(Un)known (un)knowns for the effective action of stringy partic. in the era of alternative facts

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1 (Un)known (un)knowns for the effective action of stringy particle physics models in the era of alternative facts Cluster of Excellence PRISMA & Institut für Physik, JG U Mainz based on JHEP 1704 (2017) 023 [arxiv: [hep-th]] with Isabel Koltermann & Wieland Staessens StringPheno 2017, Virginia Tech, 3 July 2017

2 Motivation: String Phenomenology Status quo: string pheno (roughly)? Cosmology many compact CY and T 6 /Γ models with SM-like spectrum dimensional reduction of SUGRA & DBI actions provides some tree-level results: 4D R, 4D trf 2 ( local models ) already Yukawas challenging - non-holomorphic results only for some (too) simple T 6 /Γ not known for realistic N = 1 models string cosmo (roughly) assume that local models of particle physics work choose CY geometry globally to explain inflation dial term(s) (e.g. specific instanton) for moduli stabilization string-inspired field theory?? = stringy low-energy effective action... why should this hold?

3 Add stringy ingredients incompletely &/or at gusto and evaluate:

4 Motivation cont d: 2 Steps towards the Effective Action 1.) Moduli stabilization: scalars from closed & open string sector mix (& axions) F-term potential for some open string moduli Marchesano, Regalado, Zoccarato 14 in the absence of open string moduli (Wilson lines & displacements) Blaszczyk, G.H., Koltermann 15; G.H., Koltermann, Staessens 17 DBI action couples D-branes to closed strings (IIA/ΩR: cplx.str.) ΩR-even ΩR-odd flat direction D-term SO/USp(2N) U(N) partial (closed) moduli stabilization before fluxes & instantons no direct couplings to (some) flat directions (cplx.str.) flat directions in φ, (some) Kähler & (some) cplx.str. moduli allow tuning of M string, 1 /g 2 to pheno. allowed values G {SU(3),SU(2),Y...}

5 Motivation cont d: 2 Steps towards the Effective Action 2.) 1-loop corrections to 1 /g 2 G {SU(3),SU(2),Y...} : tree-level: 1 g 2 a M Planck Vol(Π a) M string Vol(CY3 ) (cplx.str.) in IIA/ΩR Mstring M Planck for LARGE (cplx.str.) volume ( seemingly ) expect additional moduli dependences via field redefinitions mixing closed/open strings - omnipresent, e.g. Stückelberg mechanism for U(1) massive 1-loop corrections to 1/g 2 a D6a SU(N a) SU(N a) D6 b D6a O6 depend on tower of KK & winding modes Kähler moduli: δ 1 g v a 2 i 1 LARGE volume: g a 2 G.H., Koltermann, Staessens 17 Control perturbative action before dialing e.g. instanton(s)!

6 Outline Motivation & punch line Deformations & moduli orbifold point Constraints (not only on LARGE volumes) from 1-loop Conclusions & Outlook

7 Deformations & Moduli Orbifold Point

8 Deformations: Hypersurface Technique for T 6 /(Z 2 Z 2M ) use P with coord. (x i, v i, y i ) to describe T 2 (i) yi 2 + F i (x i, v i ) =! 0 F i (x i, v i ) = 4v i xi 3 g (i) 2 v i 3 x i g (i) 3 v i 4 g (i) 3 = 0: square torus g (i) 2 = 0: hexagonal lattice Z take product and impose Z 2 Z 2 symmetry: y 2 i yi (T 2 ) 3 /(Z 2 Z 2 ) { y 2 + F 1 F 2 F 3 = 0} with y y 1 y 2 y 3 deform fixed points αβ T(i) 4 T 2 (j) T 2 : (k) y 2 + F 1 F 2 F 3 + ε (i) αβ F i δf (α) j i j k i αβ δf (β) k! = 0 with δf (α) j (x j, v j ) also polynomials of degree 4 deformation method based on Vafa, Witten 95

9 Deformations con t: slags on T 6 /(Z 2 Z 2M ΩR) Blaszczyk, G.H., Koltermann & G.H., Koltermann, Staessens 17 slags: probe Re Π a Ω 3 > 0 and Im Π a Ω 3 = 0 specify calibration via anti-holomorphic involution σ R in general too complicated concentrate on anti-linear maps ) ( ) x i A, y i e iβ v i s.t. AA = 1I, σ R(F i(x i, v i)) = e 2iβ F (x i, v i) v i v i ( xi two different calibrations β and β + π per given A integration path Π a specified by (A a, β a) impose SUSY products of orbifold point Z 3 ΩR symmetry: M = 3 favourable for 3 generation particle physics Z 3 : need for hexagonal tori & identification of Z 2 Z 2 deformations ε (i) αβ under Z 3 ΩR-invariance: only real deformation parameters

10 G.H., Koltermann, Staessens 17

11 Moduli Orbifold Point of Z 2 Z 6 ΩR General considerations: Blaszczyk, G.H., Koltermann (i) Π frac = Πbulk + i ΠZ 2 4 couples to (some) twisted moduli N D6-branes U(N) U(1) D-term vanishes Π frac is slag related twisted moduli stabilised FI term for ζj 0 not all couplings generate D-term: e.g. USp(2N) or SO(2N) no D-term Π frac is slag for any deformation Orientifold symmetry: IIA requires anti-holomorphic involution R: (h+ 11, h 11 ) = (4 Z, 3 bulk + 8 Z Z ) for T 6 /(Z 2 Z 6 ΩR) w/ discrete torsion 6 - one Z 6 sector does not have any scalars that could resolve the singularities (h ) ΩR-even/-odd 3-cycles Z 2 deformation SUSY only ΩR-even contribution to Π Z(i) 2 h 21 = 1 bulk + [ ] Z2 + [2 + 2] Z3 +Z 6 for T 6 /(Z 2 Z 6 ΩR) w/ d.t.

12 Deformations cont d: Examples on Z 2 Z 6 Backgrd. Ecker, G.H., Staessens extensive computer scans boil down to prototypes SU(3) a USp(2) b U(1) Y SU(4) h Z 3 { Z 3 I SU(3) a USp(2) b USp(2) c Ũ(1) B L II { SU(6) I SU(4) a USp(2) b USp(2) c SU(2) II U(3) a USp(2) b U(1) c U(1) d U(4) h U(3) a USp(2) b USp(2) c U(h) 2 U(4) a USp(2) b USp(2) c U(h) orbifold point: closed & open string spectrum gauge couplings (tree + 1-loop) Counting of stabilised complex structure moduli & flat directions with 1/g 2 D6x,x {a,b,c,d,h 1,2 } dependence Z 2 Z 6 ϱ ε (1) 0,1,2 ε (1) 3 ε (1) 4+5 ε (1) 4 5 MSSM none a, c, h [b, d] flat c, d none a, d, h none 6 L-R I none h 1,2 a, d, h 1,2 a, d [b, c] flat h 1,2 none a, d none 9 L-R II none h 1,2 a, d, h 1,2 a, d [b, c] flat [a, b, c, d, h 1,2 ] flat none a, d, h 1,2 none 7 L-R IIb none h 1,2 a, d, h 1,2 a, d [b, c] flat [a, b, c, d] flat [h 1,2 ] flat a, d h 1,2 9 L-R IIc none a, d a, d [b, c, h 1,2 ] flat h 1 h 2 a, d, h 1 h 2 10 PS I none a, h [b, c] flat [a, b, c, h] flat none a, h none 4 PS II none h a, h a [b, c] flat [a, b, c, h] flat none a, h none 7 ε (2) 1,2 ε (2) 3,4 ε (3) 1,2 ε (3) 3,4 # stab

13 Deformations: Flat Direction of MSSM Example G.H., Koltermann, Staessens 17 USp(2) b and U(1) d U(1) Y feel flat direction ε (1) 4 5 < 0 ( ) Π frac b = 3ρ1 ɛ (1) 4 ɛ(1) 5 4 Π frac ( ) d = 3ρ1+ ɛ (1) 4 ɛ(1) 5 weak interaction on b slightly stronger for ε (1) a 5 0 hypercharge slightly weaker remember Caveat on stabilised moduli: compensation by charged matter vev in D-terms only bifundamentals under U(N) U(N) (cd ( ) ) MSSM only if string derived field theory couplings exist 4 1 g 2 x Vol(Π frac x )... need for new CFT computations [no results for Z 2 Z 2M w/ discrete torsion available]

14 Constraints from 1-Loop

15 1-Loop Corrections to Gauge Orbifold Point Why? deformations by twisted moduli change tree-level untwisted moduli enter 1-loop corrections only known at orbifold point independent of absolute choice of Z 2 Z 2 eigenvalues can - in principle - contribute to hierarchy M string M GUT Unusual feature here: net O6-plane charge along one direction on T 2 (1) Einstein-Hilbert M2 Planck M 2 string DBI action 8π 2 g 2 SU(3)a/SU(4) h,tree = 4π g 2 string = π 2 6 G.H., Ripka, Staessens 12 v 1 v 2 v 3 with v 1 = R (1) 1 R(1) 2 R (1) 1 M Planck R (1) M string 2 M string M Planck can be compensated by R (1) 2 R (1) 1... spoilt by 1-loop effects??

16 Constraints on Kähler Moduli Kähler moduli v i enter via parallel D6-branes Λ τ,σ(v) 1 ( 4π ln e πσ2 v 4 Λ 0,0(v) 1 v ln(η(iv)) v 4π 48 ϑ 1( τ iσv 2, iv) η(iv) ) v [ 3(1 σ) 2 1 ] v sign depends on discrete brane data (τ, σ) A+M,MSSM SU(3) a 2 48 v 1 2π v 1 + π ( v 2 + c 1 ( ) ) 2 3 1,1 v2 v 3 ln (KK & windings) δ σ,0 ln[2 sin( πτ 2 )] 4π sum over sector-by-sector (Annulus + Möbius strip) e.g. ( ( (1) R 1 A+M,MSSM SU(4) h 2 v 1 2π v 1 + π ( v 2 + c 1 ( ) ) 2 3 1,1 v2 v 3 ln R (1) 2 v 1 ) 6v2 ) 12 ( ( ) R1 6v 1 v 1 2 R 2 v1 1 disfavoured by 1 /g 2 a O(1) R (1) 1 R (1) 2 LARGE vol. for v 2 1, SU(4) h more strongly coupled than SU(3) a for v1 1 and v 2 = v 3 6: M string M GUT, A+M,MSSM SU(3) a < 0 ) G.H., Koltermann, Staessens 17 7

17 Conclusions & Outlook more closed string moduli stabilized by D-branes than usually assumed - others without direct coupling to particle physics very limited knowledge about stringy effective action: T 6 results do not carry over to T 6 /(Z 2 Z 2M ) w/ d.t. But: not even Möbius strip contributions to G complete e.g. N = 2 vanishing argument on some Yukawas e.g. vanishing 1-loop corrections to Kähler metrics by Berg, Haack, Kang (Sjörs) 11 ( 14) not valid for 3 generation particle physics models urgent need to develop (CFT) computational tools simultaneously field theoretical studies of pheno/cosmo potential pressing need for reliable results: string-inspired field theory stringy low-energy effective action in the era of precision in phenomenology & cosmology

18 Appendix

19 Ex: MSSM on rigid D6-branes on T 6 /(Z 2 Z 6 ΩR) Ecker, G.H., Staessens 15 D6-brane configuration of a global 5-stack MSSM model on the aaa lattice Angle wrapping # π Z (i) 2 eigenv. ( τ) ( σ) gauge gr. a (1, 0; 1, 0; 1, 0) (0, 0, 0) (+ + +) (0, 1, 1) (0, 1, 1) U(3) b (1, 0; 1, 2; 1, 2) (0, 1 2, 1 2 ) ( +) (0, 1, 0) (0, 1, 0) USp(2) c (1, 0; 1, 2; 1, 2) (0, 1 2, 1 2 ) ( + ) (0, 1, 1) (0, 1, 1) U(1) d (1, 0; 1, 2; 1, 2) (0, 1 2, 1 2 ) (+ ) (0, 0, 1) (0, 0, 1) U(1) h (1, 0; 1, 0; 1, 0) (0, 0, 0) ( +) (0, 1, 1) (0, 1, 1) U(4) Green-Schwarz mech.: x {a,c,d,h} U(1) x U(1) Y Z 3 perturbatively U(1) 3 massive U(1) PQ = U(1) c U(1) d non-pert. only: SU(3) SU(2) U(1)Y Z 3 SU(4) hidden

20 Ex: MSSM spectrum on T 6 /(Z 2 Z 6 ΩR) Ecker, G.H., Staessens 15 MSSM matter of (SU(3) a USp(2) b SU(4) h ) (U(1) PQ),Z 3 U(1) Y : 3 [ (3, 2, 1) (0),0 ] 1/6 + 2 (3, 1, 1)(1),1 1/3 + (3, 1, 1)(1),1 1/3 +(3, 1, 1)(1),1 2/3 }{{} + 3 [ (1, 2, 1) (1),1 1/2 + 2 (1, 2, 1)(1),1 1/2 +(1, 1, 1)( 2),1 0 + (1, 1, 1) (0),0 ] 1 }{{} = 3 [ Q L + {}}{ ] [{}}{ ] 2 d R + d R +u R + 3 H u/l + 2 L + ν R + e R [ ] [ ] Higgses: 3 (1, 2, 1) (1),1 1/2 + h.c. + 2 (1, 2, 1) ( 1),2 1/2 + h.c. [ ] axions: 3 [Σ cd + Σ cd ] = 3 (1, 1, 1) ( 2),1 0 + h.c. SM vector-like states: (5 Antib + 4 Adjc + 5 Adjd ) (1, 1, 1) (0),0 [ ] + 2 (3, 1, 4) (0),1 + (3, 1, 4)(0),2 + 2 (3 1/6 1/6 A, 1, 1) (0),0 + h.c. 1/3 [ ] + 3 (1, 1, 1) (0),0 1 + (1, 1, 1) ( 2), (1, 1, 6 A ) (0),1 0 + h.c. + 3 (1, 2, 4) (0), (1, 1, 4) ( 1),0 1/ (1, 1, 4)( 1),0 + 3 (1, 1, 4) ( 1),1 1/2 1/2