An Axion-induced SM/MSSM Higgs Landscape and the Weak Gravity Conjecture

Transcription

1 An Axion-induced SM/MSSM Higgs Landscape and the Weak Gravity Conjecture Alvaro Herraez Universidad Autónoma de Madrid & Instituto de Fisica Teórica UAM-CSIC Based on: A.H., L. Ibáñez [ ] String Pheno 2017, Blacksburg July 6, 2017

2 Outline Introduction and motivation à 4-forms & axions (in ST) Higgs landsacape from 3-formsà A minimal toy model Higgs landsacape from 3-formsà A SUSY toy model SUSY breaking scale, string scale & the WGC Conclusions

3 Minkowski 3-forms Consider the action for a 3-form in 4d Z S = d 4 x 1 Z 2 F q C 3 + S bound D 3 [Brown, Teitelboim, Bousso, Polchinski, Dvali ]

4 Minkowski 3-forms Consider the action for a 3-form in 4d Z S = d 4 x 1 Z 2 F q C 3 + S bound D 3 [Brown, Teitelboim, Bousso, Polchinski, Dvali ] Kinetic term Coupling to membranes Equation of motion (away from the membranes) F µ = f 0 µ No propagating dof in 4d

5 Minkowski 3-forms Consider the action for a 3-form in 4d Z S = d 4 x 1 Z 2 F q C 3 + S bound D 3 [Brown, Teitelboim, Bousso, Polchinski, Dvali ] Kinetic term Coupling to membranes Equation of motion (away from the membranes) F µ = f 0 µ No propagating dof in 4d f 0 Membranes separate diferent vacua labeled by the value of q f 0 f 0 + q At this level à q is undetermined

6 Minkowski 3-forms & axions Consider an axion w/ a discrete shift symmetry! + nf Couple it to the 4-form [Dvali, Kaloper, Sorbo, Lawrence] L = 1 2 F (@ µ ) 2 + µ F Axion coupling

7 Minkowski 3-forms & axions Consider an axion w/ a discrete shift symmetry! + nf Couple it to the 4-form Integrating out the 4-form L = 1 2 F (@ µ ) 2 + µ F V = 1 2 f 0 + µ 2 [Dvali, Kaloper, Sorbo, Lawrence] Axion coupling

8 Minkowski 3-forms & axions Consider an axion w/ a discrete shift symmetry! + nf Couple it to the 4-form Integrating out the 4-form L = 1 2 F (@ µ ) 2 + µ F V = 1 2 f 0 + µ 2 µ The axion gets a mass but the symmetry is still respected when! + nf [Dvali, Kaloper, Sorbo, Lawrence] Axion coupling f 0! f 0 nµf

9 Minkowski 3-forms & axions Consider an axion w/ a discrete shift symmetry! + nf Couple it to the 4-form Integrating out the 4-form L = 1 2 F (@ µ ) 2 + µ F V = 1 2 f 0 + µ 2 µ The axion gets a mass but the symmetry is still respected when! + nf f 0 [Dvali, Kaloper, Sorbo, Lawrence] q Axion coupling f 0! f 0 nµf f 0 + q Related

10 Axion/3-form structure in ST Can always describe this kind of shift symmetric potentials for the axions in terms of 4-forms coupled to them.

11 Axion/3-form structure in ST Can always describe this kind of shift symmetric potentials for the axions in terms of 4-forms coupled to them. String Theory à Plenty of 4-forms and axions. Is there any similar description for the 4d scalar potential for the axions?

12 Axion/3-form structure in ST Can always describe this kind of shift symmetric potentials for the axions in terms of 4-forms coupled to them. String Theory à Plenty of 4-forms and axions. Is there any similar description for the 4d scalar potential for the axions? Yes!

13 Axion/3-form structure in ST Can always describe this kind of shift symmetric potentials for the axions in terms of 4-forms coupled to them. String Theory à Plenty of 4-forms and axions. Is there any similar description for the 4d scalar potential for the axions? The full scalar potential of type IIA with fluxes can be written in terms of 4-forms coupling to polynomials in the axions: V = g ab F a 4 ^ F b 4 +2F a 4 a ( i )+V loc Field dependent (axion independent) Polynomial in axions (only) [Bielleman, Ibáñez, Valenzuela]

14 Axion/3-form structure in ST Can always describe this kind of shift symmetric potentials for the axions in terms of 4-forms coupled to them. String Theory à Plenty of 4-forms and axions. Is there any similar description for the 4d scalar potential for the axions? The full scalar potential of type IIA with fluxes can be written in terms of 4-forms coupling to polynomials in the axions: V = g ab F a 4 ^ F b 4 +2F a 4 a ( i )+V loc Field dependent (axion independent) Polynomial in axions (only) [Bielleman, Ibáñez, Valenzuela] MOTIVATION à Can these kind of axion/4-form systems be used to generate a landscape for the Higgs mass?

15 Higgs landscape. Minimal model L Consider just the SM Higgs field coupled to the axion/4-form system 1 2 (F a) (F b) 2 + (µf a + µ h F h )+ F h H 2 m 2 H 2 + H 4 3-forms kinetic term Axion/4-forms coupling HIggs/4-forms coupling SM Higgs potential

16 Higgs landscape. Minimal model L Consider just the SM Higgs field coupled to the axion/4-form system 1 2 (F a) (F b) 2 + (µf a + µ h F h )+ F h H 2 m 2 H 2 + H 4 3-forms kinetic term Axion/4-forms coupling HIggs/4-forms coupling SM Higgs potential Unprotected (~cutoff scale)

17 Higgs landscape. Minimal model L Consider just the SM Higgs field coupled to the axion/4-form system 1 2 (F a) (F b) 2 + (µf a + µ h F h )+ F h H 2 m 2 H 2 + H 4 3-forms kinetic term Axion/4-forms coupling HIggs/4-forms coupling SM Higgs potential Integrate out 4-forms à Scalar potential for axions+higgs respecting shift symmetries à charges are related to axion parameters. q a = µf q h = µ h f

18 Higgs landscape. Minimal model L Consider just the SM Higgs field coupled to the axion/4-form system 1 2 (F a) (F b) 2 + (µf a + µ h F h )+ F h H 2 m 2 H 2 + H 4 3-forms kinetic term Axion/4-forms coupling HIggs/4-forms coupling SM Higgs potential Integrate out 4-forms à Scalar potential for axions+higgs respecting shift symmetries à charges are related to axion parameters. q a = µf q h = µ h f Minimize the potential to find the Higgs vev H 2 = m2 cos 2 f h 0 sin cos f a cos 2 (H 2 ) ' µf = q a

19 Higgs landscape. Minimal model L Consider just the SM Higgs field coupled to the axion/4-form system 1 2 (F a) (F b) 2 + (µf a + µ h F h )+ F h H 2 m 2 H 2 + H 4 3-forms kinetic term Axion/4-forms coupling HIggs/4-forms coupling SM Higgs potential Integrate out 4-forms à Scalar potential for axions+higgs respecting shift symmetries à charges are related to axion parameters. q a = µf q h = µ h f Minimize the potential to find the Higgs vev H 2 = m2 cos 2 f h 0 sin cos f a cos 2 Cancelationà 4-form vevs ~ cutoff scale (H 2 ) ' µf = q a Steps must be below 170GeV

20 MSSM landscape Toy N =1no-scale supergravity model with 2 fields & 2 fluxes K = 2 log(u + U ) 3 log(t + T ) W = e 0 + ih 0 U U = u + ib Re(T )=t

21 N =1 MSSM landscape Toy no-scale supergravity model with 2 fields & 2 fluxes K = 2 log(u + U ) 3 log(t + T ) W = e 0 + ih 0 U U = u + ib Re(T )=t Shift symmetries for the axion again mantained if combined with shifts of the fluxes b! b + n e 0! e 0 + h 0 n

22 MSSM landscape N =1 Toy no-scale supergravity model with 2 fields & 2 fluxes K = 2 log(u + U ) 3 log(t + T ) W = e 0 + ih 0 U U = u + ib Re(T )=t Shift symmetries for the axion again mantained if combined with shifts of the fluxes b! b + n e 0! e 0 + h 0 n e 0 quantized in terms of h 0 f 0! f 0 + nq

23 MSSM landscape N =1 Toy no-scale supergravity model with 2 fields & 2 fluxes K = 2 log(u + U ) 3 log(t + T ) W = e 0 + ih 0 U U = u + ib Re(T )=t Shift symmetries for the axion again mantained if combined with shifts of the fluxes. b! b + n e 0! e 0 + h 0 n e 0 quantized in terms of h 0 The scalar potential is V = e K K UU D U W 2 =2e K e 0 h 0 b 2

24 MSSM landscape N =1 Toy no-scale supergravity model with 2 fields & 2 fluxes K = 2 log(u + U ) 3 log(t + T ) W = e 0 + ih 0 U U = u + ib Re(T )=t Shift symmetries for the axion again mantained if combined with shifts of the fluxes. b! b + n e 0! e 0 + h 0 n e 0 quantized in terms of h 0 The scalar potential is V = e K K UU D U W 2 =2e K e 0 h 0 b 2 Minimum m 2 3/2 = h2 0 2t 3 b = e 0 h 0

25 MSSM landscape (with 4-forms) In terms of 4-forms, consider the following Lagrangian L = e K F F 4 (e 0 h 0 b)

26 MSSM landscape (with 4-forms) In terms of 4-forms, consider the following Lagrangian L = e K F 4 2 eom +2F 4 (e 0 h 0 b) F 4 = e K (e 0 h 0 b) Recover the same potential as before

27 MSSM landscape (with 4-forms) In terms of 4-forms, consider the following Lagrangian L = e K F 4 2 eom +2F 4 (e 0 h 0 b) F 4 = e K (e 0 h 0 b) Recover the same potential as before 4-forms have no extra propagating dof à Related to SUSY auxiliary fields F U = e K/2 2u(e 0 h 0 b)=2ue K/2 F 4

28 MSSM landscape (with 4-forms) In terms of 4-forms, consider the following Lagrangian L = e K F 4 2 eom +2F 4 (e 0 h 0 b) F 4 = e K (e 0 h 0 b) Recover the same potential as before 4-forms have no extra propagating dof à Related to SUSY auxiliary fields F U = e K/2 2u(e 0 h 0 b)=2ue K/2 F 4 N =1 The model is consistent with a description in terms of 3-forms

29 MSSM landscape (with 4-forms) Consider this as a hidden sector for the matter fields. In a model with minimal kinetic terms for the Higgs. m 2 H u = m 2 H d = m 2 3/2 = h2 0M 4 s 2M 2 P t3 (Reinserting string and Planck masses)

30 MSSM landscape (with 4-forms) Consider this as a hidden sector for the matter fields. In a model with minimal kinetic terms for the Higgs. m 2 H u = m 2 H d = m 2 3/2 = h2 0M 4 s 2M 2 P t3 (Reinserting string and Planck masses) With h 0 quantized, the Higgs masses scan in a landscape. h 0 arbitrarily high e.g. (10 10 GeV) 2 the Higgs mass is OK as it is Planck suppressed and then it can be around 100 GeV.

31 MSSM landscape (with 4-forms) Consider this as a hidden sector for the matter fields. In a model with minimal kinetic terms for the Higgs. m 2 H u = m 2 H d = m 2 3/2 = h2 0M 4 s 2M 2 P t3 (Reinserting string and Planck masses) With h 0 quantized, the Higgs masses scan in a landscape. h 0 arbitrarily high e.g. (10 10 GeV) 2 the Higgs mass is OK as it is Planck suppressed and then it can be around 100 GeV. The point à If the relevant auxiliary fields of the hidden sector contain quantized 4-forms there will be a landscape for the Higgs mass, as in ST examples.

32 SUSY breaking scale, string scale & WGC WGC suggests bounds for the SUSY breaking scale in certain classes of string compactifications. m 3/2 & M 2 s M P

33 SUSY breaking scale, string scale & WGC WGC suggests bounds for the SUSY breaking scale in certain classes of string compactifications. m 3/2 & M 2 s M P h 0 = q = µf WGC applied to the brane 2 q T M P [Arkani-Hamed, Motl, Nicolis, Vafa]

34 SUSY breaking scale, string scale & WGC WGC suggests bounds for the SUSY breaking scale in certain classes of string compactifications. m 3/2 & M 2 s M P h 0 = q = µf WGC applied to the brane 2 q T M P [Arkani-Hamed, Motl, Nicolis, Vafa] When the goldstino multiplet contains a monodromy axion the axion mass is ~ gravitino mass m 3/2 ' µ T fm P

35 SUSY breaking scale, string scale & WGC WGC suggests bounds for the SUSY breaking scale in certain classes of string compactifications. m 3/2 & M 2 s M P h 0 = q = µf WGC applied to the brane 2 q T M P [Arkani-Hamed, Motl, Nicolis, Vafa] When the goldstino multiplet contains a monodromy axion the axion mass is ~ gravitino mass m 3/2 ' µ T fm P f M s T Ms 3

36 SUSY breaking scale, string scale & WGC WGC suggests bounds for the SUSY breaking scale in certain classes of string compactifications. m 3/2 & M 2 s M P h 0 = q = µf WGC applied to the brane 2 q T M P [Arkani-Hamed, Motl, Nicolis, Vafa] When the goldstino multiplet contains a monodromy axion the axion mass is ~ gravitino mass m 3/2 ' µ T fm P LOWER BOUND FOR THE SUSY BREAKING SCALE!!! f M s T Ms 3

37 Conclusions The quantisation properties of the 4-forms can be translated into a landscape for Higgs masses Minimal SM extension o Detailed cancelation among 4- form vevs & cutoff scale o Small quanta compared to these vevs o Still generates a lanscape for the masses that are protected by the shift symmetry (technically natural) SUSY case o No need for small quanta since gravity mediation implies Planck suppression o Generation of a landscape for Higgs masses that are again protected by the shift symmetries The WGC can give a lower bound for the SUSY breaking scale depending on the string scale for a large class of string compactifications

38

39 BACK-UP SLIDES

40 SM Higgs landscape. Stability The obtained vacua must be stable (B>1) against bubble nucleation P ' e B B = 27 2 T 4 2( V ) 3 [Coleman, De Lucia] f 0 + q f 0 V ' qf 0 ' q m2 From the WGC for membranes coupling to 3-froms [Arkani-Hamed, Motl, Nicolis, Vafa] T apple 2 qm P Combining both 0.3 3/2 (m 2 Hm 2 ) 3/4. T. 2 m2 H M P