Sakura Schafer-Nameki- F-theory GUTs in Three Steps

8/3/2019 Sakura Schafer-Nameki- F-theory GUTs in Three Steps

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Outline

0. Bottom-up: Local, semi-local, global

1. Local F-theory GUTs

2. Semi-local Model

Embedding into local E8 singularity

Monodromies Phenomenological requirements3. Global Models

4. Phenomenological Implications

Gauge coupling unification Gauge-mediation with non-GUT messenger sector

5. Conclusions and Outlook


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Motivation

Aims:

Phenomenologically viable models from string theory

Imprint of UV completion upon low energy theorybuild complete string models

main challenge:

generic statements, valid for a large class of models

Method: Bottom-up approach

[Aldazabal, Ibanez, Quevedo, Uranga], [Verlinde, Wijnholt]

Systematically build models starting with effective theory on branes Incorporate constraints from embeddability into compact model

Three-step strategy


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Bottom-up Three-Step Strategy

Step 3: Step 2: Step 1:

Global Model: Semi-local Model: Local Model:

Compact Geometry+ Fluxes

Embeddability

strong phenorestrictions

Effective field theory

on D-branes:SU(5) GUT


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Three-Step Strategy with F-theory

Step 1. Local Models:

Effective field theory on 7-branes: SU(5) GUT

Step 2. Semi-local Model:

Impose general conditions for embedding into local CY4

Embeddability implies strong phenomenological restrictions

Step 3. Global Model:

Construction of elliptically fibered CY4 realizing semi-local models

Global model: Talk by Saulina


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References

Step 1. Local Models:

[Donagi, Wijnholt], [Beasley, Heckman, Vafa]: GUTs[Marsano, Saulina, SS-N ], [Heckman, Marsano, Saulina, SS-N, Vafa]: SUSY-breaking

[Heckman, Vafa + Bouchard, Cecotti, Cheng, Kane, Seo, Shao,Tavanfar,... ], [Font,

Ibanez], [Li, Nanopoulos + ....] [Watari, Tatar + Hayashi, Kawano, Toda, Tsuchiya,

Yamazaki], [Marchesano, Martucci]: Cosmology, Neutrinos, Flavour.

Step 2. Semi-local Model:

[Andreas, Curio], [Hayashi, Kawano, Tatar, Watari], [Donagi, Wijnholt],

[Marsano, Saulina, SS-N], [Dudas, Palti]

Step 3. Global Model:

[Marsano, Saulina, SS-N]: compact geometry for F-theory GUTs

[Blumenhagen, Braun, Jurke, Krause, Weigand], [Cordova]


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1. Local Models

Low energy gauge dofs decoupling gravity dofs MGUTMPl 103:SU(5) SUSY GUT

3 generations of

10M =

Q (3,2)+1/6Uc (3,1)2/3Ec (1,1)+1

, 5M =

Dc (3,1)+1/3

L (1

,2

)1/2

Higgses: lifting triplets

5H=

Hu (1, 2)+1/2

H(3)u (3, 1)1/3, 5H=

Hd (1,2)1/2

H(3)d (3,1)+1/3

W

u 5H

10M

10M + d 5H

5M

10M

SUSY-breaking, flavour, neutrino physics, etc.


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F-theory

F-theory [Vafa][Morrison, Vafa] =Type IIB [Green, Schwarz] vacua with varying

axio-dilaton:

= C0 + ie

Geometrize consistent with SL2Z

compactify to d = 4 on elliptically fibered CY4 with base B6:

E

X4

B S


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Gauge degrees of freedom/D-branes in F-theory

F-theory: realizes branes in terms of geometric singularities.

Singularity type:

An: y2= x2 + zn+1

Dn: y2= x2z+ zn1

E6: y2 = x3 + z4

Perturbative interpretation:

An: IIB with D7-branes

Dn: IIB orientifolded with D7

and O-planes

En: no perturbative IIB picture,

exceptional 7-branes


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Matter fields

[BHV I, II], [Donagi, Wijnholt]

7-branes inside B6 wrapping surfaces, which intersect over a curve :

=

Bifundamental matter is localized along curves

G

SU(5)

U(1), in particular: SU(6) : 5, 5, SO(10) : 10,10

Chiral matter from additional gauge fluxes


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Yukawa couplings from Triple-Intersections


Yukawa couplings from triple intersection of matter curves:

Gp SU(5)U(1)1 U(1)2Such as

SO(12) : 5H 5M 10M E6 : 10M 10M 5H SU(7) : 5 5 1


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SU(5) F-theory GUT[BHV II]

A4 singularity 3 10M: SO(10) enhancement

3

5M, 5H and 5H:

SU(6) enhancement

Top/bottom Yukawa:E6 and SO(12) enhancement

SO(10)

SU(5)

SU(6)

E6 SO(12)

Wbi j H5i5j10 + ti j H5i10j10 QDHd+ LEHd+ QUHu


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GUT breaking


GUT-breaking by hypercharge flux FY:

SU(5)

SU(3)

SU(2)

U(1)Y

24 (8,1)0 (1,3)0 (1,1)0 (3,2)5 (3,2)+5Gauge Fields Exotics

Choose U(1) gauge bundle LY such that

S(L5Y ) = 0

Also solves doublet-triplet splitting by lifting triplets:

FY|M = 0 , FY|5H =+1 , FY|5H = 1Masslessness ofU(1)Y: [Buican, Malyshev, Morrison, Verlinde, Wijnholt]

FY is dual in SGUT to 2-cycle, that is homologically trivial in B


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Further Properties

Favourable flavour structure [Heckmann, Vafa +...][Font, Ibanez] all generations on same curve: leading order rank 1 Yukawas B-field: realistic mixing (CKM) Minimal flavour mixing

zu zd E6 and SO(12) points combine to E8

Absence of dimension 5 proton decay operators:

Naively QQQL absent ifHu and Hd localize on different curves

Models with U(1)PQ symmetry [Marsano, Saulina, SS-N][Heckman, Vafa]

10M 5M 5H 5H

PQ 1 1 2 2

Absence of tree-level -term H H, dim 5 and 4 proton decay ops

(10M 5M 5M and 10M 5H 5H) high-scale gauge mediation scenarios


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Step 1: Local model Summary

Phenomenologically viable SU(5) SUSY GUTs realized

in local F-theory 7-brane intersections.

Including: promising flavour and SUSY-breaking

phenomenology


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2. Semi-local Model

Impose constraints arising from embedding into local CY4:

Global CY4: Local CY4:

elliptic fibration over B ALE-fibration over SGUT

y2

= x3

+ f x+g

E X4B SGUT

ALE X4

SGUT

For SU(5) GUTs: Deformed E8 singularity over SGUT


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E8 singularity over SGUT

Local geometry around F-theory 7-branes is a deformed E8 singularity

y2 = x3 + b5xy+ b4x2z+ b3yz

2+ b2xz

3+ b0z

5

E8 gauge theory broken to SU(5) by adjoint VEVs


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Local E8 singularity over SGUT

y2 = x3+ b5xy+ b4x2z+ b3yz

2+ b2xz

3+ b0z

5

E8 singularity: b2,3,4,5 = 0 SU(5) GUT:

SU(5) : bm = 0

SO(10): 10 matter 0 = b5

SU(6) : 5 matter 0 = P = b0b25 b2b3b5+ b23b4

SO(12) : Bottom: 0=

b5=

b3E6 : Top: 0 = b5 = b4

bn depend on embedding ofSGUT into B (KSGUT and NSGUT|B)

Semi-local model: No need to specify these Generality of analysis


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Breaking the E8 gauge theory

Breaking E8:

E8 SU(5) SU(5)GUT248 (24,1)+ (1,24)+ (10,5)+ (5, 10)

+ (10,5)+ (5,10)

Breaking via:

SU(5)

adjoint vev

diag (1, 2, 3, 4, 5)

Geometrically: i = i(bn) where bn = coefficients in

deformed E8 singularity

5

4

3

2

1

6

7

5 8

SU(5)GUT

4

3

2

1

SU(5)


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Matter

All matter arises from E8 and i give masses to SU(5)GUT multiplets

E8 SU(5) SU(5)GUT248 (24,1)+ (1,24)+ (10,5)+ (5,10)+ (10,5)+ (5, 10)

SU(5) weights:5: i

10: i + j

GUT Matter/Higgses:10: i

5: i + j


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Main idea

How do constraints from embedding into semi-local model to arise?

E8 SU(5)GUT U(1)4

U(1)4 = max torus ofSU(5)

Naively (aka local model):GUT-fields carry 4 independent U(1) charges: (1, , 5)Superpotential couplings dictated by 4 independent U(1)s

Semi-local model:U(1)s get identified by monodromies

Highly constrains embeddable models


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Monodromies

Local model:

E8 SU(5)U(1)4

4 independent U(1) charges labeled by iSemi-local model:

Geometry ofE8 sing. given by bn:

bn(i) = b0 Pn(i)

b5 b012345 , b4 b0 i


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Spectral Cover[Donagi, Wijnholt]

Spectral cover C10 = auxiliary space tokeep track of monodromies

C10 is 5-fold cover ofS

GUT:

b0U5+ b2V

2U3+ b3V3U2+ b4V

4U+ b5V5= 0

Monodromy group G S5 acts onsheets and identifies U(1)s.

S

5

4

3

2

1

GUT


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Monodromies and independent U(1)s[Tatar, Tsuchiya, Watari], [Marsano, Saulina, SS-N]

Independent gauged U(1)s are encoded in # orbits of monodromy group

U(1) gauge bosons are elements in CSA:

G = transitive subgroup ofS5:only invariant combination is 5i=1 i = 0

no gauged U(1)

i in reducible representation ofG:Norbits of10s labeled by i

(N

1) gauged U(1)s

2

3

4

1

5

SGUT

2

3

4

1

5

SGUT


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Possible Monodromy Groups

Monodromy groups G S5 and orbits ofi i.e. 10s:

10 orbits # of U(1)s Monodromy Group(1)(2)(3)(4)(5) 4 id

(ij)(k)(l)(m) 3 Z2

(ij)(kl)(m) 2 Z2 Z2(ijk)(l)(m) 2 S3 , Z3

(ijk)(lm) 1 S3 Z2 , Z3 Z2(ijkl)(m) 1 S4 , D4 , Z4 , Klein4

(12345) 0 S5

Remark: in the case of 4+ 1 orbits the refinement to D4 , Z4 , Klein4

yields different orbits for 5 i.e. i + j.


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Putting Monodromies to use:

Phenomenological wish list:

no exotics, 3-generations

top and bottom Yukawas

flavor structure no tree-level -term...

What do these constraints imply for the allowed monodromy groups?


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Constraints on the Monodromy Group

[Marsano, Saulina, SS-N]

Constraint arises from the requirement ofno exotics:

FY10 = 0

FY

5

In particular: no net FY flux on 5 curves

Hu and Hd have to be on the same curve

monodromy group

S4

No tree-level -term Monodromy group is D4 or Z4 or Klein4

2

3

4

1

5

(4)C

10

SGUT

C(1)

10


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Constraints on the Monodromy Group (cont.)

Summary: [Marsano, Saulina, SS-N]

SU(5), Yukawas, no exotics and no tree-level

Allowed monodromy groups: G = D4 or Z4 or Klein4 U(1) symmetry contraining superpotential:

10M 5M 5H 5H

U(1) 1 3 2 2

No gauged U(1)PQ

Main trouble: Dim 5 proton decay:

1

10M 10M 10M 5M QQQL

cannot realize neutrino scenarios of[Bouchard, Heckman, Seo, Vafa]

Minimal Model ruled out (modulo tuning)


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Semi-local Model: Relaxing minimality

[Marsano, Saulina, SS-N]

1. Model with U(1):

No exotics Monodromy group: D4,Z4,Klein4

Problem with proton decay

tune further to get approximate global U(1)PQ2. Model with U(1)PQ:

No problems with proton decay Hypercharge flux on matter curves Non-GUT exotics

3. Beyond E8:

More general structure than E8?


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Summary: Semi-Local Models

F-theory GUT embedded into E8 singularity Minimal SU(5) GUT embedding into semi-local model:

Monodromy groups: Z4, D4, Klein4

Ruled out because of dim 5 proton decay Relaxed constraints:

Non-GUT exotics from 10 and 5

Key point: Analysis is generic and independent of specific CY4


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3. Global Model [Marsano, Saulina, SS-N] Talk by SaulinaExplicit realization in compact CY4: Proof of principle

Recall: X4 = elliptically fibered CY4 with three-fold base B:

E X4B SGUT

Constraints on B6:

X4 Calabi-Yau: B almost Fanoi.e. K1B3 semi-ample Hypercharge constraint: FY dual in SGUT to a

2-cycle that is trivial in BSGUT

3

e2e1


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4. Phenomenological Imprints

We have seen: Semi-local F-theory GUTs with gauged U(1)PQ

automatically non-GUT exotics

this is generic: independent of the specifics of the CY4Exotics arising from FY restricting non-trivially to 10 matter curves:

Field Multiplicity

(3,2)+1/6 M

(3,1)2/3 M+ N

(1,

1)+1 MN

(3,2)1/6 M(3,1)+2/3 M+ N

(1,1)1 MN


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What is the phenomenology of such models?

[Dolan, Marsano, SS-N] in progress

1. Running of gauge couplings:

Gauge-coupling unification?

2. SUSY breaking and phenomenology of non-GUT exotics:

non-GUT exotics as gauge messengers:

W FXX+ Xf f

coupling from 10 10 1 2483


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On gauge-coupling unification in F-theory GUTs[Donagi, Wijnholt], [Blumenhagen], [Conlon], [Dolan, Marsano, SS-N]

Important scales in the problem:

mz MExotics MKK MWinding

Scales in the compactification: RS, RB

M = M(MPl , RB) measures 7-brane tension

M4

KK=

GUTM

4

R = R(RS, RB) = size of direction transverse to SGUT MWinding = RM2


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MSSM-running:

With 1 = 3, 2 =1, 3 = 335

1i (MGUT) = 1i (mz)

i2

ln

MKKmz

KK-thresholds: [Wijnholt 10, private conversation] 8d theory 7-brane worldvolume theory has divergence log

External contribution (from bulk) to cancel log divergence:divergence is capped off at winding scale [Conlon]

MWinding = Winding scale > MKK

Can be written in a 4d looking way

1i 1i KK

2ln

MWinding

MKK


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Non-GUT exotic contribution:

1i 1i Exotics

2ln

MKKMExotics

In summary:

1i = 1i (mz)

i2

lnMKKmz

KK

2ln

MWinding

MKK Exotics

2ln

MKKMExotics


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Room for Unification

[Dolan, Marsano, SSN]

Condition for consistency with gauge couplings at mz

i2

lnMKKmz

+

KK

2lnMWinding

MKK

+

Exotics

2ln

MKKMExotics

0

Non-negligible effects as MWinding

M2KKRGUT

and R

large

seem difficult to satisfy without Exotics Can be achieved for reasonable range of scales

MKK 1015

GeV , MExotics 1014

GeV , MWinding 1018

GeV

Gauge-mediation with non-GUT exotics as messengers andMMess = MExotics > 10

13GeV

High-scale gauge mediation with PQ symmetry


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5. Summary...

Bottom-up approach: Global Semi-local Local

Local F-theory GUTs: 2/3 splitting, flavour, SUSY-breaking, . . .

Requiring Embedding into Semi-local Model: E8 gauge theory andhighly constraining:

Non-minimal SU(5) GUT with non-GUT exotics

Non-minimal gauge mediation models


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... and Outlook

Pheno: generic signatures of F-theory GUTs, in particular ofnon-GUT exotics

Lift ofsemi-local models to global models:fate of the U(1) symmetries

Detailed analysis of SUSY-breaking in the compact case

Moduli stabilization


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Sakura Schafer-Nameki- F-theory GUTs in Three Steps