Supersymmetry II. Hitoshi Murayama (Berkeley) PiTP 05, IAS

Transcription

1 Supersymmetry II Hitoshi Murayama (Berkeley) PiTP 05, IAS

2 Plan Mon: Non-technical Overview what SUSY is supposed to give us Tue: From formalism to the MSSM Global SUSY formalism, Feynman rules, soft SUSY breaking, MSSM Wed: SUSY breaking how to break SUSY, mediation mechanisms Thu: SUSY at colliders basic reactions, signatures, and how do we know it is SUSY? Fri: SUSY as a telescope supersymmetry breaking, GUT, string

3 Global SUSY

4 SUSY algebra In addition to the momentum P μ and Lorentz generators M μν, add spinorial charges Q αi, i=1,.., N [P µ,q i α] = 0 [M µν,q i α] = 1 2 (σµν ) β α Q i β {Q i α, Q j β} = 2P µ (σ µ ) i j α βδ {Q i α,q j β } = ε αβc i j C ij =-C ji central charges, possible only if N>1

5 Massless particle Specialize to P μ =E(1,0,0,1) helicity: h = ε 0ijk P i M jk /2P 0 = M 12 P µ = E(1,0,0,1) {Q i α, Q j β} ( = 2P µ (σ µ ) α βδ i j 2 = 2E 0) creation/annihilation operators (C ij =0) b i = Q i 1/ 4E, b i = Q i 1/ 4E {b i,b j } = δ i j {h,b i } = 1 2 bi {h,b i } = 1 2 bi δ i j

6 N=1 multiplets Restrict h 1 for renormalizability Multiplet structure: { 0>, b 0>} chiral multiplet: pick h 0>=0 or -1/2 h={0, 1/2} (anti-chiral) or {-1/2, 0} (chiral) i.e., a Weyl fermion and a complex scalar vector multiplet: pick h 0>=1/2 0> or -1 h={1/2, 1} or {-1, -1/2} i.e, a gauge field and a Weyl (Majorana) fermion must be in adjoint (real) rep non-chiral

7 N=2 multiplets Multiplet: { 0>, b 1 0>, b 2 0>, b 1 b 2 0>} hypermultiplet: pick h 0>=-1/2 h={-1/2, 0, 0, 1/2} under N=1, sum of chiral and anti-chiral a Dirac fermion and two complex scalars vector multiplet: pick h 0>=0 0> h={0, 1/2, 1/2, 1} under N=1, sum of chiral and vector a gauge field and two Weyl (Majorana) fermions, and a real scalar Neither multiplet chiral: can t be used to supersymmetrize the Standard Model

8 N=1 superspace chiral superfield introduce Grassman-odd coordinates θ α superspace: Covariant derivative Can place a constraint Note solution: (x µ,θ α, θ α ) P µ = i µ M µν = i(x µ ν x ν µ ) Q α = θ α + i θ βσ µ βα µ D α = θ α i θ βσ µ βα µ {D α,q β } = {D α, Q β} = 0 {D α, D β} = 2i µ (σ µ ) α β D αφ(x,θ, θ) y µ = x µ iθσ µ θ, = 0 D α y µ = 0 φ(x,θ, θ) = φ(y,θ) = A(y) + 2θψ(y) + θ 2 F(y)

9 vector superfield introduce Grassman-odd coordinates θ α real (vector) superfield: (x,θ, θ) = C + θχ + θ χ + θ 2 M + θ 2 M + θσ µ θa µ + θ 2 θ λ + θ 2 θλ + θ 2 θ 2 D gauge transformation: Λ is a chiral superfield Can eliminate C, χ, M: Wess-umino gauge remaining dof: Aμ, λ, D V V + iλ i Λ Field strength chiral superfield: W α = D 2 D α V = λ α (y) + θ β σ µν βα F µν + θ α D

10 Non-abelian gauge symmetry Generalization to non-abelian case V = V a T a, Λ = Λ a T a e V e i Λ e V e iλ φ e iλ φ φ e V φ : invariant Field strength chiral superfield W α = D 2 e V D α e V W α e iλ W α e iλ

11 Kähler and superpotentials Two ways to construct invariants (up to total derivatives) full superspace integral of general superfield Kähler potential d 4 θφ φ = µ A 2 + ψiσ µ µ ψ + F F chiral superspace integral of chiral superfield superpotential d 2 θw(φ) = W φ i Fi W 2 φ i φ jψi ψ j Renormalizable theory is fixed by W

12 gauge theory (Wess-umino gauge) matter kinetic term d 4 θφ e V φ = D µ A 2 + ψi Dψ + F F + D a A T a A + ( 2A λ a T a ψ + h.c.) gauge kinetic term d 2 θ 1 g 2W a αw aα = 1 superpotential 4g 2Fa Solve for auxiliary fields µνf µνa + 1 g 2 λ a i Dλ a + 1 2g 2Da D a d 2 θw(φ) = W φ i Fi W 2 φ i φ jψi ψ j D a = g 2 A T a A, F i = W φ i V = 1 2g 2Da D a + F i 2 = g2 2 (A T a A) 2 + W φ i 2

13 Feynman rules Single gauge coupling constant gives all of these Feynman vertices f f ~ f A g g T a D a T a g 2 f ~ ~ f f ~ f ~ f

14 Feynman rules Single Yukawa coupling constant gives all of these Feynman vertices A i y j j y A j j y j k k A k A i A i A j A j A k A k y F k y y F i y y F j y A j A j A k A k A i A i

15 Fayet-Illiopoulos D-term Only for U(1) gauge factors, there is another possible term d 4 θv d 4 θ(v + iλ i Λ) = d 4 θv + surface terms constant term: ξd changes the D-term potential: g 2 2 (A i Q i A i ξg 2 ) 2 not consistent with supergravity by itself unless U(1) R is gauged or Green-Schwarz mechanism is employed I will not discuss it any further

16 Renormalization Start with the Wilsonian action at scale μ non-renormalization theorem, holomorphy, and transivity says at scale μ = μe -t, d 4 θ i d 4 θ i b 0 = 3C A -Σ i T F i to identify coupling constants, need to rescale fields to canonical normalization However, rescaling fields yield anomalous Jacobians φ i e V φ i + i φ i e V φ i + ( ) 1 d 2 θ g 2 W α W α + λ i jk 0 φ iφ j φ k 0 (( 1 d 2 θ g 2 b 0 0 8π 2t ) W α W α + λ i jk 0 φ iφ j φ k )

17 Renormalization Konishi anomaly Dφ i = D(e σ φ i )e R d 2 θt i F 1 8π 22σW αw α rescaling anomaly DV = D(e σ V )e +R d 2 θc A 1 8π 22σW αw α d 4 θ i φ i e V φ i + first rescale matter fields d 2 θ (( 1 g 2 0 b 0 8π 2t T i 1 F i 8π 2ln i then rescale the gauge field V g c V 1 g 2 = 1 c g 2 0 ) W α W α + 1/2 i 1/2 b 0 8π 2t T i 1 F i 8π 2 ln 1 i C A 8π 2 lng2 c d 2 θ 1 g 2 c( D 2 e g cv D α e g cv ) 2 = 1 4 F µνf µν + j 1/2 k λ i jk 0 φ iφ j φ k )

18 The Minimal Supersymmetric Standard Model (MSSM)

19 supersymmetrize it All quarks and leptons are Weyl fermions chiral superfields have left-handed Weyl fermions Use charge conjugation to make them all left-handed: Q, L, u c, d c, e c Promote them to chiral superfields, namely add their scalar partners Naming convention: add s as a prefix, which stands for supersymmetry or scalar terrible convention! e.g.: selectron, smuon, stop, sup, sstrange

20 supersymmetrize it All gauge fields are promoted to vector multiplets namely add massless Weyl=Majorana fermions gauginos Naming convention: add ino as a suffix, which doesn t mean small in any sense terrible convention! e.g.: gluino, wino, photino, zino, bino

21 supersymmetrize it Minimal Standard Model has only one Higgs doublet It gives mass to both up- and down-type fields L Yukawa = Y i j u Promote it to a chiral superfield, namely add a Higgsino But having only one higgsino makes the SU(2) xu(1) anomalous Also complex conjugation not allowed in a superpotential Q i u j H +Y i j d Q i d j H +Y i j l solution: introduce two Higgs doublets H u (1,2,+1/2), H d (1,2,-1/2) L i e j H, H = iσ 2 H

22 The MSSM SU(3) C xsu(2) L xu(1) Y gauge theory Q d c u c L e c Hu Hd g B W SU(3) C 3 3 * 3 * SU(2) L U(1) Y +1/6 +1/3-2/3-1/2-1 +1/2-1/ mltplt χ χ χ χ χ χ χ V V V flavor

23 The superpotential The terms we want W MSSM = Yu i j Q i u c jh u +Y i j d Q id c jh d +Y i j L l i e c jh d + µh u H d The terms we don t want (violates B or L) = λ W Rp i jk u c i d c jdk c + λ i jkq i d c jl k + λ i jkl i L j e c k + µ i L i H u Impose 2 symmetry ( matter parity ) that all matter chiral superfields are odd, Higgs even combined with 2π rotation of space (-1) 2S (is equivalent to θ -θ), it gives R-parity

24 The Higgs potential Without supersymmetry breaking effects, the superpotential W MSSM = Yu i j Q i u c jh u +Y i j d Q id c jh d +Y i j L l i e c jh d + µh u H d gives the potential for the Higgs field V = µ 2 (H uh u + H d H d) + g2 8 (H u τh u + H d τh d ) 2 + g 2 8 (H uh u H d H d) 2 which has only one ground state <H u >=<H d >=0 Namely the electroweak SU(2)xU(1) is unbroken unless supersymmetry is broken

25 Breaking Supersymmetry

26 Auxiliary fields SUSY is broken if the auxiliary component of a superfield has an expectation value F-term breaking: <z>=θ 2 f, f 0 D-term breaking: <W α >=θ α d, d 0 Irrespective of dynamics that breaks supersymmetry, its effect can be parameterized in terms of these order parameters ( spurion ) assume f and d dimension 1 for this purpose A spurion does not change the UV behavior of the theory, i.e. reintroduce quadratic divergences

27 Soft SUSY breaking Take W = λφ 3 + µφ 2 + m 2 φ Using the spurion <z>=θ 2 f, we can write the most general SUSY breaking terms in Kähler d 4 θ(α z φ φ + h.c. + β z z φ φ) = (α f φ F + α f φf ) + β f f φ φ Solving for the auxiliary component F F + (W F + α f φ F + h.c.) = W + α f φ 2 = W 2 (α f φw + h.c.) α f 2 φ φ = W 2 α f (3λφ 2 + 2µφ 2 + m 2 φ + h.c.) α f 2 φ φ From the superpotential d 2 θ z (aλφ 3 + bµφ 2 + cm 2 φ) = a f λφ 3 + b f µφ 2 + c f m 2 φ bottomline: any terms of type (Aλφ 3 + Bµφ 2 +Cm 2 φ + h.c.) m 2 φφ φ

28 Soft SUSY breaking Similarly for the gauge multiplet, namely gaugino masses If there is a chiral superfield in the adjoint rept, another possible term with D-term spurion is d 2 θ z W α W α = f λλ d 2 θφw α W α = dψλ Now the complete set of soft SUSY breaking Aλφ 3, Bµφ 2, Cm 2 φ, m 2 φφ φ, m λ λλ, m d λψ

29 Sofly broken MSSM

30 Soft SUSY breaking terms in the MSSM m 2 Qi j Q i For each term in the superpotential W MSSM = Yu i j Q i u c jh u +Y i j d Q id c jh d +Y i j L l i e c jh d + µh u H d we can have the A-terms and B-term A i j u Y i j u Q i u c jh u + A i j d Y i j d Q id c jh d + A i j l Y i j l L i e c jh d + BµH u H d scalar masses for all scalars Q j + m 2 ui jũ i ũ j + m 2 di j d i d j + m 2 L Li j i L j + m 2 ei jẽ i ẽ j + m 2 H d H d 2 + m 2 H u H u 2 gaugino mass for all three gauge factors M 1 B B + M 2 W a W a + M 3 g a g a A(18x3)+B(2)+m(9x5+2)+M(2x3)+μ(2)=111 U(1) R xu(1) PQ removes only two phases cf. SM has two params in the Higgs sector 107 more parameters than the SM!

31 Higgs potential The Higgs potential in the MSSM is V = m 2 1H d H d + m 2 2H uh u m 2 3(H u H d + c.c.) + g2 8 (H d τh d + H u τh u ) 2 + g 2 8 (H d H d H uh u ) 2 where m 2 1 = m 2 H d + µ 2, m 2 2 = m 2 H u + µ 2, m 2 3 = Bµ Leaving on the neutral components V = (H 0 d H 0 u ) ( m 2 1 m 2 3 m 2 3 m2 2 )( H 0 d H 0 u ) + g2 + g 2 ( H 0 8 d 2 Hu 0 2 ) 2 breaks SU(2)xU(1) if m 12 m 2 2 < m 3 4 stable along the D-flat direction H d0 =H u 0 if m 12 +m 2 2 > 2m 3 2 i.e., m 12 m 2 2 < m 3 4 < (m 12 +m 22 )/2 which is possible

32 Higgs particles V = (H 0 d H 0 u ) ( m 2 1 m 2 3 m 2 3 m2 2 )( H 0 d H 0 u Two Higgs doublets = 8 Klein-Gordon fields 3 of them eaten by the Higgs mechanism 8-3=5 physical degrees of freedom h 0, H 0, A 0, H +, H - 3 parameters m 12, m 22, m 3 2 one of them fixed by <H d > 2 +<H u > 2 =(174GeV) 2 two remaining parameters ) + g2 + g 2 ( H 0 8 d 2 Hu 0 2 ) 2 tanβ=<h u >/<H d > and m A

33 V = (H 0 d Higgs particles H 0 u ) ( m 2 1 m 2 3 m 2 3 m2 2 )( H 0 d H 0 u Solve for mass eigenvalues of physical states Lightest Higgs below m : excluded! However, the quartic coupling evolves differently from the gauge coupling below the scalar top threshold ) + g2 + g 2 ( H 0 8 d 2 Hu 0 2 ) 2 m 2 A = 2m 2 3/sin2β, m 2 0 H = m 2 ± A + mw 2 m 2 h 0,H = 1 ( ) m 2 0 A + m ± (m 2 A + m 2 0 )2 4m 2 A m 2 0 cos2 2β Need stop > 500GeV to evade limits, m h m! m 2 h 0 3 4π 2 m 4 t v 2 ln m t 1 m t 2 m 2 t

34 gauginos, higgsinos charged ones charginos ( )( ) ( W H d ) M 2 2mW sinβ W + 2mW cosβ µ H + u neutral ones neutralinos M 1 0 m s W c β m s W s β ( B, W 0, H d, 0 H u) 0 0 M 2 m c W c β m c W s β m s W c β m c W c β 0 µ m s W s β m c W s β µ 0 B W 0 H d 0 H u 0

35 Sfermions e.g., stop ( m ( t L t R) 2 Q 3 + mt 2 (A t µ cotβ)m t (At µcotβ)m t m 2 t + m2 t )( t L t R )

36 one-loop RGE GUT prediction of gaugino masses d M i dt g 2 = 0 i M 1 : M 2 : M 3 1 : 2 : 7 at m gauge interaction boosts scalar masses d dt m2 = 1 16π 28C Fg 2 M 2 Yukawa interaction suppresses scalar masses 16π 2 d dt m2 H u =3X t 6g 2 2M g2 1M π 2 d dt m2 t R =2X t 32 3 g2 3M g2 1M π 2 d dt m2 t L =X t 32 3 g2 3M 2 3 6g 2 2M g2 1M 2 1 X t =2Y 2 t (m 2 H u + m 2 t R + m 2 t L + A 2 t ) H u mass-squared most likely to get negative!

37 RGE running of soft SUSY breaking parameters can be inferred from the RGE of coupling constants i i = i (1 θ 2 A i )(1 θ 2 A i )(1 θ 2 θ 2 m 2 i ) 1 g 2 S = 1 ( 1 θ 2 ) g 2 m λ i (t)= i (0) g2 C i F 4π 2 t + j,k λ i jk λ i jk 16π 2 t i (t)= i (0) Ci F 4π 2 2 S + S t + j,k λ 0 i jk λ 0 i jk 16π 2 1 i 1 j 1 k t