超弦理論のコンパクト化での フレーバー構造について

京産大益川塾セミナー 15年１月２６日(月)

Kavli IPMU 渡利泰山

超弦理論のコンパクト化での フレーバー構造について

苦しい状況をどのように乗り越えていくか

苦しい状況にあって、何ができるのか

苦しさ、、、

• Local effective field theory model building

– Lagrangian の係数については、自由でありすぎる。

– 「言い換え」レベルのゲームならできるのだけれど。

– 理論屋として何ができるのか、、

• String theory には解がたくさんありすぎる

–一瞬夢を見た、というのが妥当 (今のところ)。

–言い換え、分類学といった作業ならできるけど。

• この環境になって、すでに１５年ほどが経つ。

Plan of this Talk

• 質量から混合角へ

• （超弦理論）革命のあと

• ’08—’09 の仕事から分かったこと

• ’14 F-理論の landscape について分かったこと

’06 Tatar TW; Proton decay, Yukawa couplings and underlying gauge symmetry in string theory

‘08 may; ’09 jan; ’09 may; ’09 oct (incl. myself) + Heckman Vafa et.al. ’08 nov; ’09 oct ;

’13 dec. + ’14 jan A.Braun, Y.Kimura, TW ’14 aug. A.Braun, TW

質量から混合角へ

’98 Sato Yanagida, ’99 Hall Murayama Weiner ’01(?) 柳田さんの捕足コメント@ニュートリノ研究会

generation structure -- conventional picture in the past--

ud

cs b

t

e

1 2 3

‘97 1st generation 2nd generation 3rd generation

10.Lq

10 10 (5)ij kl m

ijklmh 10 5 (5)ij

Mi jh 5 (5)5 (5)h h

u-Yukawa d/e-Yukawa nu Majorana mass

strong hierarchy weaker hierarchy (not known yet)

[ ]c

L Lu u q [ ] c

L Ld q d

[ ]c

L Le e l2[ ]L Ll

L CKM Lu W V dL LFM Le W U

small mixing large mixing

5Ll

3 distinct hierarchical 10’s less structured ‘s. 5

(超弦理論の)革命のあと

hep-th/0602238 Tatar and Watari

string duality

3Het /CY2 3M/ or IIA/G CY

4 3F/ or IIB/CY CY

(assumes N=1 SUSY in the following)

たとえば、、ゲージ群が

• Het. string: E8 x E8 or SO(32) の部分として

• IIA: 6-brane on 3cycle, IIB: 7-branes on 4cycle

• 11D sugra: degenerate S^1 fibre

• F-theory: degenerate T^2 fibre

弦理論の定式化の違いは、必ずしも 低エネルギーの物理の違いに直結しない。

3Het /CY 2M /G

4F/CY

6 (2) (5)E U SU

can be realized in Het, M and F.

Yukawa: generated. Qualitative similarlity is not suprising. That’s duality.

All the rest is quantitative problem.

Matter repr. and Yukawa couplings: Lie algebra Multiplicity (# of generations): topology Flavor pattern: geometry moduli

(backup slide)

•

• Lie algebra:

• Covariant derivative interactions of super YM become up-type Yukawa couplings:

2. [(2,10) ( 2,5)] . . ( .,1) (1, .)adj h c adj adj

6 (2) (5) .GUTE U SU breaks U(2) completely.

10 5ab ematter fields:

6E;

5; 10; ; 10;[ , ] .B cd

e A ab AB abcdet t i t

E6tr [ , ] 10 10 5 .m ab cd e

m abcdeA

6E

Tatar TW ‘06

up-type Yukawa phenomenology 中間まとめ

• 11D sugra / G_2 holonomy mfd:

– does not work without doing something

• F-theory / elliptic CY_4:

– study motivated progress in ‘08—’09

• Het / CY_3:

– expected not to work generically (triple overlap)

– OK near the toroidal orbifold pts (generalization?)

– whole picture far from clear even now.

( overlap integral over a 6-dim mfd. )

’06 Tatar TW

’08—’09 の仕事で分かったこと

0805 Hayashi, Tatar, Toda, TW, Yamazaki 0901 Hayashi, Kawano, Tatar, TW, 0905 Tatar, Tsuchiya, TW 0910 Hayashi, Kawano, Tsuchiya, TW 0811 Heckman Vafa 0910 Cecotti, Cheng, Heckman, Vafa

What’s happening at 7-brane singularities?

6E6D

6A

4 .singlA 10 .repr

5 .repr

Charged matter 0-modes localized comp’nts? No.

how many?

how to determine “wavefunctions” divisors of are determined by codim-3 singl. &

Hayashi et.al. 08 may

0 ( ; .)H

0 ( ; .)h

(4).GYukawa couplings: computed by overlap integral around codim-3 singl. points.

Hayashi et.al. 09 jan

Andreas Curio ’99, Hayashi et.al. ’08 may

Each singl. point approx rank-1 Yukawa matrix

Immediate consequences

• dim-6 proton decay enhancement

– Friedman Witten ’02: linear div. in M / G_2

– F-theory: no local cmp’nts logarithmic div.

• Yukawa matrices in the low-energy eff. theory

– contrib. from all the singl. pts. rank-1 problem

– #(E_6 pts) is always even.

† † † †10 10 10 10K qqu e † † † †10 10 5 5K qu ld

’08 Wijnholt

’09 oct Hayashi et.al., ’09 oct Cordova

0

/( ) / ( ).L Rp e p

Right-handed Neutrinos

• Observation:

• Cpx str moduli: have couplings w/ 5+5bar.

• 4 obs. by 3 parameters.

– RH nu masses post-dicted properly.

(atmospheric neutrino oscil.)

4

GUT

1( ) ,GUT sR M

,GUT

GUT

cM

R

2 6 8

6(4 ) ;Pl sM R M

3 2. .6

1 .cpx strs

MR M ’02 Kachru Schulz Trivedi

’09 may Tatar Tsuchiya TW

2 solutions to the rank-1 problem

• 09 oct Hayashi et.al. – use Arkani-Hamed Schmaltz ’01

– fit well with landscape / anarchy ’99 Hall Murayama Weiner

– tuning one complex parameter required

• ’08 nov Heckman-Vafa + follow up papers

– make sure that only 1 singl. point contributes

– (anomalous) Peccei-Quinn U(1) symmetry required

– small mixing angles among q’s: extra fine tuning(?)

’07 Hall Salem TW

• The solution in Hayashi et.al. ’09 oct.

Im 1(10)Cfor

hierarchy among 10’s ’01 Arkani-Hamed Schmaltz ’04 Inabez Marchesano ’07 Hall Salem TW

2 solutions to the rank-1 problem

• 09 oct Hayashi et.al. – use Arkani-Hamed Schmaltz ’01

– fit well with landscape / anarchy ’99 Hall Murayama Weiner

– tuning one complex parameter required

• ’08 nov Heckman-Vafa + follow up papers

– make sure that only 1 singl. point contributes

– (anomalous) Peccei-Quinn U(1) symmetry required

– small mixing angles among q’s: extra fine tuning(?)

’07 Hall Salem TW

• The solution in ’08 nov Heckman-Vafa + others

Tune cpx str so that there is anomalous PQ U(1) symmetry.

1 non 210' 10 ' & 10 ;s s

2 2 † 3 non( 2, 3)5 ' 5 ' ( ) 5 'u ds s H H s

Make sure that just one singl. pt contributes to the eff. theory Yukawa matrix

approximately rank-1 Yukawa matrices; Froggatt-Nielsen like hierarchy

Quark mixing angles: will be small (hopefully), if the E6 singl pt (up Yukawa) and D6 singl pt (d Yukawa) are close.

(extra tuning in cpx str ?)

2 solutions to the rank-1 problem

• 09 oct Hayashi et.al. – use Arkani-Hamed Schmaltz ’01

– fit well with landscape / anarchy ’99 Hall Murayama Weiner

– tuning one complex parameter required

• ’08 nov Heckman-Vafa + follow up papers

– make sure that only 1 singl. point contributes

– (anomalous) Peccei-Quinn U(1) symmetry required

– small mixing angles among q’s: extra fine tuning(?)

’07 Hall Salem TW

’14 F-理論の真空解の統計学

’03 Ashok Douglas ’04 Denef Douglas ’08 Denef ’13 dec A. Braun, Y.Kimura, TW (math) ’14 jan A. Braun, Y.Kimura, TW (phys) ’14 aug A. Braun, TW ’14 aug A. Braun, TW

Type IIB の flux コンパクト化

• topological flux を入れると、cpx str moduli は特定の値のみで supersymm

• 可能な topological flux の配位

–可能な supersymmetric 真空の ensemble.

• flux の連続近似で、真空解の分布関数を導くことが可能。 ’03 Ashok Douglas, ’04 Denef Douglas

ADD formula for F-theory

• IIB flux： fix both CY3 cpx str AND D7 config.

– an ensemble of (7-brane) gauge group generated.

• Ashok-Denef-Douglas formula

– applicable as it is even in F-theory/CY4.

’08 Denef ’14 Braun Kimura TW.

’14 Braun, Kimura TW ’14 Braun, TW

3,1( ).m h X

(4) 2

* gen

( ) 1( ) .

24 2

XL G

2,22( 1) ( ).HK m h X

#(flux vacua) distrb. on the moduli space

often 1,1 3,1( ) ( ),h X h X2,2 2,2

, ( ) ( );V RM Hh X h X

* *( ) 24 .K X L L

statistics of gauge group and

• Because

• -dependence

• Gauge Group Dependence

– different by for gauge groups with different rank

– rank-4 gauge group : in probability.

* *( ) 24 .K X L L

/22 *

*

(2 )(dim. , rad )

( / 2)!

KLvol K L

K

#(flux vac) ~

*

*3.

* !

LLK

eL

’14 Braun, TW

2

* (1000) (1) .genL N Gaussian distr. in genNgenN

genN

*L (1000)

100010