F-theory Family Unification: A New Geometric Mechanism for Unparallel Three Families and Large Lepton-flavor Mixings

Transcription

1 F-theory Family Unification: A New Geometric Mechanism for Unparallel Three Families and Large Lepton-flavor Mixings Shun ya Mizoguchi KEK, Theory Center! Large Lepton-flavor Mixings from E8 Kodaira Singularity: Lopsided! F-theory Family Unification : S.M., JHEP 147 (214) 18 arxiv: [hep-th] Texture via F-theory Family Unification : S.M. arxiv: [hep-th] Jul.23, 214 YITP Workshop Strings and Fields

2 Copyright (c) by Gen-yu SASAKI Star Sand

3 Star sand is not sand Star sand Iriomote by Geomr Copyright (c) by Gen-yu SASAKI Tests of faraminfers : Some kind of shells of tiny creatures

4 You ll Being NEVER many does find star not sand mean at you can near-by find what beaches you want Katase-higashihama beach Copyright(c) Fujisawa City Tourist Association

5 How can we find star sand? Copyright (c) by Gen-yu SASAKI Creatures

6 How can we find star sand? Copyright (c) by Gen-yu SASAKI Creatures Coral reefs ( )

7 Being many does not mean String Landscape you can find what you want Dnally Dreamstime Stock Photos

8 How can we find the Standard Model? Dnally Dreamstime Stock Photos

9 What are characteristic features of the Standard Model?

10 What are characteristic features of the Standard Model?

11 Top is heavy m top 1, m up

12 Zenith-angle dependence of the atomospheric neutrino Minakata,

13 , 13 =

14

15 National Palace Museum. Three are not on equal footing

16 Standard-model-like models? Most (if not all) of the previous string phenomenology models REALIZE these structures by imposing artificial requirements and/or by tuning of parameters, but never EXPLAIN them In particular, in most cases the three generations obtained there are on EQUAL footing, and the hierarchical structure is one arranged by hand

17 What are characteristic features of the Standard Model?

18 How can we find the Standard Model? Dnally Dreamstime Stock Photos Kugo-Yangida E 7 /( U(1) 3 ) Family unification model F-theory

19 How can we find the Standard Model? Dnally Dreamstime Stock Photos Kugo-Yangida E 7 /( U(1) 3 ) Family unification model F-theory

20 Produced by Shun'ya Mizoguchi Local F-theory naturally realizes Kugo-Yanagida ew geometric mechanism EXPLAINING WHY THREE generations

21 Plan 1. Introduction 2. Family unification 3. F-theory 4. F-theory family unification 5. Summary

22 2 FAMILY UNIFICATION

23 Family unification Family unification is the idea that the quarks and leptons are the fermionic partners of the scalars of some coset supersymmetric non-linear sigma model Buchmuller,Peccei,Yanagida; Kugo,Yanagida; Irie,Yasui; Ong The importance of an unparallel family structure was first emphasized by Yanagida, and later by Bando, Kugo and others Bando,Kuramoto,Maskawa,Uehara; Itoh,Kugo,Kunitomo

24 Family unification Remarkably, the Kugo- Yanagida model automatically realizes precisely three UNPARALLEL generations of matter fields needed for the GUT * 5* U(1) 1 U(1) 1 5* 5 1 U(1)

25 Quarks and Leptons in one generation are grouped into 5,1 and 1 of

26 * 5* 1 1 U(1) 5* U(1) 5 1 U(1)

27 * 5* 1 1 U(1) 5* 5 U(1) 1 U(1) /( U(1) 3 ) Kugo-Yanagida model realizes almost minimal necessary matter content for an GUT in an amazingly economical way

28 Family unification in F-THEORY We will show that F-THEORY can naturally realize such a group coset structure To my knowledge this is the first string-theory realization of the old idea of family unification * 5* U(1) 1 U(1) 1 5* 5 1 U(1)

29 Family unification in F THEORY We are interested in some local geometric structure that can realize precisely three unparallel" families This is because if the realization of the SM were a consequence of the global details of the entire compactification space, it would be very hard, if not impossible, to find any reason or explanation for what we observe now * 5* U(1) 1 U(1) 1 5* 5 1 U(1)

30 3 F-THEORY

31 Four essential aspects of F-theory 1. Instead of considering a configuration of the IIB complex scalar = C + ie, one considers a configuration of a FICTITIOUS torus whose modulus equals Vafa

32 A complex scalar z depending only on A family of tori whose shapes vary from point to point

33 Elliptic fibration ( ) The total space is represented as a fiber bundle whose fiber is a torus

34 Four essential aspects of F-theory 1. Instead of considering a configuration of the IIB complex scalar = C + ie, one considers a configuration of a FICTITIOUS torus whose modulus equals Vafa 2. 7-branes are located where an elliptic (=torus) fiber degenerate and becomes singular

35 Degenerate torus : Singular fiber A donut becomes a donut hole

36 cooking/deep-fried-buttermilk- doughnut-holes-with-cinnamonand-sugar

37 The shape of a torus: the complex structure modulus 1

38 D7-brane where a donut hole sits = 1 2 i log z, f(z) = 1 12 log z Let z = e i = 1 2 i (log + i ) C = Im = 2 z = Magnetic flux: 2 d C = = 1 d Magnetically charged object

39 Four essential aspects of F-theory 1. Instead of considering a configuration of the IIB complex scalar = C + ie, one considers a configuration of a FICTITIOUS torus whose modulus equals Vafa 2. 7-branes are located where an elliptic (=torus) fiber degenerate and becomes singular 4. The Kodaira singularities are described by 3. Singularities of elliptic fiberations were classified according to their types investigated by Kodaira Kodaira joining/parting of 7-branes, which involves not only D-branes but general (p,q) branes DeWolfe,Hauer,Iqbal,Zwiebach

40 Monodromy where a donut hole sits = C + ie = 1 2 i log z z e 2 i z, z z =

41 A,B,C: 7-branes with a different monodromy K [p,q] = A = K [1,] = pq p2 q 2 1 pq B = K [1, 1] = C = K [1,1] = All the singularity types in Kodaira s classification are described by a coalesce of A,B and C branes DeWolfe,Hauer,Iqbal,Zwiebach(1998)

42 Collapsible set of 7-branes are classified: Kodaira s classification Fiber&type Singularity&type 71branes Brane&type In An$1 A n An$1 II A AC H III A1 A 2 C H1 IV A2 A 3 C H2 I* D4 A 4 BC D4 In* Dn+4 A n+4 BC Dn+4 II* E8 A 7 BC 2 E8 III* A 6 BC 2 IV* E6 A 5 BC 2 E6

43 Four essential aspects of F-theory 1. Instead of considering a configuration of the IIB complex scalar = C + ie, one considers a configuration of a FICTITIOUS torus whose modulus equals Vafa 2. 7-branes are located where an elliptic (=torus) fiber degenerate and becomes singular 4. The Kodaira singularities are described by 3. Singularities of elliptic fiberations were classified according to their types investigated by Kodaira Kodaira joining/parting of 7-branes, which involves not only D-branes but general (p,q) branes DeWolfe,Hauer,Iqbal,Zwiebach

44 What happens if there are B- and C-branes? Ordinary D-brane case A (1,)string A [1,]brane [1,]brane A A Short strings between the close branes yields light W (p,q) string can end on [p,q] branes bosons

45 Monodromy around B,C branes cut cut A A (1,) string turns into (-1,) string different [1,]brane state (1,)string B C (-1,)string [1,]brane

46 String junction: (p,q) analogue of open string cut cut B C A A (-1,1) and (-1,-1) strings are pulled out when the [1,]brane (-1,1)string (-1,-1)string (,1)string (1,)string (-1,)string string crosses over the B and C branes [1,]brane

47 What happens if there are B- and C-branes? When N D-branes come on top of each other, one gets U(N) gauge symmetry Witten. In this case the relevant massless W-bosons are supplemented by the excitations of light open strings ending on different D-branes Likewise, the extra massless fields needed for the gauge symmetry enhancement to an exceptional group can be thought of as coming from the string junctions connecting the coinciding 7-branes Gaberdiel,Zwiebach U(1) U(2) U(3)

48 Coalesced branes and singularities:

49 Coalesced branes and singularities: SO(1)

50 Coalesced branes and singularities: E 6

51 Coalesced branes and singularities: E 7

52 Matter from string junction T.Tani( ) Nucl.Phys.B62(21)434 Enhanced gauge symmetries in F theory are accounted for by the light string junctions between 7- branes DeWolfe, E6 27 Hauer,Iqbal,Zwiebach(1998) Likewise, the appearance of massless hypermultiplets was shown to be understood U(1) in terms of junctions localized at the intersection connecting gauge and matter 7-branes Tani(21) This is the same as the supersymmetric coset!!! Gives a perfectly consistent picture E6

53 E6 27 U(1) Extra singularity

54 SO(1) 16 E6 U(1) Extra singularity

55 1 SO(1) U(1)

56 4 F-THEORY FAMILY UNIFICATION

57 Three new contributions in: F-theory Family Unification SM, JHEP 147(214) 18, arxiv: [hep-th] Consider a set of coalesced local 7-branes of a particular Kodaira singularity type and allow some of the branes to bend and separate from the rest, so that they meet only at an intersection point

58 Three new contributions in: F-theory Family Unification SM, JHEP 147(214) 18, arxiv: [hep-th] I have shown that 1. The six-dimensional matter spectrum coincides (after an orbifold projection) with that of a supersymmetric coset sigma model whose target space is a corresponding homogeneous Kahler manifold

59 Table 2.SummaryofmatterfieldsinF-theory/heteroticdualityinsixdimensions.Onlythecases for the split type with rank 2 are listed, where n is ±(the number of instantons 12) inoneof E 8 s on the heterotic side, and r specifies how they are distributed when the commutant group is a direct product [6]. In addition to the data shown in [6], the corresponding 7-brane configurations as well as the homogeneous Kähler manifolds are also displayed. Gauge group Neutral hypers Charged matter 7-branes Homogeneous Kähler manifold n+8 E 7 2n A+ A6 BCC E 8 /(E 7 U(1)) E 6 3n + 28 (n + 6)27 A+ A 5 BCC E 7 /(E 6 U(1)) SO(1) 4n + 33 SO(8) 6n + 44 SU(4) 8n + 51 SO(4) 1n + 54 (n + 4)16 A 5 BC+ C E 6 /(SO(1) U(1)) (n + 6)1 A+ A 5 BC SO(12)/(SO(1) U(1)) (n + 4)8 c (n + 4)8 s A 4 BC+ C E 5 /(SO(8) U(1)) (= SO(1)/(SO(8) U(1))) (n + 4)8 v A+ A 4 BC SO(1)/(SO(8) U(1)) (4n + 16)4 A 3 BC+ C E 4 /(SO(6) U(1)) (= /(SU(4) U(1))) (n + 2)6 A+ A 3 BC SO(8)/(SO(6) U(1)) (4n + 16)((1, 2) +(2, 1)) A 2 BC+ C E 3 /(SO(4) U(1)) (= SU(3)/(SU(2) U(1))) n(2, 2) A+ A 2 BC SO(6)/(SO(4) U(1)) SU(3) 12n + 66 (6n + 18)3 A+ A 3 SU(4)/(SU(3) U(1)) SO(12) 2n + 18 r n+4 r A 6 BC+C E 7 /(SO(12) U(1)) (n + 8)12 A+ A 6 BC SO(14)/(SO(12) U(1)) r 2 2 A6 +X [2, 1] +C E 6 /(SU(6) U(1)) SU(6) 3n r + 21 (2n r)6 A+ A 6 SU(7)/(SU(6) U(1)) For all (the split cases) the patterns of gauge symmetry breaking in the F/heterotic duality investigated by Bershadsky et.al., the six-dimensional charged matter content corresponds to a homogeneous Kahler manifold of the relevant type 5n + 36 (n +2 r)15 A 6 +B+C SO(12)/(SU(6) U(1)) (3n + 16)5 A+ A 5 SU(6)/( U(1)) (n + 2)1 A 5 +B+C SO(1)/( U(1))

60 Three new contributions in: F-theory Family Unification SM, JHEP 147(214) 18, arxiv: [hep-th] 2. Such a brane configuration can preserve N=1 sixdimensional SUSY, at least locally

61 Proof of supersymmetry ds 2 4 = e dzd z + e (dw + dz)(d w + d z) = (z,w), : real functions : complex function w = x 6 + ix 7 Any hermitian metric can be written in this form ei e µ = a e ā ī µ = i, ī; i = z,w; ī = z, w; = a, ā; a =1, 2; ā = 1, 2 e a i e 8 i + ie 9 i e 6 i + ie 7 i = e ā ī e 8 ī ie 9 ī e 6 ī ie 7 ī = 1 2 I e 2 e 2 e 2 e 2 = 1 2 I, I = 1 1 g µ = e µ e, ds 4 = g µ dx µ dx e 2 e 2,.

62 1 8 + i 9 = 1 8 i 9 = i 7 = Choice of gamma matrices 2 2 i 7 = 3 2 I = I = = = C A, 1 C A, Due to the holomorphic assumption we have, again, P ī = (ī = z, w). The dilatino variation thus reads P i ea i a Since the leftmost columns of a (a =1, 2) are zero, this vanishes for = 1 C A C A, 1 C A

63 Gravitino variation µ : Since the nonzero component of is only the first one, we are only concerned with the first columns of 1 = 2 = 1 = 2 = e 2 ( w w + z ) 2e 2 e ( w z )+e w e 2 e ( w ) z w 2e 2 ( w + w z ) ((1, 1) component of 1 ) ((1, 1) component of 2 ), Since the ``Bismut-like" connection contains, besides the spin connection, only Q µ which is U(1), the gravitino variations vanish only if the off-diagonal components (of the first columns) do,,

64 SUSY transformations: µ = 1 µ = i P µ µ 1 i 4 µ 2 Q µ Since the Bismut-like connection contains, besides the spin connection, only Q µ which is U(1), the gravitino variations vanish only if the off-diagonal components (of the first columns) do e ( w z )+e w = and w + w z = They are equivalent to w(e + e ) = z (e ) and or w(e ) = z e ig jī = j g iī, īg ji = jg īi Kähler (1) (2) That the system of equations (1) (2) has a solution can be confirmed by expanding them in the coordinates and showing that the coefficients are determined order by order in this expansion

65 Using the solutions, and satisfying (1) (2) Q i = i 2 i log( ), Q ī = + i 2 ī log( ) for some holomorphic functions A Killing spinor exists if + = log( )+F (z i )+ F ( zī)

66 Three new contributions in: F-theory Family Unification SM, JHEP 147(214) 18, arxiv: [hep-th] 3. If one starts from the E 7 singularity, one obtains the same chiral matter content as the E 7 / (xu(1) 3 ) Kugo- Yanagida model yielding precisely three generations with an UNPARALLEL family structure!!!

67 Kugo-Yanagida in F-theory * 5* 1 1 U(1) 5* 5 U(1) 1 U(1) To get a 4D theory, we still need to compactify on T 2 and take an orbifold

68 Large Lepton-flavor Mixings from E8 Kodaira Singularity SM, arxiv: [hep-th] Sato-Yanagida s scenario using the Frogatt-Nielsen mechanism Assume THREE PAIRS s i, s i (i =, 1, 2) of singlet scalar fields with particular U(1) charges in the E 7 /(xu(1) 3 ) model Hierarchical Yukawa couplings for both the quark and lepton sectors, qualitatively in agreement Large lepton / small quark mixing angles hs 1 i provided that, and hs i.5 hs 2 i hs 1 i.5 tan 1

69 Sato-Yanagida s scenario is naturally realized in F-theory family unification! Necessary Frogatt-Nielsen fields naturally arise if we consider the branching of E8 singularity!

70 5 SUMMARY

71 Summary We have shown that a certain local 7-brane system in F theory can realize, already at the level of six dimensions, the same quantum numbers as that of the SUSY nonlinear sigma model considered in family unification If half of the spectrum is projected out, then it becomes precisely what we observe in nature The massless spectrum was studied by investigating We considered a set of coalesced local 7-branes of a particular singularity type and allowed some of the branes to bend and separate from the rest string junctions near the intersection and shown to be the same as the sigma model