Phases of Axion Inflation

Transcription

1 Phases of Axion Inflation Wieland Staessens (JdC) based on 17xx.xxxxx ( , [hep-th]) with G. Shiu Instituto de Física Teórica UAM/CSIC Madrid StringPheno 2017, IFT-UAM/CSIC, 04 July 2017

2 Inflationary axions & String Theory Controlled perturbative corrections to V inf Linde (1988) for axions with a a + ε: prerequisite for large field inflation (LFI) natural inflation Freese-Frieman-Olinto (1990) with V = Λ 4 [ ( s 1 cos a )] f slow-roll for f > M Pl and V 1/4 Λ s GeV In StringPheno, we ask questions like: (1) How to embed LFI? (McAllister) Multi-axions N-flation (2005), aligned natural (2004), kinetic mixing (2015) different mass mechanism axion monodromy (2008, 2014) (2) How are models constrained by quantum gravity? f > M Pl No-go theorems for single axion Banks et al (2003), Svrček-Witten (2006) WGC + swampland conjectures (Shiu, Heidenreich, Rudelius, Valenzuela, Blumenhagen, Ooguri, Hebecker, Wolf, Kläwer, Herraez, Montero, Stout) Here: Explore richness of EFT for stringy axions in Type II with D-branes & mass generation mechanisms

3 Inflationary axions & String Theory Controlled perturbative corrections to V inf Linde (1988) for axions with a a + ε: prerequisite for large field inflation (LFI) natural inflation Freese-Frieman-Olinto (1990) with V = Λ 4 [ ( s 1 cos a )] f slow-roll for f > M Pl and V 1/4 Λ s GeV In StringPheno, we ask questions like: (1) How to embed LFI? (McAllister) Multi-axions N-flation (2005), aligned natural (2004), kinetic mixing (2015) different mass mechanism axion monodromy (2008, 2014) (2) How are models constrained by quantum gravity? f > M Pl No-go theorems for single axion Banks et al (2003), Svrček-Witten (2006) WGC + swampland conjectures (Shiu, Heidenreich, Rudelius, Valenzuela, Blumenhagen, Ooguri, Hebecker, Wolf, Kläwer, Herraez, Montero, Stout) Here: Explore richness of EFT for stringy axions in Type II with D-branes & mass generation mechanisms

4 Inflationary axions & String Theory Controlled perturbative corrections to V inf Linde (1988) for axions with a a + ε: prerequisite for large field inflation (LFI) natural inflation Freese-Frieman-Olinto (1990) with V = Λ 4 [ ( s 1 cos a )] f slow-roll for f > M Pl and V 1/4 Λ s GeV In StringPheno, we ask questions like: (1) How to embed LFI? (McAllister) Multi-axions N-flation (2005), aligned natural (2004), kinetic mixing (2015) different mass mechanism axion monodromy (2008, 2014) (2) How are models constrained by quantum gravity? f > M Pl No-go theorems for single axion Banks et al (2003), Svrček-Witten (2006) WGC + swampland conjectures (Shiu, Heidenreich, Rudelius, Valenzuela, Blumenhagen, Ooguri, Hebecker, Wolf, Kläwer, Herraez, Montero, Stout) Here: Explore richness of EFT for stringy axions in Type II with D-branes & mass generation mechanisms

5 Inflationary axions & String Theory Controlled perturbative corrections to V inf Linde (1988) for axions with a a + ε: prerequisite for large field inflation (LFI) natural inflation Freese-Frieman-Olinto (1990) with V = Λ 4 [ ( s 1 cos a )] f slow-roll for f > M Pl and V 1/4 Λ s GeV In StringPheno, we ask questions like: (1) How to embed LFI? (McAllister) Multi-axions N-flation (2005), aligned natural (2004), kinetic mixing (2015) different mass mechanism axion monodromy (2008, 2014) (2) How are models constrained by quantum gravity? f > M Pl No-go theorems for single axion Banks et al (2003), Svrček-Witten (2006) WGC + swampland conjectures (Shiu, Heidenreich, Rudelius, Valenzuela, Blumenhagen, Ooguri, Hebecker, Wolf, Kläwer, Herraez, Montero, Stout) Here: Explore richness of EFT for stringy axions in Type II with D-branes & mass generation mechanisms

6 Inflationary axions & String Theory Controlled perturbative corrections to V inf Linde (1988) for axions with a a + ε: prerequisite for large field inflation (LFI) natural inflation Freese-Frieman-Olinto (1990) with V = Λ 4 [ ( s 1 cos a )] f slow-roll for f > M Pl and V 1/4 Λ s GeV In StringPheno, we ask questions like: (1) How to embed LFI? (McAllister) Multi-axions N-flation (2005), aligned natural (2004), kinetic mixing (2015) different mass mechanism axion monodromy (2008, 2014) (2) How are models constrained by quantum gravity? f > M Pl No-go theorems for single axion Banks et al (2003), Svrček-Witten (2006) WGC + swampland conjectures (Shiu, Heidenreich, Rudelius, Valenzuela, Blumenhagen, Ooguri, Hebecker, Wolf, Kläwer, Herraez, Montero, Stout) Here: Explore richness of EFT for stringy axions in Type II with D-branes & mass generation mechanisms

7 Stringy Axions from Type II compactifications reviews: Baumann (2009), Baumann-McAllister (2009,2014), Westphal (2014), Grimm-Louis (2005), Grimm-Lopes (2011), Kerstan-Weigand (2011), Grimm-Louis (2004), Jockers-Louis (2005), Haack-Krefl-Lust-Van Proeyen-Zagermann (2006) Closed string axions a i from dim. red. of p-forms C (p) on M 1,3 X 6 /ΩR (C (p) RR-forms + NS 2-form in Type II) a i (2π) 1 Σ i C (p), p cycle Σ i X 6, i {1,..., h 11 h } Kinetic terms for p-forms C (p) kinetic terms for a i Stack of N D-branes on M 1,3 Σ i U(N) gauge theory in 4 + p dim Reduction of D-brane CS-action couplings for a i C 3 Tr(G G) anomal. coupling a i Tr(G G) IIA C 5 F Stückelberg-coupling (da i k i A) 2 under U(1) Subset of axions are eaten by anomalous U(1) s survive as global symmetries Also possible in IIB for C 2,4 using D7-branes with magn. fluxes

8 Stringy Axions from Type II compactifications reviews: Baumann (2009), Baumann-McAllister (2009,2014), Westphal (2014), Grimm-Louis (2005), Grimm-Lopes (2011), Kerstan-Weigand (2011), Grimm-Louis (2004), Jockers-Louis (2005), Haack-Krefl-Lust-Van Proeyen-Zagermann (2006) Closed string axions a i from dim. red. of p-forms C (p) on M 1,3 X 6 /ΩR (C (p) RR-forms + NS 2-form in Type II) a i (2π) 1 Σ i C (p), p cycle Σ i X 6, i {1,..., h 11 h } Kinetic terms for p-forms C (p) kinetic terms for a i Stack of N D-branes on M 1,3 Σ i U(N) gauge theory in 4 + p dim Reduction of D-brane CS-action couplings for a i C 3 Tr(G G) anomal. coupling a i Tr(G G) IIA C 5 F Stückelberg-coupling (da i k i A) 2 under U(1) Subset of axions are eaten by anomalous U(1) s survive as global symmetries Also possible in IIB for C 2,4 using D7-branes with magn. fluxes

9 Stringy Axions from Type II compactifications reviews: Baumann (2009), Baumann-McAllister (2009,2014), Westphal (2014), Grimm-Louis (2005), Grimm-Lopes (2011), Kerstan-Weigand (2011), Grimm-Louis (2004), Jockers-Louis (2005), Haack-Krefl-Lust-Van Proeyen-Zagermann (2006) Closed string axions a i from dim. red. of p-forms C (p) on M 1,3 X 6 /ΩR (C (p) RR-forms + NS 2-form in Type II) a i (2π) 1 Σ i C (p), p cycle Σ i X 6, i {1,..., h 11 h } Kinetic terms for p-forms C (p) kinetic terms for a i Stack of N D-branes on M 1,3 Σ i U(N) gauge theory in 4 + p dim Reduction of D-brane CS-action couplings for a i C 3 Tr(G G) anomal. coupling a i Tr(G G) IIA C 5 F Stückelberg-coupling (da i k i A) 2 under U(1) Subset of axions are eaten by anomalous U(1) s survive as global symmetries Also possible in IIB for C 2,4 using D7-branes with magn. fluxes

10 Stringy Axions from Type II compactifications reviews: Baumann (2009), Baumann-McAllister (2009,2014), Westphal (2014), Grimm-Louis (2005), Grimm-Lopes (2011), Kerstan-Weigand (2011), Grimm-Louis (2004), Jockers-Louis (2005), Haack-Krefl-Lust-Van Proeyen-Zagermann (2006) Closed string axions a i from dim. red. of p-forms C (p) on M 1,3 X 6 /ΩR (C (p) RR-forms + NS 2-form in Type II) a i (2π) 1 Σ i C (p), p cycle Σ i X 6, i {1,..., h 11 h } Kinetic terms for p-forms C (p) kinetic terms for a i Stack of N D-branes on M 1,3 Σ i U(N) gauge theory in 4 + p dim Reduction of D-brane CS-action couplings for a i C 3 Tr(G G) anomal. coupling a i Tr(G G) IIA C 5 F Stückelberg-coupling (da i k i A) 2 under U(1) Subset of axions are eaten by anomalous U(1) s survive as global symmetries Also possible in IIB for C 2,4 using D7-branes with magn. fluxes

11 Effective Action & Effective Decay Constant w/ Shiu-Ye , [hep-th] Type II String Theory compactifications w/ D-branes 4d EFT with mixing axions S eff axion = 1 2 ( N G ij (da i k i A) 4 (da j k j A) 1 N ) 8π 2 r i a i Tr(G G) + L gauge i,j=1 i=1 metric mixing U(1) mixing k i 0 anomalous coupling Diagonalisation of kinetic and potential terms effective decay constant f eff with moduli dependence different from N-enhancement mechanisms: f eff N p f with p 1/2, Dimopoulos-Kachru-McGreevy-Wacker (2005), Choi-Kim-Yung (2014), Bachlechner-Long-McAllister (2014/15), Junghans (2015)

12 Effective Action & Effective Decay Constant w/ Shiu-Ye , [hep-th] Type II String Theory compactifications w/ D-branes 4d EFT with mixing axions S eff axion = 1 2 ( N G ij (da i k i A) 4 (da j k j A) 1 N ) 8π 2 r i a i Tr(G G) + L gauge i,j=1 i=1 U(1) mixing k i 0 metric mixing anomalous coupling Diagonalisation of kinetic and potential terms effective decay constant f eff with moduli dependence θ GeV f eff GeV G22 G 11 2k 1 = k 2 = 2r 1 = 2r 2 different from N-enhancement mechanisms: f eff N p f with p 1/2, Dimopoulos-Kachru-McGreevy-Wacker (2005), Choi-Kim-Yung (2014), Bachlechner-Long-McAllister (2014/15), Junghans (2015)

13 Effective Action & Effective Decay Constant w/ Shiu-Ye , [hep-th] Type II String Theory compactifications w/ D-branes 4d EFT with mixing axions S eff axion = 1 2 ( N G ij (da i k i A) 4 (da j k j A) 1 N ) 8π 2 r i a i Tr(G G) + L gauge i,j=1 i=1 U(1) mixing k i 0 metric mixing anomalous coupling Diagonalisation of kinetic and potential terms effective decay constant f eff with moduli dependence θ GeV f eff GeV G22 G 11 2k 1 = k 2 = 2r 1 = 2r 2 different from N-enhancement mechanisms: f eff N p f with p 1/2, Dimopoulos-Kachru-McGreevy-Wacker (2005), Choi-Kim-Yung (2014), Bachlechner-Long-McAllister (2014/15), Junghans (2015)

14 Consistency & 4-Fermion Couplings w/ Shiu-Ye , [hep-th] U(1) gauge invariance requires presence of chiral fermions ψ U(1) ζ G G Integrating out massive U(1) boson 1 axion ξ + 1 non-abelian gauge group + chiral fermions S = 1 2 dξ 4dξ 1 ξ 8π 2 Tr(G G) C f ξ f2 2 J ψ 4 J ψ +L ψ }{{} 4 fermion

15 Consistency & 4-Fermion Couplings w/ Shiu-Ye , [hep-th] U(1) gauge invariance requires presence of chiral fermions ψ U(1) ζ G + U(1) ψ ψ G = 0 G ψ G reversed GS mechanism Aldazabel-Franco-Ibáñez-Rábadan-Uranga ( 01) Integrating out massive U(1) boson 1 axion ξ + 1 non-abelian gauge group + chiral fermions S = 1 2 dξ 4dξ 1 ξ 8π 2 Tr(G G) C f ξ f2 2 J ψ 4 J ψ +L ψ }{{} 4 fermion

16 Consistency & 4-Fermion Couplings w/ Shiu-Ye , [hep-th] U(1) gauge invariance requires presence of chiral fermions ψ Integrating out massive U(1) boson 1 axion ξ + 1 non-abelian gauge group + chiral fermions S = 1 2 dξ 4dξ 1 ξ 8π 2 Tr(G G) C f ξ f2 2 J ψ 4 J ψ +L ψ }{{} 4 fermion ψ U(1) ψ ψ ψ = + global U(1) anom ψ ψ ψ ψ N-JL type couplings see StringPheno 2016

17 U(1) Breaking & Mass Generation Shiu-W.S. (work in progress) at strong coupling for SU(N) non-perturbative effects important Instantons ( G G = 0): U(1) Z nf interactions between ψ and instantons t Hooft (1976) L t Hooft = C e 8π2 g 2 +iθ det(ψ L ψ R ) + h.c. Fermion condensate ( ψ L ψ R = 0): Z nf Z 2 Casher (1979) fermion mass M 1 Mst 2 ψ L ψ R E < Λ s: bound state ψψ mass spectrum in vacuum EFT for composite scalar Φ(x) = σ(x)e i η f f ξ f f ξ f f f ξ m η < m σ m ξ m ξ, m η < m σ m ξ m η < m σ

18 U(1) Breaking & Mass Generation Shiu-W.S. (work in progress) at strong coupling for SU(N) non-perturbative effects important Instantons ( G G = 0): U(1) Z nf interactions between ψ and instantons t Hooft (1976) L t Hooft = C e 8π2 g 2 +iθ det(ψ L ψ R ) + h.c. Fermion condensate ( ψ L ψ R = 0): Z nf Z 2 Casher (1979) fermion mass M 1 Mst 2 ψ L ψ R E < Λ s: bound state ψψ mass spectrum in vacuum EFT for composite scalar Φ(x) = σ(x)e i η f f ξ f f ξ f f f ξ m η < m σ m ξ m ξ, m η < m σ m ξ m η < m σ

19 Shiu-W.S. (work in progress) U(1) Breaking & Mass Generation at strong coupling for SU(N) non-perturbative effects important Instantons ( G G = 0): U(1) Z nf interactions between ψ and instantons t Hooft (1976) L t Hooft = C e 8π2 g 2 +iθ det(ψ L ψ R ) + h.c. Fermion condensate ( ψ L ψ R = 0): Z nf Z 2 Casher (1979) fermion mass M 1 Mst 2 ψ L ψ R E < Λ s: bound state ψψ V = µ 2 Φ Φ+ λ 2 (Φ Φ) 2 +Λ 2 s EFT for composite scalar Φ(x) = σ(x)e i η f ) (MΦ + MΦ + κe i ξ f ξ det(φ) + κe i ξ f ξ det(φ ) mass spectrum in vacuum f ξ f f ξ f f f ξ m η < m σ m ξ m ξ, m η < m σ m ξ m η < m σ

20 U(1) Breaking & Mass Generation Shiu-W.S. (work in progress) at strong coupling for SU(N) non-perturbative effects important Instantons ( G G = 0): U(1) Z nf interactions between ψ and instantons t Hooft (1976) L t Hooft = C e 8π2 g 2 +iθ det(ψ L ψ R ) + h.c. Fermion condensate ( ψ L ψ R = 0): Z nf Z 2 Casher (1979) fermion mass M 1 Mst 2 ψ L ψ R E < Λ s: bound state ψψ EFT for composite scalar Φ(x) = σ(x)e i η f V = µ 2 σ 2 + λ 2 σ4 + 2Λ 2 s M σ cos η ( η f + 2κΛ2 s σ cos f + ξ ) + θ f ξ with vacuum σ = f Λ s, η = 0 = ξ mass spectrum in vacuum f ξ f f ξ f f f ξ m η < m σ m ξ m ξ, m η < m σ m ξ m η < m σ

21 U(1) Breaking & Mass Generation Shiu-W.S. (work in progress) at strong coupling for SU(N) non-perturbative effects important Instantons ( G G = 0): U(1) Z nf interactions between ψ and instantons t Hooft (1976) L t Hooft = C e 8π2 g 2 +iθ det(ψ L ψ R ) + h.c. Fermion condensate ( ψ L ψ R = 0): Z nf Z 2 Casher (1979) fermion mass M 1 Mst 2 ψ L ψ R E < Λ s: bound state ψψ EFT for composite scalar Φ(x) = σ(x)e i η f V = µ 2 σ 2 + λ 2 σ4 + 2Λ 2 s M σ cos η ( η f + 2κΛ2 s σ cos f + ξ ) + θ f ξ with vacuum σ = f Λ s, η = 0 = ξ mass spectrum in vacuum massive (σ, η) = INFLADRONS f ξ f f ξ f f f ξ m η < m σ m ξ m ξ, m η < m σ m ξ m η < m σ

22 Phases of Axion Inflation Shiu-W.S. (work in progress) α SU(N) α = 1 Witten, Veneziano ( 79, 98) SU(N) vacuum degeneracy monodromy (η) multi-field (?) natural (ξ) Dilute Instanton Gas g 2 SU(N) > M string Λ s f GeV Honecker- Koltermann-W.S. N-JL inflation (σ) gauged Higgs-Yukawa model depends on UV boundary conditions chaotic inflation, α-attractor with α = 2 Inagaki-Odintsov-Sakamoto ( 15-17) see StringPheno 2016 log f ξ f

23 Weak Gravity & Strong Dynamics WGC = criterium for field theory to be coupled to gravity electric WGC for U(1): particle with m qm Pl In UV (E > M st): 2 charged, massles axions In IR (E < M st): only global U(1) electric WGC for axions: instanton such that S O(1) M Pl f

24 Weak Gravity & Strong Dynamics WGC = criterium for field theory to be coupled to gravity electric WGC for U(1): particle with m qm Pl In UV (E > M st): 2 charged, massles axions In IR (E < M st): only global U(1) electric WGC for axions: instanton such that S O(1) M Pl f

25 Weak Gravity & Strong Dynamics WGC = criterium for field theory to be coupled to gravity electric WGC for U(1): particle with m qm Pl In UV (E > M st): 2 charged, massles axions In IR (E < M st): only global U(1) electric WGC for axions: instanton such that S O(1) M Pl f

26 Weak Gravity & Strong Dynamics WGC = criterium for field theory to be coupled to gravity electric WGC for U(1): particle with m qm Pl In UV (E > M st): 2 charged, massles axions In IR (E < M st): only global U(1) electric WGC for axions: instanton such that S O(1) M Pl f

27 Weak Gravity & Strong Dynamics WGC = criterium for field theory to be coupled to gravity electric WGC for U(1): particle with m qm Pl In UV (E > M st): 2 charged, massles axions In IR (E < M st): only global U(1) electric WGC for axions: instanton such that S O(1) M Pl f In UV (E > M st): (non dominant) D2-brane instantons In IR (E < Λ s): gauge instantons coupled to ξ appear to violate WGC

28 Conclusions Conclusions and Outlook Type II String compactifications rich 4dim EFT for axions At lower energy scales N-JL interactions, instantons and fermion condensates generating dynamical masses a phase (parameter) space for inflationary models WGC fine in UV, violated in IR analogous to Saraswat (2016) Open issues Verification of magnetic WGC see also Hebecker-Henkenjohann-Witkowski (2017), Dolan et al (2017) Detailed analysis for axion monodromy realisation Kaloper-Lawrence-Sorbo (2010) Full String Theory construction including moduli stabilisation Blumenhagen, Valenzuela, Zoccarato

29 Conclusions Conclusions and Outlook Type II String compactifications rich 4dim EFT for axions At lower energy scales N-JL interactions, instantons and fermion condensates generating dynamical masses a phase (parameter) space for inflationary models WGC fine in UV, violated in IR analogous to Saraswat (2016) Open issues Verification of magnetic WGC see also Hebecker-Henkenjohann-Witkowski (2017), Dolan et al (2017) Detailed analysis for axion monodromy realisation Kaloper-Lawrence-Sorbo (2010) Full String Theory construction including moduli stabilisation Blumenhagen, Valenzuela, Zoccarato

30

31

32 Natural Inflation Phase Inflaton = closed string axion Viable inflationary model requires control over perturbative (& non-perturbative) quantum corrections: EFT for Infladrons Φ receives non-renormalisable corrections: derivative terms: M 4 UV Φ Φ 2, M 2 UV Φ 2 Φ 2 additionally suppressed potential & mixed terms by powers of f fξ 10 3 Loop-corrections to Φ ( 1-loop effective action V 1 loop ( µ 2 + 3λ Φ 2 ) {ln 2 µ 2 +3λ Φ 2 ) } Λ Colemanr Weinberg ( 73) proper resummation of eff. potential maintains vacuum structure Induced non-minimal coupling to gravity g ξ Φ 2 R Hill-Salopek ( 92) Solving RGE for ξ IR-fixed point ξ = 0 Infladron-backreaction: hierarchy m ξ m η < m σ has to prevail along Buchmüller et al ( 15) inflationary trajectory OK when f 10 f 3 ξ

33 Natural Inflation Phase Inflaton = closed string axion Viable inflationary model requires control over perturbative (& non-perturbative) quantum corrections: EFT for Infladrons Φ receives non-renormalisable corrections: derivative terms: M 4 UV Φ Φ 2, M 2 UV Φ 2 Φ 2 additionally suppressed potential & mixed terms by powers of f fξ 10 3 Loop-corrections to Φ ( 1-loop effective action V 1 loop ( µ 2 + 3λ Φ 2 ) {ln 2 µ 2 +3λ Φ 2 ) } Λ Colemanr Weinberg ( 73) proper resummation of eff. potential maintains vacuum structure Induced non-minimal coupling to gravity g ξ Φ 2 R Hill-Salopek ( 92) Solving RGE for ξ IR-fixed point ξ = 0 Infladron-backreaction: hierarchy m ξ m η < m σ has to prevail along Buchmüller et al ( 15) inflationary trajectory OK when f 10 f 3 ξ

34 Natural Inflation Phase Inflaton = closed string axion Viable inflationary model requires control over perturbative (& non-perturbative) quantum corrections: EFT for Infladrons Φ receives non-renormalisable corrections: derivative terms: M 4 UV Φ Φ 2, M 2 UV Φ 2 Φ 2 additionally suppressed potential & mixed terms by powers of f fξ 10 3 Loop-corrections to Φ ( 1-loop effective action V 1 loop ( µ 2 + 3λ Φ 2 ) {ln 2 µ 2 +3λ Φ 2 ) } Λ Colemanr Weinberg ( 73) proper resummation of eff. potential maintains vacuum structure Induced non-minimal coupling to gravity g ξ Φ 2 R Hill-Salopek ( 92) Solving RGE for ξ IR-fixed point ξ = 0 Infladron-backreaction: hierarchy m ξ m η < m σ has to prevail along Buchmüller et al ( 15) inflationary trajectory OK when f 10 f 3 ξ

35 Natural Inflation Phase Inflaton = closed string axion Viable inflationary model requires control over perturbative (& non-perturbative) quantum corrections: EFT for Infladrons Φ receives non-renormalisable corrections: derivative terms: M 4 UV potential & mixed terms Φ Φ 2, M 2 UV Φ 2 Φ 2 Loop-corrections to Φ ( 1-loop effective action V 1 loop ( µ 2 + 3λ Φ 2 ) {ln 2 µ 2 +3λ Φ 2 Λ 2 r additionally suppressed by powers of f fξ 10 3 ) } 2 3 Coleman- Weinberg ( 73) proper resummation of eff. potential maintains vacuum structure Induced non-minimal coupling to gravity g ξ Φ 2 R Hill-Salopek ( 92) Solving RGE for ξ IR-fixed point ξ = 0 Infladron-backreaction: hierarchy m ξ m η < m σ has to prevail along Buchmüller et al ( 15) inflationary trajectory OK when f 10 f 3 ξ ξ ξ σ η

36 Stringy Axions and Stückelberg-mechanism reviews: Baumann (2009), Baumann-McAllister (2009,2014), Westphal (2014), Grimm-Louis (2005), Grimm-Lopes (2011), Kerstan-Weigand (2011), Grimm-Louis (2004), Jockers-Louis (2005), Haack-Krefl-Lust-Van Proeyen-Zagermann (2006) Closed string axions a i from dim. red. of p-forms C (p) on M 1,3 X 6 /ΩR (C (p) RR-forms + NS 2-form in Type II) a i (2π) 1 Σ i C (p), p cycle Σ i X 6, i {1,..., h 11 h } Kinetic terms for p-forms C (p) kinetic terms for a i Stack of N D-branes on M 1,3 Σ i U(N) gauge theory in 4 + p dim Reduction of D-brane CS-action couplings for a i C 3 Tr(G G) anomal. coupling a i Tr(G G) IIA C 5 F Stückelberg-coupling (da i k i A) 2 under U(1) Subset of axions are eaten by anomalous U(1) s survive as global symmetries Also possible in IIB for C 2,4 using D7-branes with magn. fluxes

37 2 Mixing Axions minimal set-up: 2 axions + 1 U(1) + 1 Non-Abelian gauge group 1 axion eaten by U(1) gauge boson, axion ξ with decay constant: λ+λ M st f ξ = cos θ 2 (λ+k+ r 2 + λ k r 1 ) + sin θ 2 (λ k r 2 λ +k + r 1 ) with λ ± eigenvalues of G ij and M st λ +(k + ) 2 + λ (k ) 2 cos θ = G 11 G 22 λ + λ, sin θ = 2G 12 λ + λ, Contour plot representation of f ξ (in units G 11 ) ( ) ( k + cos θ sin θ k = 2 2 sin θ cos θ 2 2 ) ( k 1 k 2 ) 2k 1 = k 2 = 2r 1 = 2r 2 k 1 = 2k 2 = r 1 = 2r 2 k 1 = 2k 2 = r 1 = 2r 2 G22 G 11 G22 G 11 G22 G 11 θ θ θ

38 Axions & String Theory: Example reviews: Blumenhagen-Cvetič-Langacker-Shiu ( 05); Blumenhagen-Körs-Lüst-Stieberger ( 06); Ibañez-Uranga ( 12) e.g. Type IIA D6-branes on CY 3 /ΩR Σ i = Σ i + + Σi Σ i C (5) F 0 Stückelberg coupling for a i T 6 /ΩR with 4 ΩR-even 3-cycles Σ i=0,1,2,3 + and 4 ΩR-odd 3-cycles Σ i=0,1,2,3 }{{} 4 axions a i ΩR Π a Π a = Σ 0 + Σ 3 + }{{} + Σ1 Σ 2 }{{} a 0 F a F a (da 1 A a) 2 a 3 F a F a (da 2 + A a) 2

39 The road to NJL Models Shiu-WS-Ye( 15), Shiu-WS (work in progress) U(1) gauge invariance: A A + dχ, a a + kχ S sub = M2 st 2 da ka 2 1 4gU(1) 2 F 2 1 a Tr( G G) 8π2 }{{} not U(1) invariant U(1) ζ G intersection of two D-brane stacks with U(N) U(1) chiral matter in bifund. rep. EFT for generation-indep. U(1) charges: L = M2 st 2 da ka 2 1 4gU(1) 2 F 2 + i N f ψ i N L / ψi L + i f ψ i R / ψi R i=1 i=1 + q L ψ i L /Aψi L + q R ψ i R /Aψi R + coupling to U(N)

40 The road to NJL Models Shiu-WS-Ye( 15), Shiu-WS (work in progress) U(1) gauge invariance: A A + dχ, a a + kχ Ibáñez-Rábadan- S sub = M2 st 2 da ka 2 1 4gU(1) 2 F 2 1 a Tr( G G) 8π2 }{{} not U(1) invariant + chiral fermions ψ Antoniadis-Kiritsis- G ψ G Rizos ( 02) U(1) a + U(1) ψ Aldazabel-Franco- G ψ reversed GS intersection of two D-brane stacks with U(N) U(1) chiral matter in bifund. rep. G Uranga ( 01) EFT for generation-indep. U(1) charges: L = M2 st 2 da ka 2 1 4gU(1) 2 F 2 + i N f ψ i N L / ψi L + i f ψ i R / ψi R i=1 i=1 + q L ψ i L /Aψi L + q R ψ i R /Aψi R + coupling to U(N)

41 The road to NJL Models Shiu-WS-Ye( 15), Shiu-WS (work in progress) U(1) gauge invariance: A A + dχ, a a + kχ S sub = M2 st 2 da ka 2 1 4gU(1) 2 F 2 1 a Tr( G G) 8π2 }{{} + chiral fermions ψ not U(1) intersection of two D-brane stacks with U(N) U(1) chiral matter in bifund. rep. (, q L ) EFT for generation-indep. U(1) charges: (, q R ) ψ i L ψ i R with q L q R L = M2 st 2 da ka 2 1 4gU(1) 2 F 2 + i + i q L ψ i L /Aψi L + i N f ψ i N L / ψi L + i f ψ i R / ψi R i=1 i=1 q R ψ i R /Aψi R + coupling to U(N)

42 The road to NJL Models (II) Shiu-WS-Ye( 15), Shiu-WS(work in progress) Integrating out Stückelberg U(1) through solving Lorenz gauge condition: A 1 ( Mst 2 q L ψ i L Γψi L + q Rψ i ) R Γψi R L 4ψ q2 Mst 2 (ψγψ) 2 ψ U(1) ψ ψ ψ = + global U(1) anom ψ ψ ψ ψ Using Fierz-identities NJL-type models with N f = 1 L 4ψ = q [ ] Lq R 2Mst 2 (ψψ)(ψψ) (ψγ 5 ψ)(ψγ 5 ψ) +...

43 NJL & Dynamical Mass Generation Nambu-Jona-Lasinio ( 61), review: Vogl-Weise ( 91) NJL = U(1) chiral invariant 4ψ-interactions ψ e iαγ5 ψ L NJL = ψi / ψ + q [ ] Lq R 2Mst 2 (ψψ)(ψψ) (ψγ 5 ψ)(ψγ 5 ψ) ψψ ψψ cos(2α) +iψγ 5 ψ sin(2α) ψγ 5 ψ ψγ 5 ψ cos(2α) +iψψ sin(2α) two phases: Wigner phase: ψψ = 0 U(1) chiral unbroken and ψ massless Nambu-Goldstone-phase : ψψ = 0 m ψ = q Lq R Mst 2 ψψ and U(1) chiral NG-phase requires satisfied self-consistency condition: GAP: m ψ = 4iq Lq R N M 2 st d 4 p (2π) 4 and m ψ 0 at strong coupling: α U(1) (Λ) > π N m ψ p 2 m 2 ψ for Λ < Mst

44 NJL & Dynamical Mass Generation (II) Verify minimum in NG-phase in large N limit Gross-Neveu ( 74) ( ) ( ) ( σ cos 2α sin 2α σ auxiliary fields σ, π: π sin 2α cos 2α π L σ = ψi / ψ 1 ( ) 2g 2 (σ2 + π 2 ) + σψψ + iπψγ 5 ψ, g 2 = q Lq R Mst 2 d 4 p effective potential V eff (σ) = 1 2g 2 σ 2 2N (2π) 4 ln minimum: σ=σmin = 0 GAP eq dv eff dσ (1 + σ2 p 2 ) V eff (σ) ) V eff (σ min ) < 0 Bound (scalar) states: (Salpeter-Bethe ( 51) or poles of G (4) 4ψ ) 0 + state σ = gψψ m = 4m 2 ψ σ min 0 state π = gψiγ 5 ψ m 2 0 = 0

45 First Inflationary Steps Gradient expansion EFT for (σ, π) with inflaton= σ L (σ,π) = 1 2 ( σ) ( π) m2 ψ σ2 λ 4 σ RGE-analysis with conformal coupling to gravity R 2 -type inflation Hill-Salopek ( 92), Inagaki-Odintsov-Sakamoto ( 15) ( V(σ,π) RGE = 1 B 1 + D 6 (σ2 ) 1/(1+ αc α ) 2 σ2 + C ) 4 (σ4 ) 1/(1+ α αc ) compatible with Planck 2015 data: n s = 0.961, r = weak coupling α 1 Strong SU(N) dynamics can spoil (σ, π) set-up: t Hooft (1976) SU(N) gauge instantons produce U(1) coupling κ det(ψ(1 + γ 5 )ψ) + h.c. mπ 2 κ 2 < mσ 2 = 4m2 ψ + m2 π π = inflaton? remaining string axions coupling to SU(N) can play rôle of inflaton?

46 First Inflationary Steps Gradient expansion EFT for (σ, π) with inflaton= σ L (σ,π) = 1 2 ( σ) ( π) m2 ψ σ2 λ 4 σ RGE-analysis with conformal coupling to gravity R 2 -type inflation Hill-Salopek ( 92), Inagaki-Odintsov-Sakamoto ( 15) ( V(σ,π) RGE = 1 B 1 + D 6 (σ2 ) 1/(1+ αc α ) 2 σ2 + C ) 4 (σ4 ) 1/(1+ α αc ) compatible with Planck 2015 data: n s = 0.961, r = weak coupling α 1 NJL inflation (N=50)