Anomalies and Remnant Symmetries in Heterotic Constructions. Christoph Lüdeling

Transcription

1 Anomalies and Remnant Symmetries in Heterotic Constructions Christoph Lüdeling bctp and PI, University of Bonn String Pheno 12, Cambridge CL, Fabian Ruehle, Clemens Wieck PRD 85 [arxiv: ] and work in progress C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

2 Motivation MSSM superpotential contains (potentially) bad terms: W bad µh u H d +QLd c +u c d c d c +LLe c +QQQL+u c u c d c e c + Forbid/constrain these operators by (discrete) symmetries, (matter parity, proton hexality, R 4,...) [many people here] C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

3 Motivation MSSM superpotential contains (potentially) bad terms: W bad µh u H d +QLd c +u c d c d c +LLe c +QQQL+u c u c d c e c + Forbid/constrain these operators by (discrete) symmetries, (matter parity, proton hexality, R 4,...) Anomaly constraints on such symmetries (bottom-up) from consistency and grand unification? [many people here] Appearance of discrete symmetries as remnants of gauge or Lorentz symmetries in string constructions? Simple example: Consider heterotic 3 orbifold, its blowup and GLSM realisation C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

4 Contents 1 Green Schwarz Mechanism and Universality 2 Heterotic Models 3 Remnant Discrete Symmetries 4 Conclusion C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

5 Green Schwarz Mechanism Requires [Green, Schwarz 84] a) factorisation of anomaly polynomial, I 6 = X 4 Y 2, i.e. Y 2 = F 2 = da 1 ( anomalous U(1) ) b) axion field a with shift gauge transformation δa = λ δa 1 = dλ c) axion coupling to X 4 1 S GS = 2 da+a 1 2 +ax 4 = δs GS = I (1) 4 = A C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

6 Green Schwarz Mechanism Requires [Green, Schwarz 84] a) factorisation of anomaly polynomial, I 6 = X 4 Y 2, i.e. Y 2 = F 2 = da 1 ( anomalous U(1) ) b) axion field a with shift gauge transformation δa = λ δa 1 = dλ c) axion coupling to X 4 1 S GS = 2 da+a 1 2 +ax 4 = δs GS = I (1) 4 = A Shift alone Stueckelberg Mass C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

7 Green Schwarz Mechanism Requires [Green, Schwarz 84] a) factorisation of anomaly polynomial, I 6 = X 4 Y 2, i.e. Y 2 = F 2 = da 1 ( anomalous U(1) ) b) axion field a with shift gauge transformation δa = λ δa 1 = dλ c) axion coupling to X 4 1 S GS = 2 da+a 1 2 +ax 4 = δs GS = I (1) 4 = A Shift alone Stueckelberg Mass Shift & coupling Anomaly cancellation C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

8 Green Schwarz Mechanism Requires [Green, Schwarz 84] a) factorisation of anomaly polynomial, I 6 = X 4 Y 2, i.e. Y 2 = F 2 = da 1 ( anomalous U(1) ) b) axion field a with shift gauge transformation δa = λ δa 1 = dλ c) axion coupling to X 4 1 S GS = 2 da+a 1 2 +ax 4 = δs GS = I (1) 4 = A Shift alone Stueckelberg Mass Shift & coupling Anomaly cancellation Sum of factorised anomalies I 6 = a Ya X a : Cancelled by set of axions C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

9 Green Schwarz Mechanism: Axion or 2-Form Dual description axion a 2-form C 2 shift δa Y 2 coupling C 2 Y 2 coupling ax 4 shift δc 2 X 4 In particular: Kalb Ramond B 2 dual to universal axion a 0 always present in heterotic compactifications C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

10 Green Schwarz Mechanism: Axion or 2-Form Dual description axion a 2-form C 2 shift δa Y 2 coupling C 2 Y 2 coupling ax 4 shift δc 2 X 4 In particular: Kalb Ramond B 2 dual to universal axion a 0 always present in heterotic compactifications X 4 contains field strengths of other gauge group factors G i, weighted with (arbitrary) anomaly coefficients: X 4 = A grav U(1) trr 2 + i A G 2 i U(1) A trfi 2 Universal axion a 0 couples universally via X uni 4 (fixed by transformation of B 2 ) this is a special case! C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

11 Anomaly Universality from Underlying SU(5)? Take SU(5) U(1) X, flavour-blind matter charges, R or non-r symmetry Matter gauginos Higgses 5,10 SM X,Y 3 s 2 s A 5 = 3 ( 2 3q10 +q 5 4R ) +5R +C H A 3 = 3 ( 2 3q10 +q 5 4R ) +3R +2R +C H A 2 = 3 ( 2 3q10 +q 5 4R ) +2R +3R +C H A 1 = 3 ( 2 3q10 +q 5 4R ) +5R C H C H (C H = 1 2 (q H u +q Hd 2R)) Breaking by (discrete) orbifold twists of fluxes changes zero mode spectrum C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

12 Anomaly Universality from Underlying SU(5)? Take SU(5) U(1) X, flavour-blind matter charges, R or non-r symmetry Matter gauginos Higgses 5,10 SM X,Y 3 s 2 s A 5 = 3 ( 2 3q10 +q 5 4R ) +5R +C H A 3 = 3 ( 2 3q10 +q 5 4R ) +3R +2R +C H A 2 = 3 ( 2 3q10 +q 5 4R ) +2R +3R +C H A 1 = 3 ( 2 3q10 +q 5 4R ) +5R C H C H (C H = 1 2 (q H u +q Hd 2R)) Breaking by (discrete) orbifold twists of fluxes changes zero mode spectrum SU(5) breaking removes X,Y gauginos non-universality for R symmetries C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

13 Anomaly Universality from Underlying SU(5)? Take SU(5) U(1) X, flavour-blind matter charges, R or non-r symmetry Matter gauginos Higgses 5,10 SM X,Y 3 s 2 s A 5 = 3 ( 2 3q10 +q 5 4R ) +5R +C H A 3 = 3 ( 2 3q10 +q 5 4R ) +3R +2R +C H A 2 = 3 ( 2 3q10 +q 5 4R ) +2R +3R +C H A 1 = 3 ( 2 3q10 +q 5 4R ) +5R C H C H (C H = 1 2 (q H u +q Hd 2R)) Breaking by (discrete) orbifold twists of fluxes changes zero mode spectrum SU(5) breaking removes X,Y gauginos non-universality for R symmetries doublet-triplet splitting generic non-universality C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

14 Implications for G SM N 2 N or 3 N anomalies not meaningful No need for N G 2 anomalies to be universal [Ibanez, Ross 91; Banks, Dine 92; Araki et al. 08] Less restrictions on phenomenologically interesting symmetries E.g. non-r X 6 with q 10 = 1, q 5 = 5, q Hu = 4, q Hd = 0 allows Yukawas and Weinberg operator, forbids µ term and B, L operators (up to dimension five) In string models, possibly remnants of universality exist C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

15 T 6 / 3 Orbifold and its Blowup For illustration, consider simple T 6 / 3 orbifold model: V = 1 ( 3 1,1, 2,0 5 )( 0 8), no Wilson lines Gauge group E 6 SU(3)[ E 8 ] spectrum 3 ( 27,3 ) +27 [ (27,1)+3(1,3) ] No anomalous U(1), i.e. universal axion does not shift, no FI term Blowup: Smooth out geometry Heuristically, VEVs for twisted states ( blow-up modes ) volumes of resolution cycles All volumes large description in supergravity regime, i.e. compactification on smooth CY with gauge bundle Gauge bundle data representations of blow-up modes C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

16 Blowup as CY with Bundle [Groot Nibbelink et al ] Toric resolution, replace singularities by exceptional divisors (È 2 s) E r Line bundles, i.e. Abelian fluxes given by bundle vectors: F = V I r H I E r C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

17 Blowup as CY with Bundle [Groot Nibbelink et al ] Toric resolution, replace singularities by exceptional divisors (È 2 s) E r Line bundles, i.e. Abelian fluxes given by bundle vectors: F = V I r H I E r Simple model, twice breaking E 6 SO(10) U(1) V (1) V (2) gauge group SO(8) U(1) A U(1) B SU(3) C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

18 Blowup as CY with Bundle [Groot Nibbelink et al ] Toric resolution, replace singularities by exceptional divisors (È 2 s) E r Line bundles, i.e. Abelian fluxes given by bundle vectors: F = V I r H I E r Simple model, twice breaking E 6 SO(10) U(1) V (1) V (2) gauge group SO(8) U(1) A U(1) B SU(3) Massless spectrum depends on distribution of flux among exceptional divisors (given by (k,p,q) with k +p +q = 27) Flux gauges shift symmetries of new axions both U(1) s are massive, even though flux is Abelian C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

19 Blowup Anomalies anomaly polynomial I 6 = X I 12 with backgrounds inserted, or from triangle diagrams ( ) ( ) k 6 k 18 I 6 F A3 +F A FB [ +F A trfsu(3) 2 +trf2 SO(8) + 7 ] ( ) k 9 48 trr2 2 [ 1 +F B 8 F2 B F2 A +trf2 SU(3) +trf2 SO(8) + 7 ( ) p q ] 48 trr2 2 C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

20 Blowup Anomalies anomaly polynomial I 6 = X I 12 with backgrounds inserted, or from triangle diagrams ( ) ( ) k 6 k 18 I 6 F A3 +F A FB [ +F A trfsu(3) 2 +trf2 SO(8) + 7 ] ( ) k 9 48 trr2 2 [ 1 +F B 8 F2 B F2 A +trf2 SU(3) +trf2 SO(8) + 7 ( ) p q ] 48 trr2 2 For p = q, U(1) B is omalous, while U(1) A is always anomalous C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

21 Blowup Anomalies anomaly polynomial I 6 = X I 12 with backgrounds inserted, or from triangle diagrams ( ) ( ) k 6 k 18 I 6 F A3 +F A FB [ +F A trfsu(3) 2 +trf2 SO(8) + 7 ] ( ) k 9 48 trr2 2 [ 1 +F B 8 F2 B F2 A +trf2 SU(3) +trf2 SO(8) + 7 ( ) p q ] 48 trr2 2 For p = q, U(1) B is omalous, while U(1) A is always anomalous Remnant universality among non-abelian groups from one E 8 C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

22 Remnant Gauge Symmetry Remnant non-r symmetries: discrete subgroup of U(1) A U(1) B which leaves VEVs invariant Blow-up modes: 1 4,0, 1 2, 2, 1 2,2 discrete remnant 4 4, generated by { } 2πi T ± : φ (qa,q B ) exp 4 (q A ±q B ) φ (qa,q B ) However: Charges of all massless fields are even under both 4 s only 2 2 realised on massless spectrum Both 2 factors are omalous C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

23 R Symmetries from Orbifolds R transformations from sublattice rotations act as R : Φ e 2πivR Φ ( ) where v = 1 3,0,0, and our blow-up modes have R = q sh N = ( 1 3, 1 3, 1 ) 3 C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

24 R Symmetries from Orbifolds R transformations from sublattice rotations act as R : Φ e 2πivR Φ ( ) where v = 1 3,0,0, and our blow-up modes have R = q sh N = ( 1 3, 1 3, 1 ) 3 Symmetry conventions somewhat tricky upshot: 18 symmetry with charges for (bosons,fermions,θ) = 1 18 (2n,2n 3,3) Remnant R symmetries: unbroken combinations of the three sublattice rotations and U(1) A,B only a trivial 2 R symmetry survives in blow-up C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

25 Model: GLSM Description [Witten 93;Groot Nibbelink 10; Blaszczyk et al. 11] [See talk by Fabian Ruehle on Thursday] Algebraically, describe the orbifold by (È 2 [3] is a T 2 ) È 2 [3] È 2 [3] È 2 [3] Blowup (crepant resolution) in (0, 2) GLSM description: 3 2D SUSY field theory with U(1) symmetries, fields coordinates flows to worldsheet CFT in IR Geometry cut out by F and D term equations, GLSM FI terms become CY Kähler parameters To resolve singularities, introduce extra coordinates (exceptional divisors) and U(1)s Gauge bundle given by chiral-fermi superfields Λ I related to bundle vectors, i.e. fluxes C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

26 R Symmetries in the GLSM Set of F and D terms fixes geometry. discrete transformations of the fields which leave F and D terms invariant R symmetries if holomorphic three-form Ω transforms [Witten 85] Ω η T Γ ijk η Q R (Ω) = Q R (W) = 2 Different types of R symmetries: Phases z e 2πi/3 z: always possible (but see next slide) 6 R symmetries Permutations of fields: Only possible for special values of Kähler parameters corresponds to groupwise exchange of exceptional divisors C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

27 R Symmetry breaking in GLSM È 2 coordinates z iα only appear as z 3 iα or z iα 2 unbreakable 3 rotations? (Presumably) broken by marginal deformations of Kähler potential terms in presence of gauge bundles ( massless charged 4D matter) d 2 θ + φ 4d (x µ )N(z) }{{} ΛΛ gauge bundle fields [Groot Nibbelink 10] Fits with orbifold: Bundle corresponds to blowup Presence of deformations depends on the geometric phase controlled by Kähler (FI) parameters which may force certain fields to vanish Generically, no R symmetry in blow-up (all FI terms large), but enhanced at certain loci of parameter space C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

28 Conclusions Anomalies are generically not universal: Not required for anomaly cancellation, not generic from unification Orbifold anomalous U(1) is the exception because it is cancelled purely by the universal axion For generic string compactifications, many axions possible Non-R symmetries can forbid the µ term C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

29 Conclusions Anomalies are generically not universal: Not required for anomaly cancellation, not generic from unification Orbifold anomalous U(1) is the exception because it is cancelled purely by the universal axion For generic string compactifications, many axions possible Non-R symmetries can forbid the µ term Line bundles do reduce the rank via the axion shift also omalous U(1)s can become massive Blow-ups can leave gauged discrete subgroups unbroken important for phenomenology On orbifold, R symmetries exist but are broken by the blow-up C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

30 Outlook Geometry part of GLSM generically has many unbreakable R-like symmetries seem to be broken by the gauge bundle, but better understanding of their breaking required Linked to determination of charged massless spectrum Non-generic type of R symmetries: Exchange symmetries, appearing for certain loci in Kähler moduli space, e.g. exchange of exceptional divisors if their volumes are equal Study these symmetries for more realistic models, including Wilson lines etc. Orbifold blowups are only a small part of CY landscape... C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

31 Backup: Discrete Symmetries and Higher Anomalies [Ibanez, Ross; Banks, Dine; Araki; Araki et al.] Discrete (finite) symmetries have some subtleties (as compared to U(1)s): Charges defined mod N (Majorana) mass terms possible for charged states, so anomaly could be influenced by heavy states Path integral Jacobian involves field strengths only linear N anomalies well-defined for non-abelian discrete groups, only N subgroups relevant C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

32 Backup: Table of Operators and Charges Operator (LH u ) 2 H u H d LH u H d LH u H d H u C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

33 Backup: Table of Operators and Charges Operator SM Yukawas Weinberg Op. (LH u ) 2 4R 4q 10 +2q 5 2R H u H d 4R 3q 10 q 5 5R 5q 10 +k N 2 LH u 2R 2q 10 +q 5 R +k N q 10 +2q 5 2R +5q q 10 +q 5 R +5q 10 +k N H d 4R +2q 10 q 5 3R +k N 2 LH u H d H u 6R 5q 10 6R 5q 10 C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

34 Backup: Table of Operators and Charges Operator SM Yukawas Weinberg Op. X 6 (LH u ) 2 4R 4q 10 +2q 5 2R 0 H u H d 4R 3q 10 q 5 5R 5q 10 +k N 2 4 LH u 2R 2q 10 +q 5 R +k N q 10 +2q 5 2R +5q q 10 +q 5 R +5q 10 +k N H d 4R +2q 10 q 5 3R +k N 2 3 LH u H d H u 6R 5q 10 6R 5q 10 1 forbid all bad terms e.g. by non-r X 6 with q 10 = 1, q 5 = 5, q Hu = 4, q Hd = 0 2 matter parity as subgroup C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

35 Backup: Table of Operators and Charges Operator SM Yukawas Weinberg Op. X 6 SO(10) (LH u ) 2 4R 4q 10 +2q 5 2R 0 2R H u H d 4R 3q 10 q 5 5R 5q 10 +k N LH u 2R 2q 10 +q 5 R +k N 2 3 R 1055 q 10 +2q 5 2R +5q R q 10 +q 5 R +5q 10 +k N 2 2 4R H d 4R +2q 10 q 5 3R +k N 2 3 3R LH u H d H u 6R 5q 10 6R 5q 10 1 R forbid all bad terms e.g. by non-r X 6 with q 10 = 1, q 5 = 5, q Hu = 4, q Hd = 0 2 matter parity as subgroup Requiring SO(10) relations for matter, we need R = 1 C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17

36 Backup: Group Theory Issues in Blowup [Groot Nibbelink et al.; Blaszczyk et al.] Orbifold picture: Switch on VEVs of twisted states Higgs-type breaking of gauge symmetries, including rank reduction Smooth CY: Abelian gauge bundle does not reduce rank Solution: Flux creates axion shifts which break U(1)s Stueckelberg-style axions related to phases of VEVved states ( twisted axions ) Matching of states involves field redefinitions relates VEVs to anomalies Orbifold blowups rare among CYs generically, one cannot redefine away anomalies into orbifold VEV Blowup CYs: largish h 1,1 O(20 50), small h 2,1 (none a few) C. Lüdeling (bctp) Anomalies and Discrete Symmetries String Pheno, June 26, / 17