Introduction to MSSM
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1 Introduction to MSSM Shrihari Gopalakrishna Institute of Mathematical Sciences (IMSc), Chennai SUSY & DM 013 IISc, Bangalore October 013
2 Talk Outline Supersymmetry (SUSY) Basics Superfield formalism Constructing SUSY invariant theory SUSY breaking Minimal Supersymmetric Standard Model (MSSM) SUSY preserving Lagrangian and soft-breaking terms R-parity Superpartner Mixing Implications Dark Matter 15 GeV Higgs
3 SUSY invariant Theory Supersymmetry (SUSY) Reviews: [Wess & Bagger] Symmetry: Fermions Bosons Q Φ = Ψ ; Q Ψ = Φ Q α is a spinorial charge SUSY algebra: { Qα, Q β } {Qα, Q β = } = σ µ Pµ { α β Q α, Q β } = 0 [ P µ, Qα ] = [ P µ, Q α ] = 0
4 SUSY invariant Theory Superfield SUSY algebra realized through a Superfield Φ ( x µ, θ, θ ) θ are fermionic (Grassmann) coordinates that anticommute: θ α, α = {1, } {θ, θ} = { θ, θ } = { θ, θ } = 0 = (θ 1 ) = (θ ) = 0 Define θθ θ α θα ; θ θ θ α θ α ; θ α ɛ αβ θ β ; θ α ɛ α β θ β ; ɛ 1 = ɛ1 = +1 Chiral Superfield DΦ L = 0 ; DΦ R = 0 Φ L = φ(y) + θψ L (y) + θθf (y) F : auxiliary field δ ξ φ = ξψ δ ξ ψ =i σ m ξ mφ + ξf δ ξ F =i ξ σ m mψ Vector Superfield V = V V (x, θ, θ) = θσ µ θa µ (x) + iθθ θ λ(x) θ θθλ(x) + 1 θθ θ θd(x) D : auxiliary field δ ξ A mn =i [( ξσ n m λ + ξ σ n mλ ) (n m) ] δ ξ λ =iξd + σ mn ξa mn δ ξ D = ξ σ m mλ ξσ m m λ
5 SUSY invariant Theory Constructing a SUSY theory Under a SUSY transformation, F and D transform into total derivatives So they can be used for constructing SUSY invariant L
6 SUSY invariant Theory Constructing a SUSY theory Under a SUSY transformation, F and D transform into total derivatives So they can be used for constructing SUSY invariant L A SUSY gauge invariant theory L = Φ θθ i egv Φ i + ( W(Φ i ) θθ + h.c. ) + 1 θ θ 3g W αw α θθ Superpotential W(Φ), a Holomorphic function Gauge Kinetic Function W α = 1 4 D De V D αe V
7 SUSY invariant Theory Constructing a SUSY theory Under a SUSY transformation, F and D transform into total derivatives So they can be used for constructing SUSY invariant L A SUSY gauge invariant theory L = Φ θθ i egv Φ i + ( W(Φ i ) θθ + h.c. ) + 1 θ θ 3g W αw α θθ Superpotential W(Φ), a Holomorphic function Gauge Kinetic Function W α = 1 4 D De V D αe V Eliminating the Auxiliary fields L = D µφ i i ψ i σ µd µ ψ i g ) (φ i T a ψ i λ a + λ a ψ T a φ i ( 1 ) W(φ i ) ψ j ψ k + h.c. W(φ i ) φ j φ k φ j 1 4 F µνf a a µν 1 gφ i T ij a φ j iλ a σ µd µ λ a a
8 SUSY invariant Theory Consequences Solution to gauge hierarchy problem h i y t t L, t R h + h i y t t L, t R Λ divergence cancelled h = 0 [Romesh Kaul s talk] (Similarly W ±, Z divergences cancelled by λ)
9 SUSY invariant Theory Consequences Solution to gauge hierarchy problem h i y t t L, t R h + h i y t t L, t R Λ divergence cancelled h = 0 [Romesh Kaul s talk] (Similarly W ±, Z divergences cancelled by λ) Lightest SUSY Particle (LSP) stable dark matter (if R p conserved) Gauge Coupling Unification - SUSY SO(10) GUT Includes ν R Neutrino mass via seesaw
10 SUSY breaking SUSY breaking Exact SUSY = M ψ = M φ ; M A = M λ So experiment = SUSY must be broken SUSY broken if and only if 0 H 0 > 0 Spontaneous SUSY breaking O Raifeartaigh F -term breaking Fayet-Iliopoulos D-term breaking STr(M ) = 0 = cannot break SUSY spontaneously using SM superfield Hidden sector breaking Communicated to SM Mediation Spectrum depends on Mediation type + RGE In effective low-energy theory Explicit soft-breaking terms, i.e., with dimensionful parameters
11 MSSM The Minimal Supersymmetric Standard Model (MSSM) Reviews: [Martin] [Drees] [Drees,Godbole,Roy] [Baer,Tata]
12 Ingredients MSSM fields To every SM particle, add a superpartner (with spin differing by 1/) Matter fields (Chiral Superfields) Components (SU(3), SU()) U(1) Q (3, ) 1/6 ( q L, q L, F Q ) ; ( ( ) ũl ul q L = ); q d L = L dl U c ( 3, 1) /3 (ũ R, uc R, F U ) D c ( 3, 1) 1/3 ( d R, d c R, F D) L (1, ) 1/ ( l L, l L, F L ) ; ll = ( νl ẽ L ) ; l L = ( νl e L ) E c (1, 1) 1 (ẽ R, ec R, F E ) (N c ) (1, 1) 0 ( ν R, νc R, F N ) Gauge fields (Vector Superfields) Components SU(3) (g µ, g, D 3) SU() (W µ, W, D ) U(1) (B µ, B, D 1) Higgs fields (Chiral Superfields) (SU(3), SU()) U(1) Hu (1, ) 1/ (h u, h u, F Hu ) ; hu = Components H d (1, ) 1/ (h d, h d, F Hd ) ; h d = ( h 0 d h d ( h + u h 0 u ) ) ; hu = ( h + u h 0 u ; h d = ( h 0 d h d ) )
13 Ingredients MSSM Superpotential Write most general W consistent with SU(3) c SU() L U(1) Y W = U c y uqh u D c y d QH d E c y elh d + µh uh d + (N c y nlh u) W L = LH u + LE c L + QD c L ; W B = U c D c D c W L + W B induce proton decay : τ p s for m 1 TeV
14 Ingredients MSSM Superpotential Write most general W consistent with SU(3) c SU() L U(1) Y W = U c y uqh u D c y d QH d E c y elh d + µh uh d + (N c y nlh u) W L = LH u + LE c L + QD c L ; W B = U c D c D c W L + W B induce proton decay : τ p s for m 1 TeV So impose Matter Parity R M = ( 1) 3(B L) to forbid L and B terms For components = R-parity R p = ( 1) 3(B L)+s R p(particle) = +1, R p(sparticle) = 1 Consequence : The Lightest SUSY Particle (LSP) is stable Cosmologically stable Dark Matter Missing Energy at Colliders
15 Ingredients Soft SUSY breaking Effective parametrization with explicit soft-susy-breaking terms L soft SUSY Br Q m Q Q ũ R m uũ R d R m d d R L m L L ẽ R m eẽr ( ν R m ν ν R) 1 M 1 B B 1 M W W 1 M 3 g g + h.c. ũ c A u QH u + d c A d QH d + e c A e LH d (ν c A ν LH u) + h.c. m H u H u Hu m H d H d H d (BµH uh d + h.c.)
16 Ingredients Soft SUSY breaking Effective parametrization with explicit soft-susy-breaking terms L soft SUSY Br Q m Q Q ũ R m uũ R d R m d d R L m L L ẽ R m eẽr ( ν R m ν ν R) 1 M 1 B B 1 M W W 1 M 3 g g + h.c. ũ c A u QH u + d c A d QH d + e c A e LH d (ν c A ν LH u) + h.c. m H u H u Hu m H d H d H d (BµH uh d + h.c.) UV SUSY breaking and mediation dynamics will set these parameters Eg: Gravity Mediation (MSUGRA, CMSSM) Inputs m 0,M 1/, A 0, tan β, sign(µ) at GUT scale TeV scale values determined by RGE
17 Ingredients Electroweak symmetry breaking (EWSB) H u = 1 ( 0 v u ) ; H d = 1 ( vd 0 ) ; v = vu + v d vu ; tan β ; v d Physical Higgses: h 0, H 0, A 0, H ±
18 Ingredients Electroweak symmetry breaking (EWSB) H u = 1 ( 0 v u ) ; H d = 1 ( vd 0 ) ; v = vu + v d vu ; tan β ; v d Physical Higgses: h 0, H 0, A 0, H ± V = ( µ + m Hu ) hu + ( µ + m H d ) h d (b µh uh d + h.c.) (g + g )( h u h d ) Minimization and EWSB bµ sin(β) = m Hu +m + µ H and m Z = m H m d H d 1 sin (β) m Hu m H µ d d
19 Ingredients Electroweak symmetry breaking (EWSB) H u = 1 ( 0 v u ) ; H d = 1 ( vd 0 ) ; v = vu + v d vu ; tan β ; v d Physical Higgses: h 0, H 0, A 0, H ± V = ( µ + m Hu ) hu + ( µ + m H d ) h d (b µh uh d + h.c.) (g + g )( h u h d ) Minimization and EWSB bµ sin(β) = m Hu +m + µ H and m Z = m H m d H d 1 sin (β) m Hu m H µ d d So we need µ (SUSY preserving param) m (SUSY br param)! Why? This is called the µ-problem
20 Consequences 15 GeV Higgs At 1-loop m h m Z cos (β) + 3g m4 t 8π m W [ ln ( m t1 m t m t [Haber, Hempfling, Hoang,1997] [Debtosh s thesis] ) ( )] + X t 1 X t m t1 m t 1m t1 m t where X t = A t µ cot β m h = 15 GeV needs sizable loop contribution Hard! Needs large m t1 m t or large X t [ But δmh u 3g ( ) ( )] m4 t m t1 m t 8π m W ln mt + X t 1 X t m t1 m t 6m t1 m t So fine-tuning necessary to keep m Z correct (cf previous EWSB relation) Little hierarchy problem
21 Consequences Neutralino mixing Neutralino: Neutral EW gauginos ( B, W 3, H u, 0 H d 0 ) : Majorana states L 1 ( B W 3 H 0 u Diagonalizing this H 0 d B W 3 H 0 u H 0 d ) M 1 0 c β s W m Z s β s W m Z 0 M c β c W m Z s β c W m Z c β s W m Z c β c W m Z 0 µ s β s W m Z s β c W m Z µ 0 χ 0 1 χ 0 χ 0 3 χ 0 4 : the mass eigenstates B W 3 H 0 u H 0 d
22 Consequences Chargino mixing Chargino: Charged EW gauginos (W ±, H ± ), W ± = W 1 ± i W ( W Form Dirac states W + = α + ) ( H W α ; H + u = + ) α H α d ( ) ) L W (M ( + H + χp L + M χ P W + ) R H + + h.c. ( M where Mχ = sin β mw cos β mw µ ( W Diagonalizing this + ) ( ) χ + H + 1 χ + : the mass eigenstates )
23 Consequences Scalar mixing Eg. stop sector ( t L, t R ) L ( t L t ) ( m LL + m t + L (v ua t µ cot β m t) R (v ua t µ cot β m t) m RR + m t + R ) ( tl t R ) Diagonalizing this ( ) ( ) tl t1 : the mass eigenstates t R t where = (T 3 Qs W ) cos(β) m Z
24 Consequences Flavor Issues Flavor Problem In general m ij can have arbitrary flavor structure and phases (MSSM has 9 new phases + 1 CKM phase) FCNC & EDM experiments severely constrain these some deeper reason (in SUSY br mediation)? Minimal Flavor Violation (MFV) Only CKM phase
25 Dark Matter Dark Matter (DM) With R p conserved, a superpartner (odd) cannot decay to SM (even) Lightest Supersymmetric Particle (LSP) is stable LSP is in thermal equilibrium in the early universe, and left over today as Dark Matter χ 0 1 LSP with 100 GeV mass is a good DM candidate Weakly Interacting Massive Particle (WIMP)
26 Dark Matter Dark Matter (DM) With R p conserved, a superpartner (odd) cannot decay to SM (even) Lightest Supersymmetric Particle (LSP) is stable LSP is in thermal equilibrium in the early universe, and left over today as Dark Matter χ 0 1 LSP with 100 GeV mass is a good DM candidate Weakly Interacting Massive Particle (WIMP) It s number density can be depleted by annihilating with another sparticle Self-annihilation Co-annihilation χ χ χ SM χ SM Could be detected in ongoing experiments Direct detection (on earth) ψn ψn Indirect detection (cosmic rays) ψψ γγ, e + e,... Aside: Almost pure ν R, if LSP, could be non-thermal DM
27 Collider Signatures SUSY at LHC Cascade decays R p conservation = Missing energy signals q l ± g g g g g q q q q χ 0 q χ 0 q q W W W ± χ 0 Z ν χ 0 1 χ 0 1 l l + Can we determine the spin and couplings to show SUSY? Angular distributions [ATLAS Physics TDR]
28 Conclusions Conclusions Supersymmetry solves the Heirarchy Problem MSSM is an effective theory at the TeV scale SUSY breaking in hidden sector Communicated to SM via mediation mechanism (which determine the MSSM parameters) R p conservation Gravity mediation, Gauge mediation, Anomaly mediation Dark Matter Candidate Missing Energy at Colliders 14 TeV LHC run crucial for SUSY
29 BACKUP SLIDES BACKUP SLIDES
30 Standard Model (SM) problems Gauge hierarchy problem (M EW M Pl ) Electroweak scale : M EW = 10 3 GeV Gravity scale : M Pl = GeV Higgs sector unstable (quadratic divergence) Fermion mass hierarchy problem (m e m t) Flavor symmetry? What is the dark matter Inadequate source of CP violation for observed baryon asymmetry Cosmological constant problem
31 Gauge hierarchy problem in detail ( ) L SM 1 m h h + yt ht R t L + h.c Higgs mass is not protected by any symmetry! h i y t t L, t R h δm h = 3y t 8π Λ (Λ is momentum cut-off) Quadratic divergence! = unnatural (fine-tuning) m h = 15 GeV ; Λ M pl = GeV
32 Gauge hierarchy problem in detail ( ) L SM 1 m h h + yt ht R t L + h.c Higgs mass is not protected by any symmetry! h i y t t L, t R h δm h = 3y t 8π Λ (Λ is momentum cut-off) Quadratic divergence! = unnatural (fine-tuning) m h = 15 GeV ; Λ M pl = GeV New physics (BSM) restores naturalness? Below what scale (Λ) should it appear? Fine-tuning measure: f T m h δm h f T > 0.1 = Λ < TeV (for m h = 10 GeV ) So expect new physics below TeV scale
33 SM problems cured by Physics Beyond SM (BSM)? Some BSM proposals Supersymmetry Strong dynamics (Technicolor, Composite Higgs) Extra dimensions Flat extra dims Warped (AdS space) extra dim Little Higgs 5-D gravity theory in AdS DUAL 4-D conformal field theory (CFT) AdS/CFT correspondence [Maldacena 97]
34 ± 1 ± ± LHC Data (SUSY jets + MET) Entries / 50 GeV L dt ~ 35 pb Data 010 ( s = 7 TeV) SM Total QCD multijet W+jets Z+jets tt and single top SM + SUSY ref. point ATLAS Events / CMS 3 Jets -1 L dt = 35 pb, s = 7 TeV Data Standard Model QCD Multijet tt, W, Z + Jets LM0 LM DATA/SM miss E T [GeV] α T [GeV] m 1/ ATLAS q~ (400) int L -1 = 35 pb, 0 lepton combined exclusion Reference point q~ (600) q~ (800) s=7 TeV Observed 95% C.L. limit Median expected limit Expected limit ±1 σ -1 CMS α T, 35 pb ~ LEP l LEP χ D0 χ ~ g (800) ~ g (600) ~ g (400) MSUGRA/CMSSM: tanβ = 3, A = 0, µ>0. m 0 [GeV] 0, ~ χ -1 D0 ~ g, ~ q, µ<0,.1 fb CDF ~ g, ~ q, tanβ=5, µ<0, fb (GeV) m 1/ # $ = LSP LM1-1 L int = 35 pb, tan! = 3, A 0 NLO Expected Limit NLO Observed Limit LO Observed Limit q ~ (800)GeV = 0, sign(µ) > 0 q ~ (650)GeV q ~ (500)GeV LM0 s = 7 TeV -1 CDF g~, ~ q, tan!=5, fb -1 D0 g~, ~ q, tan!=3,.1 fb ± LEP " # 1 ~ ± LEP l D0 " ±, " 0 1 g ~ (800)GeV g ~ (650)GeV g ~ (500)GeV m 0 (GeV)
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