

6

2

1

4

1 2

W =

!

"XL

23

Y L23

ZL23

#

$

!

"0 Z12 !Y62

!Z12!61" 0 X12

!61"

Y62 !X12 0

#

$

!

"X36

Y31

Z36

#

$ .


W =

!

"XL

23

Y L23

ZL23

#

$

!

"0 Zu

12 !Y u62

!Zu12

!! 0 Xu

12!!

Y u62 !Xu

12 0

#

$

!

"Xu

36

Y u31

Zu36

#

$ +

!

"Xd

23

Y d23

Zd23

#

$

!

"0 Zd

12 !Y d62

!Zd12

vd! 0 0

Y d62 0 0

#

$

!

"Xd

36

Y d31

Zd36

#

$

VCKM = V †u Vd =

!

""#

Xu12

Zu12

!Y u12

Zu12"61

1!Y u

12Xu12

(Zu12)

2"61

1 ! !Y u12

Zu12"61

1 0 !Xu12

Zu12

$

%%& .

!

""#

0 0 1a !Y d

12Zd

12"61a 0

a !a !Y d12

Zd12"61

0

$

%%& .=

!

"1 ! !3

! 1 !2

!3 !4 1

#

$ .

In dP1 we get almost the right CKM.

dP1:

E




W =

!

"XL

23

Y L23

ZL23

#

$

!

"0 Zu

12 !Y u62

!Zu12

!! 0 Xu

12!!

Y u62 !Xu

12 0

#

$

!

"Xu

36

Y u31

Zu36

#

$ +

!

"Xd

23

Y d23

Zd23

#

$

!

"0 Zd

12 !Y d62

!Zd12

vd! 0 0

Y d62 0 0

#

$

!

"Xd

36

Y d31

Zd36

#

$

VCKM = V †u Vd


3

2L

2L2

2R

2R2

W =

!

"XL

43

Y L23

ZL23

#

$

!

"0 Z14 !Y64

!Z14!61"42

#2 0 X12!61#

Y64"42# !X12 0

#

$

!

"X36

Y31

Z36

#

$ .

E




W =

!

"XL

23

Y L23

ZL23

#

$

!

"0 Zu

12 !Y u62

!Zu12

!! 0 Xu

12!!

Y u62 !Xu

12 0

#

$

!

"Xu

36

Y u31

Zu36

#

$ +

!

"Xd

23

Y d23

Zd23

#

$

!

"0 Zd

12 !Y d62

!Zd12

vd! 0 0

Y d62 0 0

#

$

!

"Xd

36

Y d31

Zd36

#

$

VCKM = V †u Vd


3

2L

2L2

2R

2R2

=

!

"1 ! !3

! 1 !2

!3 !2 1

#

$ .=

!

"""#

Xu12!61Y u64"

Xu12Zu

14

(Y u64)

2 1

1 !Zu14!61"Y u

64!Xu

12!61"Y u

64Zu

14!61Y u64"

1 !Xu12Zu

14

(Y u64)

2

$

%%%&.

!

"#0 0 1

a "Y d64

Zd14!61

!bZd14!61

"Y d64

0a b 0

$

%& .

W =

!

"XL

43

Y L23

ZL23

#

$

!

"0 Z14 !Y64

!Z14!61"42

#2 0 X12!61#

Y64"42# !X12 0

#

$

!

"X36

Y31

Z36

#

$ .

dP2:

E




W =

!

"XL

23

Y L23

ZL23

#

$

!

"0 Zu

12 !Y u62

!Zu12

!! 0 Xu

12!!

Y u62 !Xu

12 0

#

$

!

"Xu

36

Y u31

Zu36

#

$ +

!

"Xd

23

Y d23

Zd23

#

$

!

"0 Zd

12 !Y d62

!Zd12

vd! 0 0

Y d62 0 0

#

$

!

"Xd

36

Y d31

Zd36

#

$

VCKM = V †u Vd

3

2L

2L2

2R

2R2

=

!

"1 ! !3

! 1 !2

!3 !2 1

#

$ .=

!

"""#

Xu12!61Y u64"

Xu12Zu

14

(Y u64)

2 1

1 !Zu14!61"Y u

64!Xu

12!61"Y u

64Zu

14!61Y u64"

1 !Xu12Zu

14

(Y u64)

2

$

%%%&.

!

"#0 0 1

a "Y d64

Zd14!61

!bZd14!61

"Y d64

0a b 0

$

%& .

W =

!

"XL

43

Y L23

ZL23

#

$

!

"0 Z14 !Y64

!Z14!61"42

#2 0 X12!61#

Y64"42# !X12 0

#

$

!

"X36

Y31

Z36

#

$ .

In dP1 we get almost the right CKM. In dP2 we get the right CKM.CP violation: with correct CKM, Jarlskog invariant J≈ε6

dP2:

E

Summary

D-branes at toric singularities interesting class of models:

Upper bound of 3 families in toric singularities

Mass Hierarchies are possible, generic structure (0, m, M).

Sufficient structure for realistic CKM-matrix & CP-violation (concrete models with this structure)

Open questions: compact models, a completely realistic local model...

kdmq: 1002.1790

Seiberg duality in quivers and dimers

The zeroth Hirzebruch surface

1 2

1 4 1

4 3

3 2 3

1 2

! ! !

"

! ! !

"

!! ! 3

1

4

2

1

3 4

2

! !

" " "

! !

" " "

! !

3 4 3

3 2

1 2 1

2 3

33 4

The zeroth Hirzebruch surface

1 2

1 4 1

4 3

3 2 3

1 2

! ! !

"

! ! !

"

!! ! 3

1

4

2

1

3 4

2

! !

" " "

! !

" " "

! !

3 4 3

3 2

1 2 1

2 3

33 4

Documents

Phenomenology of D-branes at Toric · PDF filePhenomenology of D-branes at Toric Singularities. Disclaimer: This talk is somewhat independent of the LHC. ... details in Aldazabal,