Supersymmetry Highlights. Kevin Hambleton

Transcription

1 Supersymmetry Highlights Kevin Hambleton

2 Outline SHO Example Why SUSY? SUSY Fields Superspace Lagrangians SUSY QED MSSM

3 i Warm Up A Hint Of SUSY... Remember Quantum Simple Harmonic Oscillator... Canonical Coordinates w/commutation Relation [x, p] =i Input the standard Hamiltonian H = 1 2 p x2

4 Creation/Annihilation Ops H = 1 2 p x2 = 1 2 (p2 + x 2 ) = 1 2 (x ip)(x + ip) i [x, p] 2 = a a + 1 2

5 Fermions? Anticommutator instead of commutator {x, p} = i No terms like x^2 or p^2 allowed in H Try... H = ixp

6 Fermion Creation \Annihilation Ops H = ixp = i 2 (xp px)+ i {x, p} 2 = 1 2 (x ip)(x + ip)+ i {x, p} 2 = 1 2 b b 1 2

7 SUSY Transformations Boson (x,p) Fermion (ψ,π) SHOs H = 1 2 x p2 + i = a a + b b Construct operators which commute with H Q = iab, Q = iba Bosonic Modes -> Fermionic Modes, And Visa Versa [1] Generators of SUSY Transformations

8 SUSY Transformations Q = iab, Q = iba N F = {0, 1} Q N B,N F i = p (N B )(N F + 1) N B 1,N F +1i Q N B,N F i = p (N B + 1)(N F ) N B +1,N F 1i (QQ + Q Q) N B,N F i =(N B + N F ) N B,N F i Use NF={0,1}

9 More On Q From Previous Slide H = {Q, Q } The Vacuum Satisfies Q 0i = Q 0i =0 Therefore The Vacuum Energy Is h0 H 0i = h0 {Q, Q } 0i =0

10 Principle Of SUSY Equal Number Bosonic And Fermionic Degrees Of Freedom (DoFs) Find Conserved Charges Which Generate SUSY Transformations Expect Nice Cancellations e.g Vacuum Energy In SHO Example

11 Why SUSY?

12 Why? Fermion Loops Come With Extra Minus Signs This Can Cause Important Cancellations Similar To Ghosts In Non-Abelian Gauge Theory

13 Quintessential Example: Higgs Mass Fermion Loops => Quadratic Divergences New Scalar -> Same Divergence With The Opposite Sign! Dirac fermion (4 DoFs) => Two Complex Scalars

14 Explicit Loop Calculation L H ff Z ( i ) 2 tr d 4 k i(/k + m) i(/k + /p + m) (2 ) 4 k 2 m 2 (k + p) 2 m 2! 4 2 Z d 4 k 1 (2 ) 4 k 2 2 Complex Scalars Z 2 ( 2i 2 ) L d 4 k (2 ) 4 2 H 2 s s i k 2 m 2! +4 2 Z d 4 k 1 (2 ) 4 k 2

15 SUSY Fields

16 Chiral Superfield Complex Scalar (sfermion) One Weyl Fermion (fermion) Bosonic DoF = 2 Fermionic DoF = 2

17 Lagrangian L µ + i µ@ µ = = = i( µ µ = i( µ µ L = total divergence

18 Vector Superfields Gauge Invariant Real Vector (Gauge Boson) Weyl Fermion (Gaugino) Bosonic DoF = 2 Fermionic DoF = 2

19 L = 1 4 F µ F µ + i µ@ µ A µ = µ µ = i 2 ( µ ) F µ = i 2 ( µ ) F µ L = total divergence

20 Superspace

21 Superspace SUSY Transformations Can Be Unified Define Superspace Coordinates (x, a, ȧ ) ; a=1,2 The θ coordinates are Grassmann Numbers

22 Superfields Superfields Are Taylor Expansions In General Superfield: S(x,, )=a b + c + µ v µ d

23 Supertransformations Combine All Supertransformations [2] S = i( Q + Q ) @ + i[ µ + µ ]@ µ! S = S(x µ + i µ + i µ, +, + ) S(x µ, ) Supertransformations -> Translation In Superspace

24 Vacuum Energy Generically, We Can Write The Hamiltonian As [2] h H i /h {Q,Q } i In Unbroken SUSY, The Vacuum Satisfies i Q i = Q =0 Therefore We Have h H i =0

25 Chiral Superfield Useful To Use New Coordinates y = x µ + i µ We Can Write Chiral Superfield As = (y)+ p 2 (y)+ F (y) F Is An Auxiliary Field -> No On Shell DoFs

26 Vector Superfield Impose That Superfield S Be Real V = a b + b + µ v µ d Super Gauge Transformations V! V + i( ) Wess-Zumino Gauge V WZ = µ A µ D Retains Ordinary Gauge Freedom

27 Lagrangians In Superspace Superspace -> Automatic SUSY Invariance S = Superfield, Φ = Chiral Superfield The Following Are SUSY Invariant Z Z d 2 d 2 d 2 S D Term Z d 2 F Term

28 D-Term Example D-Term Only Transforms By A Total Div. ) d / µ@ µ + µ

29 Super Potential Product Of Chiral Superfields = Chiral Superfield Construct Super Potential W W = 1 2 M ij i j y ijk i j k Use F Term In Lagrangian Plus h.c L [W ] F +h.c

30 SUSY QED

31 SUSY QED Two Chiral Superfields, Opposite Charge L SQED =[ 1 ev 1] D +[ +M[ 1 2] F + M[ D-Terms = Super-Gauge Invariant Kinetic Terms F-Terms = Mass Terms 1 2 e V 2] D 2 ] F 1! e i 1, 2! e i 2, V! V i( )

32 SUSY QED L SQED = 1 4 F µ F µ i µ + (i µ M) 1 (@2 + M 2 ) 1 2 (@2 + M 2 ) 2 + g µ A µ + gi( µ 1 1@ µ 1 )Aµ + g 2 1 1A µ A µ gi( µ 2 2@ µ 2 )Aµ + g 2 2 2A µ A µ p 2g( P 1 L + 1 PR 2 P L P 2 R ) 1 2 g2 ( ) 2

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34 MSSM

35 MSSM Chiral Superfields [2]

36 MSSM Vector Superfields [2]

37 Why 2HD? We Would Like To Write H H However This Is Not Allowed In W We Require A Second Higgs Doublet W MSSM =ūy u QH u dyd QH d ēy e LH d + µh u H d

38 Soft SUSY Breaking Soft = Positive Mass Dimension Maintains EW/Plank Scale Hierarchy [2] 1 Gaugino Mass 2 m a a a 1 Holomorphic Scalar Mass 2 b ij i j Non-Holomorphic Scalar Mass m ij i 1 Tri-Scalar Coupling 6 c ijk i j k j

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40 MSSM 124 Parameters [3] SUSY Conserving Gauge Couplings Higgs Mass Parameter μ Yukawa Couplings SUSY Violating Gaugino/Sfermion/ Higgs Masses Higgs-Sfermion- Fermion Couplings tanβ=vu/vd (Note that vu 2 + vd 2 Fixed By W Mass)

41 R-Symmetry Postulate New U(1) Symmetry [2]! e i! e i Superfield Components Transform Differently Define A Supercharge rs S(x,, )! e ir s S(x, e i,e i )

42 R-Parity In MSSM aka Matter Parity Discrete Version Of R-Symmetry Ordinary Particles Have PM = +1 Super Partners (Sparticles) Have PM = -1 Implies Each Vertex Must Have Even Number of Sparticles => Lightest Sparticle Is Stable! (LSP)

43 Matter Parity Cont. Consider What Would Happen Without Matter Parity... Note that L and Hd Have The Same Quantum Numbers In WMSSM We Could Write dql Would Violate Lepton Number

44 LSP As Dark Matter [4] Neutral Higgsinos, Winos, and Binos Are Not Mass Eigenstates 4 Mass Eigenstates Are Called Neutralinos Majorana Fermions Thought To Be A Likely Candidate For DM

45 What Did We Learn? You Can Construct SUSY In Ordinary QM SUSY Gives Cancellations SUSY Fields Chiral & Vector Superspace + Superfields => SUSY Automatic SUSY QED Lagrangian And Feynman Diagrams MSSM: 2HDM & SUSY Breaking & LSP DM

46 Thank You!

47 References [1] An Introduction to Superstmmetry in Quantum Mechanical Systems by T. Wellman [2] A Super Symmetry Primer arxiv:hep-ph/ [3] PDG SUSY Review theory.pdf [4] LSP As A Candidate For Dark Matter arxiv:hep-ph/ v2