Introduction to Supersymmetry

Transcription

1 Introduction to Supersymmetry I. Antoniadis Albert Einstein Center - ITP Lecture 5 Grand Unification I. Antoniadis (Supersymmetry) 1 / 22

2 Grand Unification Standard Model: remnant of a larger gauge symmetry semi-simple group G SU(3) SU(2) U(1) Y at a scale M GUT U(1) Y : non-abelian generator => charge quantization Main consequences: G has a single coupling g => gauge coupling unification TrY = 0 in every representation of G => quarks and leptons are generally mixed => B, L violation proton decay => M GUT > GeV I. Antoniadis (Supersymmetry) 2 / 22

3 gauge coupling unification - non abelian couplings: g 2 = g 3 at M GUT but what about g Y? - U(1) Y should also be normalized as the non-abelian generators representation R: Tr R t a t b = T(R)δ ab compute TrT 2 3 and TrY2 for a complete fermion family: q 1/6 u c 2/3 d c 1/3 l 1/2 e c 1 total TrT = = TrY = = = = it follows: Y = g Y = 5 3 T 1 (non-abelian generator) => 3 5 g 1 (covariant derivative: g Y Y = g 1 T 1 ) g 1 = g 2 = g 3 => prediction: sin 2 θ W = g2 Y g 2 Y +g2 2 = 3/5 3/5+1 = 3 8 at M GUT I. Antoniadis (Supersymmetry) 3 / 22

4 Renormalization Group evolution At energies < M GUT only light SM particles contribute in the loops running with the SM beta-functions: beta-function coefficients: SM b 3 = N g dα i d lnq = b i 2π α2 i SSM = 9 2N g nb of generations b 2 = N g 1 6 N H = 6 2N g 1 2 N H b 1 = 4 3 N g 1 10 N H = 2N g 3 10 N H nb of Higgs doublets low energy data at M Z : α 1 3 = 8.50±0.14 α 1 2 = 29.57±0.02 α 1 1 = 59.00±0.02 I. Antoniadis (Supersymmetry) 4 / 22

5 gauge coupling evolution of SM versus SSM α SM α α 2 1 SSM 10 α Log 10 (Q/1 GeV) M GUT GeV I. Antoniadis (Supersymmetry) 5 / 22

6 GUT prediction of QCD coupling input α em,sin 2 θ W => output α 3 exp value I. Antoniadis (Supersymmetry) 6 / 22

7 SU(5) grand unification Standard Model: rank 4 => rank(g) 4 TrQ of SM representations: q ( ) 3 = 1 uc 2 d c 1 l 1 e c 1 => traceless combinations: (u c qd c )(le c ),(u c d c e c )(ql),(u c qe c )(d c l) only possibility: SU(5) 10 5 SM embedding in SU(5): generators 5 5 traceless matrices ( ) ( ) ( SU(3) U(1) : T SU(2) 1 = c TrT 1 = 0 TrT1 2 = => c = 5 T 1 = cy ) I. Antoniadis (Supersymmetry) 7 / 22

8 SM embedding in SU(5) SU(3) SU(2) U(1) fermions: 5 = ( 3,1) 1/3 +(1,2) 1/2 10 = (3,2) 1/6 +( 3,1) 2/3 +(1,1) 1 d c l q u c e c d1 c 0 u d2 c 3 c u2 c u 1 d 1 u 5 = d3 c 3 c 0 u1 c u 2 d 2 10 = u e 2 c u1 c 0 u 3 d 3 u 1 u 2 u 3 0 e c ν e d 1 d 2 d 3 e c 0 adjoint 24 = (8,1) 0 +(1,3) 0 +(1,1) 0 +(3,2) 5/6 +( 3, 2) 5/6 ր ր ր տ( ) ր X gluons W ±,3 Q = 4/3 B Y Q = 1/3 X 1 Ȳ 1 0 X 2 Ȳ 2 X,Y: 12 more generators X 3 Ȳ 3 X 1 X 2 X 3 Y 1 Y 2 Y 3 I. Antoniadis (Supersymmetry) 8 / 22

9 GUT symmetry breaking Higgs in 24 SU(5) adjoint: Σ Σ = V Y 0 => SU(5) SU(3) SU(2) U(1) 12 Goldstone bosons eaten by X,Y massive massive physical higgses: color octet + weak triplet + singlet EW symmetry breaking: need a pair of 5+ 5 higgses H, H ( ) dh Higgs triplet with quantum numbers of d quark H = h Higgs doublet 0 0 H = 0 0 => SU(2) U(1) U(1) em 1 I. Antoniadis (Supersymmetry) 9 / 22

10 fermion masses SUSY SU(5): H = ( dh H 2 ) Ĥ = ( d c H H 1 ) Yukawa couplings: λ u H +λ d H 10 = (q,u c,e c ) 5 = (d c,l) 5 H = (d H,H 2 ) 5 H = (d c H,H 1) => λ u : qu c H 2 + u c e c d H +qqd H λ d : qd c H 1 +e c lh 1 + u c d c d c H +qldc H proton decay => m b = m τ at M GUT RG evolution at low energies correct prediction for m b /m τ however it fails for the first two generations m s /m µ and m d /m e I. Antoniadis (Supersymmetry) 10 / 22

11 gauge hierarchy SUSY SU(5): H = ( dh H 2 ) Ĥ = ( d c H H 1 ) general superpotential: W = M GUT TrΣ 2 +λtrσ 3 +MHĤ +ρhσĥ => SU(5) breaking: Σ = V Y 0 makes H superheavy: M(d H d c H +H 2H 1 )+ρv ( 1 3 d Hd c H H 2H 1 ) = ( M ρ 3 V) d H d c H +( M + ρ 2 V) H 2 H 1 => fine-tuning to keep the EW Higgs doublets light: ( M + ρ 2 V) = µ O(m W ) with M,V O(M GUT ) Higgs triplet d H,dH c : proton decay via dim-5 operators => keep superheavy doublet/triplet splitting problem I. Antoniadis (Supersymmetry) 11 / 22

12 SO(10) grand unification The only GUT group of rank 5: all fermions of a generation in a single representation SU(5) decomposition: 16 = ν c includes R-neutrino ν c => theory of neutrino masses EW Higgs: 10 H = 5 H + 5 H Yukawa couplings: H B L is an SO(10) generator Higgs sector becomes complicated I. Antoniadis (Supersymmetry) 12 / 22

13 Advantages of SUSY natural elementary scalars gauge coupling unification: theory perturbative up to the GUT scale LSP: natural dark matter candidate extension of space-time symmetry: new Grassmann dimensions attractive mechanism of Electroweak Symmetry Breaking prediction of light Higgs rich spectrum of new particles within LHC reach I. Antoniadis (Supersymmetry) 13 / 22

14 Problems of SUSY too many parameters: soft breaking terms SUSY breaking mechanism => dynamical aspect of the hierarchy + theory of soft terms SM global symmetries are not automatic B, L from R-parity, conditions on soft terms for FCNC suppression SUSY GUTs: no satisfactory model doublet/splitting, large Higgs reps, strong coupling above M GUT µ problem: SUSY mass parameter but of the order of the soft terms SUSY not yet discovered => already fine-tuning at a %-per mille level little hierarchy problem I. Antoniadis (Supersymmetry) 14 / 22

15 proposals for the µ problem - NMSSM: extra singlet σ coupled to higgses δw = λ 1 σh 1 H 2 +λ 2 σ 3 : σ 0 => µ-term generation - dim-5 effective operator from high-energy physics in the Kähler potential δk = 1 M d 4 θs H 1 H 2 : F S 0 => µ = F S M S = F S θ 2 e.g. M = M Planck F S 1/ GeV => µ O(TeV) However (H 1,H 2 ) is a non chiral state: why is massless in a fundamental theory? I. Antoniadis (Supersymmetry) 15 / 22

16 Little hierarchy problem minimum of the potential: m 2 Z = 2m1 1 m2 2 tan2 β tan 2 β 1 2m RG evolution: m 2 2 = m2 2 (M GUT) 3λ2 t 4π 2m2 t ln M GUT m t + m2 2(M GUT) O(1)m + 2 t On the other hand: upper bound on the Higgs mass: [ ( mh 2 < mz 2 cos2 2β + 3 mt 4 (4π) 2 v 2 ln m2 t mt 2 + A2 t m 2 t 1 A2 t 12m 2 t m h 126 GeV => m t 3 TeV or A t 3m t 1.5 TeV => % to a few %0 fine-tuning is needed in m 2 Z )] < (130GeV) 2 I. Antoniadis (Supersymmetry) 16 / 22

17 Reduce the fine-tuning minimize radiative corrections M GUT Λ : low messenger scale (gauge mediation) δm 2 t = 8α s 3π M2 3 ln Λ M 3 + increase the tree-level upper bound => extend the MSSM extra fields beyond LHC reach effective field theory approach Low scale SUSY breaking => extend MSSM with the goldstino Non linear MSSM I. Antoniadis (Supersymmetry) 17 / 22

18 Split supersymmetry { scalars : heavy squarks and sleptons sparticles fermions : light (TeV) gauginos and higgsinos natural splitting: gauginos, higgsinos carry R-symmetry, scalars do not gauge coupling unification is preserved squarks + sleptons form complete SU(5) multiplets => same contribution to all 1-loop beta-functions relative velocities of energy evolution unchanged Dark Matter candidate is kept neutralino combination of bino-wino-higgsino mass hierarchy problem comes back (stop - top) contribution to the Higgs mass becomes huge I. Antoniadis (Supersymmetry) 18 / 22

19 Split supersymmetry: benefits number of low energy parameters is reduced significantly gaugino masses M 1,M 2,M 3 and Higgs scalar masses m 1,m 2,Bµ no soft sfermion masses, no A-terms global symmetries of the Standard Model appear again: B/L symmetry, no FCNC, etc distinct experimental signatures experimentally allowed Higgs mass => moderate split m 0 few - thousands TeV e.g. gauginos: a loop factor lighter than scalars ( m 3/2 ) I. Antoniadis (Supersymmetry) 19 / 22

20 Split supersymmetry: signatures squarks superheavy => long lived gluino τ gl ( s )( m 0 ) ( ) 4 5 1TeV 10 9 GeV M 3 => displaced vertices late decays captured near the detector, etc susy unification of 5 couplings at m 0 : L = 2g u H Wψu + 2g d H Wψ d g uh Bψu 1 2 g d H Bψ d ր ր λ 2 (H H) 2 higgsinos susy relations: g u = g 2 sinβ, g d = g 2 cosβ, g u = g Y sinβ g d = g 2cosβ, λ = 1 4 (g2 2 +g2 Y )cos2 2β => 5 relations in terms of one parameter I. Antoniadis (Supersymmetry) 20 / 22

21 I. Antoniadis (Supersymmetry) 21 / 22

22 SUSY : λ = 0 => sinβ = 1 H SM = sinβh u +cosβh d λ = 1 8 (g2 2 +g 2 )cos 2 2β λ = 0 at a scale GeV => m H = 126±3 GeV #$ ( # ) * + %#&'!",#&',#&',#&',#& e.g. for universal 2m = M = M SS, A = 3/2M I. Antoniadis (Supersymmetry) 22 / 22