What s SUSY? Supersymmetry s a boson-fermon symmetry that s amed to unfy all forces n Nature ncludng gravty wthn a snge framework Q boson >= fermon > Q fermon >= boson > j = = { Q, Q } = δ ( σ ) [,] bb 0, { f, f} 0 j β P β Modern vews on supersymmetry n partcle physcs are based on strng paradgm, though low energy manfestatons of SUSY can be found (?) at modern collders and n non-accelerator experments 0.07.007 Pre-conference school, SUSY'07
Motvaton of SUSY n Partcle Physcs Unfcaton wth wth Gravty Unfcaton of gauge couplngs Soluton of the herarchy problem Dark spn matter n the spn Unverse 3/ spn 1 spn 1/ spn 0 Superstrngs Unfcaton of matter (fermons) wth forces (bosons) naturally arses from an attempt to unfy gravty wth the other nteractons j β β ε ε j { Q, Q } = δ ( σ ) P { δ, δ } = ( εσ ε) P ε = ε( x) local coordnate transformaton. Local translaton = general relatvty! Supertranslaton x x + θσ ξ ξσ θ, θ θ + ξ, θ θ + ξ 0.07.007 Pre-conference school, SUSY'07 3
Motvaton of SUSY n Partcle Physcs Unfcaton of gauge couplngs Low Energy n SU (3) SU () U (1) G (or G + symm) c L Y gluons W, Z photon gauge bosons 3 1 Hgh Energy GUT quarks leptons fermons g g g g GUT Q = ( ) = (dstance) Λ 0.07.007 Pre-conference school, SUSY'07 4 Runnng of the strong couplng
Input 1 ( M ) = 18.978± 0.07 Z sn θ = 0.3146± 0.00017 MS ( M ) = 0.1184± 0.0031 s RG Equatons Z Motvaton of SUSY d dt b1 0 4/3 1/10 SM : b = b = /3 + NFam 4/3 + NHggs 1/6 b 11 4/3 0 M M SUSY GUT -1 GUT 3 Output 3.4± 0.9± 0.4 = 10 GeV = 15.8± 0.3± 0.1 10 GeV = 6.3± 1.9 ± 1.0 = b, = / 4 π = g /16 π, t=log(q / ) b1 0 3/10 MSSM : b = b = 6 + NFam + NHggs 1/ b 9 0 3 Unfcaton of the Couplng Constants n the SM and n the MSSM 0.07.007 Pre-conference school, SUSY'07 5 SUSY yelds unfcaton!
Motvaton of SUSY Soluton of the Herarchy Problem m H v 10 GeV 16 m Σ V 10 GeV Destructon of the herarchy by radatve correctons m H -14 10 1 m Σ Cancellaton of quadratc terms bosons m = fermons m 0.07.007 Pre-conference school, SUSY'07 6
Motvaton of SUSY Dark Matter n the Unverse The flat rotaton curves of spral galaxes provde the most drect evdence for the exstence of large amount of the dark matter. Spral galaxes consst of a central bulge and a very thn dsc, and surrounded by an approxmately sphercal halo of dark matter SUSY provdes a canddate for the Dark matter a stable neutral partcle 0.07.007 Pre-conference school, SUSY'07 7
Cosmologcal Constrants New precse cosmologcal data Ω h = Ω vacuum 1 73% Ω 3± 4% Ω DarkMatter Baryon 4% ρ = ρ crt Dark Matter n the Unverse: Supernova Ia exploson CMBR thermal fluctuatons Hot DM (not favoured by galaxy formaton) Cold DM (rotaton curves of Galaxes) SUSY 0.07.007 Pre-conference school, SUSY'07 8
Lorentz Algebra ( Super) Algebra [ P, P] = 0, [ P, M ] = ( g P g P), ν ρσ ρ σ σ ρ SUSY Algebra 1 β 1 ν ν β ν Superalgebra [ M, M ] = ( g M g M g M + g M ), ν ρσ νρ σ νσ ρ ρ νσ σ νρ [ Q, P] = [ Q, P] = 0, β [ Q, M ] = ( σ ) Q, [ Q, M ] = Q ( σ ), j β P β j j β β { Q, Q } = δ ( σ ) j { Q, Q } = δ ( σ ) P, ββ,,, = 1,; j, = 1,,..., N. β ν Superspace x x, θ, θ Grassmannan parameters, = 1, 0.07.007 Pre-conference school, SUSY'07 9 Q ϑ = 0, ϑ = 0 SUSY Generators = ϑ σ θ Q = + θσ ϑ Q = 0, Q = 0 Ths s the only possble graded Le algebra that mxes nteger and half-nteger spns and changes statstcs
Quantum States Quantum states: Vacuum = E, λ > Q E, λ >= 0 [ Q, P ] = [ Q, P ] = 0 Energy helcty State Expresson # of states vacuum 1 1-partcle -partcle N-partcle E, λ > Q E, λ >= E, λ+ 1/> QQ E, λ >= E, λ+ 1> j QQ...,, / 1 QN E λ >= E λ+ N > ( N ) 1 = N ( N ) N( N 1) = ( N ) N = 1 N ( N) N N 1 N 1 k = = bosons + fermons k = 0 0.07.007 Pre-conference school, SUSY'07 10 Total # of states:
Chral multplet Vector multplet SUSY Multplets N N = 1, λ=0 = 1, λ=1/ helcty # of states helcty # of states -1/ 0 1/ 1 1-1 -1/ 1/ 1 1 1 1 1 Members of a supermultplet are called superpartners scalar spnor ( ϕ, ψ ) ( λ, A ) spnor vector N=4 SUSY YM helcty -1 1/ 0 1/ 1 λ = -1 # of states 1 4 6 4 1 N=8 SUGRA helcty - 3/ 1 1/ 0 1/ 1 3/ N 4S λ = - # of states 1 8 8 56 70 56 8 8 1 spn N 4 N 8 For renormalzable theores (YM) For (super)gravty 0.07.007 Pre-conference school, SUSY'07 11
Smplest (N=1) SUSY Multplets Bosons and Fermons come n pars ( ϕ, ψ ) ( λ, A ) ( g, g) Spn 0 Spn 1/ Spn 1/ Spn 1 Spn 3/ Spn 0.07.007 Pre-conference school, SUSY'07 1
SUSY Transformaton N=1 SUSY Chral supermultplet: spn=0 spn=1/ δ A = εψ, ε δ ψ = σ ε A+ εf, ε δ F = εσ ψ ε parameter of SUSY transformaton (spnor) Auxlary feld (unphyscal d.o.f. needed to close SYSY algebra ) SUSY multplets Φ ( y, θ) = A( y) + θψ( y) + θθf( y) superfeld = Ax ( ) + θσ θ Ax ( ) + θθθθ Ax ( ) Superfled n Superspace + θψ ( x) / θθ ψ ( x) σ θ + θθ F( x) θ = θθ, θ = θ = 0! 1 4 ( y = x+ θσθ) Expanson over grassmannan parameter component felds 1 1 0.07.007 Pre-conference school, SUSY'07 13
Gauge superfeld Gauge superfelds + V( x, θθ, ) = C( x) + θχ( x) θχ( x) + θθm( x) θθm ( x) θσ θv ( x) + θθθ[ λ( x) + σ χ( x)] θθθ [ λ( x) + σ χ( x)] + θθθθ[ Dx ( ) + Cx ( )] 1 1 Gauge transformaton C C+ A+ A χ χ ψ M M F v v A A * ( ) λ λ D D * V Feld strength tensor V +Φ+Φ Wess-Zumno gauge C = χ = M = 0.07.007 Pre-conference school, SUSY'07 14 0 physcal felds W = 1 4 V V D e De Covarant dervatves ν W = + D Fν + D D D = ϑ + σ ϑ = θσ λ θ ( ) σ σ θ θ σ λ θ
How to wrte SUSY Lagrangans 1 st step Take your favorte Lagrangan wrtten n terms of felds nd step Replace Feld ( ϕ, ψ, A ) Superfeld ( Φ, V ) 3 rd step Acton = 4 d x L x ( ) Replace d x d 4 4 θ L( x, θθ, ) Grassmannan ntegraton n superspace dθ = 0, θ dθ = δ β β 0.07.007 Pre-conference school, SUSY'07 15
Superfelds SUSY Lagrangan (Matter) L = d θd θ ΦΦ+ d θ ( λφ+ m ΦΦ + y ΦΦΦ ) + hc..] + 1 1 j j 3 jk j k Components Knetc term Superpotental * * L= ψ σψ + A A + F F no dervatves + [ λf + m ( AF ψψ ) + y ( AA F ψψ A ) + hc..] 1 j j j jk j k j k Constrant δ L δ F k = F + λ + m A + y AA = * k k k jk j 0 F k * 1 1 * j j j j L= ψ σψ+ A A mψψ m ψψ y ψψ A y ψ ψ A V( A, A ) * * jk j k jk j k j V = F F * k k 0.07.007 Pre-conference school, SUSY'07 16
SUSY Lagrangans (gauge) Gauge felds 1 1 1 ν 4 4 ν L d θ W W d θ W W D F F λσ D λ = + = Gauge transformaton gλ + + gλ + Φ e Φ, Φ Φ e, V V + ( Λ Λ ) + Gauge nvarant nteracton wth matter (covarant dervatve) + + gv Φ Φ Φ e Φ 0.07.007 Pre-conference school, SUSY'07 17
Gauge Invarant SUSY Lagrangan L = d θ Tr(W W ) + d θ Tr(W W ) SUSY YM 1 1 4 4 + d θd θ Φ ( e ) Φ + d θ W( Φ ) + d θ W( Φ) SUSY YM gv a b a b 1 a aν a a 1 a a 4 ν λσ λ L = F F D + D D a a a a a a ( A gv T A) ( A gv T A) ψ σ ( ψ gv T ψ) + a a a a a a λψ ψ λ DgAT A gat + g T A+ FF W W W W + + A A A A A A 1 1 F F ψψ j j j a a 1 a a A 0.07.007 Pre-conference school, SUSY'07 18 ψ ψ W D = ga T A, F = V= D D + F F j
Spontaneous Breakng of SUSY Energy E =< 0 H 0> j β P β j { Q, Q } = δ ( σ ) 1 j 1 4 0 {, } 0 4 0 0 = 1, = < >= > E Q Q Q E =< 0 H 0> 0 f and only f Q 0> 0 0.07.007 Pre-conference school, SUSY'07 19