F-theory GUTs with Discrete Symmetry Extensions. George Leontaris. Ioannina University GREECE

Transcription

1 Madrid June 2015 F-theory GUTs with Discrete Symmetry Extensions George Leontaris Ioannina University GREECE 1

2 Outline of the Talk Introductory remarks F-theory and Elliptic Fibration F-GUTs with discrete symmetries Mordell-WeilU(1) and GUTs Concluding Remarks 2

3 A Properties of Ordinary GU T s 3

4 interesting features Gauge coupling unification Assembling of SM fermions in a few irreps. Charge Quantisation deficiencies fermion mass hierarchy and mixing not predicted Yukawa Lagrangian poorly constrained Baryon number non-conservation... Solution requires new insights... such as: Discrete and U(1) symmetry extensions These appear naturally in F T HEORY constructions 4

5 New Ingredients from F-theory Discrete and U(1) symmetries: necessary tools to suppress or eliminate undesired superpotential terms Fluxes :... truncate GUT irreps, eliminate coloured Higgs triplets, induce chirality... Internal Geometry :... determines SM arbitrary parameters from a handful of topological properties 5

6 B F-theory and Elliptic Fibration 6

7 F-theory ( Vafa 1996) Geometrisation of Type II-B superstring II-B: closed string spectrum obtained by combining left and right moving open strings with NS and R-boundary conditions: Bosonic spectrum: (NS +,NS + ), (R,R ), (NS +,R ), (R,NS + ) (NS +,NS + ): graviton, dilaton and 2-form KB-field: g µν, φ, B µν B 2 (R,R ): scalar, 2- and 4-index fields (p-form potentials) C 0,C µν,c κλµν C p, p = 0,2,4 7

8 Definitions (F -theory bosonic part) 1. String coupling: g s = e φ 2. Combining the two scalarsc 0, φ to one modulus: τ = C 0 +ie φ C 0 + i g s IIB - action (see e.g. Denef, 0803:1194): S IIB d 10 x gr (Imτ) 2dτ d τ + 1 Imτ G 3 G F 5 F 5 +C 4 H 3 F 3 Property: Invariant under SL(2, Z) S-duality: τ aτ+b cτ+d 8

9 FIBRAT ION F-theoryR 3,1 X X, elliptically fibered CY 4-fold overb 3 a torusτ = C 0 +ı/g s at each point ofb

10 Elliptic Fibration described by Weierstraß Equation y 2 = x 3 +f(w)xz 4 +g(w)z 6 For each point ofb 3, the above equation describes a torus 1. x, y, z homogeneous coordinates 2. f(w), g(w) 8 th and 12 th degree polynomials. 3. Discriminant (w) = 4f 3 +27g 2 Fiber singularities at (w) = 0 24 roots w i 10

11 Manifold Singularities CY 4-fold: Red points: pinched torus 7-branes B 3 11

12 Kodaira classification: Type of Manifold singularity is specified by the vanishing order off(w), g(w) and (w) Singularities are classified in terms ofade Lie groups (Kodaira). Interpretation of geometric singularities CY 4 -Singularities gauge symmetries Groups SU(n) SO(m) E n 12

13 Tate s Algorithm y 2 +α 1 xyz +α 3 yz 3 = x 3 +α 2 x 2 z 2 +α 4 xz 4 +α 6 z 6 Table: Classification of Elliptic Singularities w.r.t. vanishing order of Tate s form coefficientsα i : Group α 1 α 2 α 3 α 4 α 6 SU(2n) 0 1 n n 2n 2n SU(2n+1) 0 1 n n+1 2n+1 2n+1 SU(5) SO(10) E E E

14 Basic ingredient in F-theory: D7 - brane GUTs are associated to 7-branes wrapping certain classes of internal 2-complex dim. surfaces B 3 Gauge symmetry: E 8 G GUT C G GUT = SU(5), SO(10),... C Commutant... monodromies: U(1) n,ordiscretesymmetry S n, A n, D n, Z n... acting as family or discrete symmetries 14

15 Model in this talk: SU(5) : E 8 SU(5) SU(5) C = SU(5). Spectral Cover C described by C : k b k s 5 k = 0, b 1 = 0, roots t i Matter resides in 10 and 5 along intersections with other 7-branes λ t,b -Yukawas at intersections and gauge symmetry enhancements ( Heckman et al ; Font et al ; GG Ross, GKL, ); ( Cecotti et al ; Camara et al, 1110,2206; Aparicio et al, ,... ) 15

16 C Non-Abelian Discrete Symmetries 16

17 Application: Spectral Cover splitting: C 5 C 4 C 1 Motivation: The neutrino sector (TB-mixing) C 4 C 1 implies the splitting of thec 5 polynomial in two factors b k s 5 k = (a 1 +a 2 s+a 3 s 2 +a 4 s 3 +a 5 s 4 )(a }{{} 6 +a 7 s) }{{} k C 4 C 1 Topological properties ofa i are fixed in terms of those ofb k, by equating coefficients of same powers ofs Moreover: C 1 : associated to au(1) C 4 : reduction to (i) continuous SU(4) subgroup, or (ii) to Galois group S 4 b 0 = a 5 a 7, b 5 = a 1 a 6, etc... (see Heckman et al, , Marsano et al, , I. Antoniadis and GKL ) 17

18 Properties and Residual Spectral Cover Symmetry IfH S 4 the Galois group, final symmetry of the model is: SU(5) GUT H U(1) }{{} family symmetry H S 4 is linked to specific topological properties of the polynomial coefficientsa i. a i coefficients determine useful properties of the model, such as i) Geometric symmetries R-parity ii) Flux restrictions on the matter curves Fluxes determine useful properties on the matter curves including : Multiplicities and Chirality of matter/higgs representations 18

19 Figure 1: S 4 and the relevant discrete subgroups 19

20 The Galois group inc 4 Determination of the Galois group, requires examination of (partially) symmetric functions of rootst i of the polynomialc 4. For our purposes, it suffices to examine the Discriminant and the Resolvent 1.) The Discriminant = δ 2 where δ = i<j(t i t j ) δ is invariant unders 4 -even permutations A 4 symmetric can be expressed in terms of coefficientsa i F If = δ 2, such thatδ(a i ) F, then If δ 2, (i.e. δ(a i ) / F), then (t i ) (a i ) H A 4 orv 4 (= Klein group) H S 4 ord 4 20

21 2.) To study possible reductions ofs 4, A 4 to their subgroups, we examine the resolvent: f(x) = (x x 1 )(x x 2 )(x x 3 ) x 1 = t 1 t 2 +t 3 t 4, x 2 = t 1 t 3 +t 2 t 4, x 3 = t 2 t 3 +t 1 t 4 x 1,2,3 are invariant under the three Dihedral groups D 4 S 4. Combined results of andf(x) : δ 2 = δ 2 f(x) irreducible S 4 A 4 f(x) reducible D 4,Z 4 V 4 21

22 Figure 2: S4 tod4 22 f(x) factor.

23 The induced restrictions on the coefficientsa i 1. Tracelessness conditionb 1 = 0 demands (Dudas& Palti ) 2. ForS 4 D 4, δ 2 (arxiv: ) 3. Reducibility of the functionf(x) is achieved if a 4 = a 0 a 6, a 5 = a 0 a 7 ( a2 2 a 5 a 4 2 a 1 ) 2 ( 16a1 a 5 a 2 a 4 3 f(0) = 4a 5 a 3 a 1 a 1 a 4 2 a 5 a 2 2 = 0 ) 3 23

24 Matter Parity Spectral Cover eq. k b ks 5 k, invariant under (see Hayashi et. al., ) s s,b k ( 1) k e iχ b k ForC 4 (see I. Antoniadis, GKL, ) b k = n+m=12 k a m a n a n e iψ e i(3 n) a n Defining Equs of matter curves are expressed in terms ofa n s.... a Geometric Z 2 symmetry assigned to Matter Curves 24

25 SU(5) Def. Eqn. Parity Content D 4 t κ Q L +u c L +ec L a 2 + u c L +ēc L a 2 + u c L +ēc L µ 2Q L +4e c L a a d c L b a 7 + H u c κa 7 4d c L +3L d a 2 + H d e a 2 + dc L f a 7 + 2d c L 2 1 Table 1: Full spectrum forsu(5) D 4 U(1) t5 model. 25

26 Low Energy Spectrum D 4 rep U(1) t5 Z 2 Q 3,u c 3,e c u c u c Q 1,2,e c 1,2 2 0 L i,d c i ν3 c ν1,2 c 2 0 H u H d Table 2: SM spectrum withd 4 U(1) t5 Z 2 symmetry. (Karozas et al ) 26

27 D 4 Phenomenology 27

28 Neutrino Sector (Main Motivation for Non-Abelian Discrete Symmetries) result... m ν m ν = m D M 1 R mt D 1+(z 1 2y)gz 1 (1 gyz 1 )x 2 +(z 1 y)gz 2 (1 gyz 1 )x 3 (1 gyz 1 )x 2 +(z 1 y)gz 2 x 2 2 2gyz 2 x 2 +gz 2 2 (x 2 gyz 2 )x 3 (1 gyz 1 )x 3 (x 2 gyz 2 )x 3 x

29 X3 5 X3 X X X X 2 Figure 3: Left: sin 2 θ 12 (3σ) (blue-0.270, pink-0.304, yellow-0.344); Middle: sin 2 θ 23 (3σ) (blue-0.382, pink-0.452, yellow-0.5); Right: R = m 2 23/ m 2 12 = 31.34(blue) andr = (yellow). 29

30 Baryon Number Violation eliminated by flux 10 2 (Q,u c,e c ) (,u c,e c ) parity violating term c 5 c λ dbu u c d c d c only! Neutron-antineutron oscillations d ~ b ~ W W ~ b d Figure 4: Feynman box graph for n n oscillations (Goity&Sher PLB 346(1995)69) 30

31 Λ dbu M t GeV Figure 5: λ dbu bounds for: Blue: Mũ = M c = 0.8TeV, Dashed: Mũ = M c = 1TeV, Dotted: Mũ = M c = 1.2TeV. = = 500GeV,τ = 10 8 sec.). (M bl M br 31

32 E Mordell-WeilU(1) and GUT s 32

33 A new class of Abelian Symmetries associated to Rational Sections of elliptic curves Mordell-Weil group... finitely generated: Z Z Z G }{{} r Abelian group: Rank - r (unknown) Torsion part: G : G = Z n n = 1,2,...,10,12 Z k Z 2 k = 2,4,6,8... models with new U(1) s and Discrete Symmetries from Mordell-Weil (Cvetic et al , ; Mayhofer et al, ; Borchmann et al ; Krippendorf et al, ) 33

34 Simplest (and perhaps most viable) Case: Rank-1 Mordell-Weil Sections required: [u : v : w] = [1 : 1 : 2] P (1,1,2) -weighted projective space... described by the equation: (see Morrison & Park ) w 2 +a 2 v 2 w = u(b 0 u 3 +b 1 u 2 v +b 2 uv 2 +b 3 v 3 ) 34

35 Weierstrass model obtained Birational Map v = w = u = z a 2 y b 2 3 u2 a 2 2 (b 2u 2 +x) b 3 uy b 2 3 u2 a 2 2 (b 2u 2 +x) x a 2 (2) (1) (3) 35

36 These lead to the Weierstraß equation in Tate s form y 2 +2 b 3 a 2 xyz ±b 1 a 2 yz 3 = x 3 ± ( ) b 2 b2 3 a 2 x 2 z 2 2 ( ) b 2 b2 3 a 2 z 6 2 b 0 a 2 2xz 4 b 0 a

37 but now Tate s coefficients are not all independent! y 2 +2 b 3 a 2 xyz ±b 1 a 2 yz 3 = x 3 ±... comparing with standard general Tate s form: ( ) b 2 b2 3 a 2 x 2 z 2 2 ( ) b 2 b2 3 a 2 z 6 2 b 0 a 2 2xz 4 b 0 a 2 2 y 2 +α 1 xyz +α 3 yz 3 = x 3 +α 2 x 2 z 2 α 4 xz 4 α 6 z 6 Observation: α 6 = α 2 α 4 37

38 Assume local expansion of Tate s coefficients Implications on the non-abelian structure α k = a k,0 +α k,1 ξ + Vanishing orders for SU(2n): α 2 = a 2,1 ξ + α 4 = α 4,n ξ n + α 6 = α 6,2n ξ 2n +...from SU(n) series, compatible are Only: α 6 = α 2 α 4 α 2,1 α 4,n ξ n+1 = α 6,2n ξ 2n n = 1 SU(2), and SU(3) 38

39 ... extending the analysis to exceptional groups... Viable non-abelian GUTs withu(1) MW and the vanishing order of the coefficientsa 2 a 2,m ξ m,b k b k,n ξ n Group a 2 b 0 b 1 b 2 b 3 E E This simple property... perhaps suggestive for a model E 6 U(1) MW 39

40 Remarks Spectral Cover: Analysis of model with gauge symmetry SU(5) D 4 U(1) Non-abelian discrete symmetries naturally incorporated n n oscillations, suppressed proton decay Mordell-Weil:... gauge symmetries with one abelian Mordell-Weil: E 6 U(1) MW, E 7 U(1) MW... extrau(1) MW might have interesting implications to Model building... Torsion group: possible explanation of discrete symmetries... 40

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