We consider 2 → 2 spinless scattering and take, for simplicity, p 2 i = m 2 i = m 2 . Momentum conservation implies p 1 + p 2 = p 3 + p 4 . Now let us postulate an additional external symmetry, e.g. a conserved tensor R µν = p µ p ν − 1 4 g µν m 2 . If R µν is conserved, then R 1 µν + R 2 R 3 µν + R 4 = µν µν p 1 µ p 1 ν + p 2 µ p 2 p 3 µ p 3 ν + p 4 µ p 4 and thus = ν . ν 16 / 65
Specifically, in the center-of-mass frame we have = ( E , 0 , 0 , p ) p 1 = ( E , 0 , 0 , − p ) p 2 p 3 = ( E , 0 , p sin θ, p cos θ ) p 4 = ( E , 0 , − p sin θ, − p cos θ ) Let us look at e.g. µ = ν = 4. We find 2 p 2 = 2 p 2 cos θ . ⇒ θ = 0, i.e. no scattering 17 / 65
The Haag-Lopuszanski-Sohnius theorem Tensors a µ 1 ··· µ N are combinations of Lorentz vector indices, which each transform like a vector: µ 1 ··· µ N = Λ ν 1 µ 1 · · · Λ ν N a ′ µ N a µ 1 ··· µ N → tensors are bosons This points to the loop-hole in the Coleman-Mandula “no-go” theorem: The argument of Coleman-Mandula does not apply to conserved charges transforming as spinors. 18 / 65
The Haag-Lopuszanski-Sohnius theorem Tensors a µ 1 ··· µ N are combinations of Lorentz vector indices, which each transform like a vector: µ 1 ··· µ N = Λ ν 1 µ 1 · · · Λ ν N a ′ µ N a µ 1 ··· µ N → tensors are bosons This points to the loop-hole in the Coleman-Mandula “no-go” theorem: The argument of Coleman-Mandula does not apply to conserved charges transforming as spinors. Haag, Lopuszanski & Sohnius (1975): Supersymmetry is the only possible external symmetry of the scattering amplitude beyond Lorentz symmetry, for which the scattering is non-trivial. 18 / 65
The Haag-Lopuszanski-Sohnius theorem Tensors a µ 1 ··· µ N are combinations of Lorentz vector indices, which each transform like a vector: µ 1 ··· µ N = Λ ν 1 µ 1 · · · Λ ν N a ′ µ N a µ 1 ··· µ N → tensors are bosons This points to the loop-hole in the Coleman-Mandula “no-go” theorem: The argument of Coleman-Mandula does not apply to conserved charges transforming as spinors. Haag, Lopuszanski & Sohnius (1975): Supersymmetry is the only possible external symmetry of the scattering amplitude beyond Lorentz symmetry, for which the scattering is non-trivial. How could nature have ignored this last possible external symmetry? 18 / 65
Supersymmetry What is the algebra of the SUSY generators Q α ? One can work out that [ P µ , Q α ] = 0 [ M µν , Q α ] − i ( σ µν ) β = α Q β { Q α , Q β } = 0 { Q α , Q † 2( σ µ ) αβ P µ β } = where σ µ = (1 , σ i ), ¯ σ µ = (1 , σ i ), σ µν = ( σ µ ¯ σ ν − σ ν ¯ σ µ ) / 4. 19 / 65
Supersymmetry What is the algebra of the SUSY generators Q α ? One can work out that [ P µ , Q α ] = 0 [ M µν , Q α ] − i ( σ µν ) β = α Q β { Q α , Q β } = 0 { Q α , Q † 2( σ µ ) αβ P µ β } = where σ µ = (1 , σ i ), ¯ σ µ = (1 , σ i ), σ µν = ( σ µ ¯ σ ν − σ ν ¯ σ µ ) / 4. Q raises by spin 1/2, Q † lowers by spin 1/2 19 / 65
Supersymmetry What are the immediate consequences of SUSY invariance? [ P µ , Q ] = 0 [ m 2 , Q ] = [ P µ P µ , Q ] = 0 ⇒ 20 / 65
Supersymmetry What are the immediate consequences of SUSY invariance? [ P µ , Q ] = 0 [ m 2 , Q ] = [ P µ P µ , Q ] = 0 ⇒ Thus we must have e = m e . m ˜ 20 / 65
Supersymmetry What are the immediate consequences of SUSY invariance? [ P µ , Q ] = 0 [ m 2 , Q ] = [ P µ P µ , Q ] = 0 ⇒ Thus we must have e = m e . m ˜ e charged ([ Q , T a ] = 0) scalar But we have not seen a 511 keV= m ˜ → SUSY must be broken At what scale? What is the mass of the supersymmetric particles? 20 / 65
The hierarchy problem and the scale of SUSY breaking 21 / 65
The hierarchy problem and the scale of SUSY breaking Let us first look at electrodynamics: e 2 The Coulomb field of the electron is E self = 3 r e . 5 This can be interpreted as a contribution to the electron mass: m e c 2 = m e , 0 c 2 + E self . 21 / 65
The hierarchy problem and the scale of SUSY breaking Let us first look at electrodynamics: e 2 The Coulomb field of the electron is E self = 3 r e . 5 This can be interpreted as a contribution to the electron mass: m e c 2 = m e , 0 c 2 + E self . < 10 − 17 cm (exp. bound on point-like nature) one has However, with r e ∼ m e c 2 = 511 keV = ( − 9999 . 489 + 10000 . 000) keV → fine-tuning! 21 / 65
Is there fine-tuning in quantum electrodynamics? 22 / 65
Is there fine-tuning in quantum electrodynamics? Coulomb self-energy in time-ordered perturbation theory: 22 / 65
Is there fine-tuning in quantum electrodynamics? Coulomb self-energy in time-ordered perturbation theory: But also have positron e + with Q ( e + ) = − Q ( e − ) and m ( e + ) = m ( e − ) → new diagram 22 / 65
Is there fine-tuning in quantum electrodynamics? Coulomb self-energy in time-ordered perturbation theory: But also have positron e + with Q ( e + ) = − Q ( e − ) and m ( e + ) = m ( e − ) → new diagram � � �� 1 + 3 α � → m e c 2 = m e , 0 c 2 4 π ln m e cr e 22 / 65
m e c 2 = m e , 0 c 2 � � �� 1 + 3 α � We found that 4 π ln . m e cr e So even if r e = 1 / M Planck = 1 . 6 × 10 − 33 cm, the corrections to the electron mass are small m e c 2 ≈ m e , 0 c 2 (1 + 0 . 1) . Also, if m e , 0 = 0 then m e = 0 to all orders: the mass is protected by a (chiral) symmetry 23 / 65
m e c 2 = m e , 0 c 2 � � �� 1 + 3 α � We found that 4 π ln . m e cr e So even if r e = 1 / M Planck = 1 . 6 × 10 − 33 cm, the corrections to the electron mass are small m e c 2 ≈ m e , 0 c 2 (1 + 0 . 1) . Also, if m e , 0 = 0 then m e = 0 to all orders: the mass is protected by a (chiral) symmetry Recall ’t Hooft’s naturalness argument 23 / 65
Now let us look at the scalar (=Higgs) self-energy: 24 / 65
Now let us look at the scalar (=Higgs) self-energy: � � � d 4 k 2 m 2 1 ⇒ ∆ m 2 φ = 2 N ( f ) λ 2 f + k 2 − m 2 ( k 2 − m 2 f (2 π ) 4 f ) 2 f The integral is divergent, so we introduce a momentum cut-off. R Λ dkk 3 / ( k 2 − m 2 R Λ dkk 3 / ( k 2 − m 2 f ) ∼ Λ 2 and [Recall that d 4 k ∼ k 3 dk → f ) 2 ∼ ln Λ.] 24 / 65
Now let us look at the scalar (=Higgs) self-energy: � � � d 4 k 2 m 2 1 ⇒ ∆ m 2 φ = 2 N ( f ) λ 2 f + k 2 − m 2 ( k 2 − m 2 f (2 π ) 4 f ) 2 f The integral is divergent, so we introduce a momentum cut-off. R Λ dkk 3 / ( k 2 − m 2 R Λ dkk 3 / ( k 2 − m 2 f ) ∼ Λ 2 and [Recall that d 4 k ∼ k 3 dk → f ) 2 ∼ ln Λ.] Straightforward calculation gives � � Λ 2 + m 2 � � φ = N ( f ) λ 2 Λ 2 Λ 2 + 3 m 2 ∆ m 2 + 2 m 2 f f f ln . Λ 2 + m 2 f 8 π 2 m 2 f f 24 / 65
Because of the quadratic divergence we find φ (Λ = M Planck ) ≈ 10 35 GeV 2 = (3 × 10 17 GeV) 2 ∆ m 2 25 / 65
Because of the quadratic divergence we find φ (Λ = M Planck ) ≈ 10 35 GeV 2 = (3 × 10 17 GeV) 2 ∆ m 2 and so < 1 TeV 2 = m 2 m 2 φ, 0 + ∆ m 2 φ ∼ φ implies a huge fine-tuning: Comment: it is essential that Λ < ∞ , i.e. we assume that new physics sets in at E ∼ Λ. Is this a tautology? No: we assume new physics at some very high scale Λ and find that the standard model needs new physics well below Λ. The natural mass scale of a scalar field is the highest scale in nature. 25 / 65
The SUSY solution to the hierarchy problem 26 / 65
The SUSY solution to the hierarchy problem Let us increase the particle content (as for the e − self-energy) Before we had Now we include in addition two scalars ˜ f L , ˜ f R with couplings � λ f ˜ � λ 2 2 φ 2 � f R | 2 � � f R | 2 � f L | 2 + | ˜ f L | 2 + | ˜ | ˜ − v ˜ | ˜ A f φ ˜ f L ˜ f λ 2 f ∗ √ L φ ˜ f = − f φ + R + h.c. 2 which lead to additional contributions to the self-energy: 26 / 65
The additional contributions to the Higgs mass are: � d 4 k � � 1 1 ˜ f N (˜ ∆ m 2 λ 2 = f ) + φ k 2 − m 2 k 2 − m 2 (2 π ) 4 ˜ ˜ f L f R � d 4 k � � 1 1 f v ) 2 N (˜ (˜ λ 2 + f ) f L ) 2 + ( k 2 − m 2 ( k 2 − m 2 (2 π ) 4 f R ) 2 ˜ ˜ � d 4 k 1 ( λ f A f ) 2 N (˜ + f ) ( k 2 − m 2 f L )( k 2 − m 2 (2 π ) 4 f R ) ˜ ˜ . 27 / 65
The additional contributions to the Higgs mass are: � d 4 k � � 1 1 ˜ f N (˜ ∆ m 2 λ 2 = f ) + φ k 2 − m 2 k 2 − m 2 (2 π ) 4 ˜ ˜ f L f R � d 4 k � � 1 1 f v ) 2 N (˜ (˜ λ 2 + f ) f L ) 2 + ( k 2 − m 2 ( k 2 − m 2 (2 π ) 4 f R ) 2 ˜ ˜ � d 4 k 1 ( λ f A f ) 2 N (˜ + f ) ( k 2 − m 2 f L )( k 2 − m 2 (2 π ) 4 f R ) ˜ ˜ . The first term cancels the SM Λ 2 -contribution if ˜ N (˜ λ f = λ f and f ) = N ( f ) as required in SUSY. 27 / 65
The cancellation happens because of spin-statistics: fermion loop → (-1) boson-loop → (+1) 28 / 65
The cancellation happens because of spin-statistics: fermion loop → (-1) boson-loop → (+1) Note: ◮ the cancellation of quadratic divergences is independent of m ˜ f L , m ˜ f R , A f . ◮ the term ∝ A f φ ˜ f L ˜ f ∗ R breaks SUSY but does not lead to Λ 2 divergences → ”soft” SUSY breaking 28 / 65
Let us look at the finite SM + SUSY contributions: � � � λ 2 1 − ln m 2 f ln m 2 f N ( f ) ∆ m 2 − 2 m 2 f + 4 m 2 f = φ f 16 π 2 µ 2 µ 2 � � � m 2 m 2 m 2 µ 2 − | A f | 2 ln ˜ ˜ ˜ + 2 m 2 f − 4 m 2 f f 1 − ln f ln , ˜ ˜ f µ 2 µ 2 where we have assumed m ˜ f L = m ˜ f R = m ˜ f . 29 / 65
Let us look at the finite SM + SUSY contributions: � � � λ 2 1 − ln m 2 f ln m 2 f N ( f ) ∆ m 2 − 2 m 2 f + 4 m 2 f = φ f 16 π 2 µ 2 µ 2 � � � m 2 m 2 m 2 µ 2 − | A f | 2 ln ˜ ˜ ˜ + 2 m 2 f − 4 m 2 f f 1 − ln f ln , ˜ ˜ f µ 2 µ 2 where we have assumed m ˜ f L = m ˜ f R = m ˜ f . One has ∆ m 2 φ = 0 for A f = 0 and m ˜ f = m f (SUSY) 29 / 65
Let us look at the finite SM + SUSY contributions: � � � λ 2 1 − ln m 2 f ln m 2 f N ( f ) ∆ m 2 − 2 m 2 f + 4 m 2 f = φ f 16 π 2 µ 2 µ 2 � � � m 2 m 2 m 2 µ 2 − | A f | 2 ln ˜ ˜ ˜ + 2 m 2 f − 4 m 2 f f 1 − ln f ln , ˜ ˜ f µ 2 µ 2 where we have assumed m ˜ f L = m ˜ f R = m ˜ f . One has ∆ m 2 φ = 0 for A f = 0 and m ˜ f = m f (SUSY) But SUSY is broken, i.e. m 2 f = m 2 f + δ 2 . Thus ˜ � � φ = λ 2 2 + ln m 2 f N ( f ) ∆ m 2 δ 2 + O ( δ 4 ) f 8 π 2 µ 2 f + δ 2 = O (1 TeV 2 ) To have ∆ m 2 φ small, we thus need m 2 f = m 2 ˜ 29 / 65
Supersymmetry: Summary of first lecture SUSY is great! Must have been tired yesterday. . . 30 / 65
Motivation for supersymmetry A Priori: ◮ SUSY is the unique maximal external symmetry in Nature. ◮ Weak-scale SUSY provides a solution to the hierarchy problem. A Posteriori: ◮ SUSY allows for unification of Standard Model gauge interactions. ◮ SUSY provides dark matter candidates. ◮ SUSY QFT’s allow for precision calculations. ◮ SUSY provides a rich phenomenology and is testable at the LHC. 31 / 65
Outline ◮ The supersymmetric harmonic oscillator ◮ Motivation for SUSY: Symmetry & the hierarchy problem ◮ The MSSM ◮ SUSY searches 32 / 65
The Minimal Supersymmetric extension of the SM ◮ external symmetries: Poincare symmetry & supersymmetry ◮ internal symmetries: SU(3) ⊗ SU(2) ⊗ U(1) gauge symmetries ◮ minimal particle content 33 / 65
Gauge coupling unification In QFT the gauge couplings “run”: d α i ( µ ) d ln µ 2 = β i ( α i ( µ )) The beta-functions β i depend on the gauge group and on the matter multiplets to which the gauge bosons couple. Only particles with mass < µ contribute to the β i and to the evolution of the coupling at any given mass scale µ . The Standard Model couplings evolve with µ according to SU ( 3 ) : β 3 , 0 = (33 − 4 n g ) / (12 π ) SU ( 2 ) : β 2 , 0 = (22 − 4 n g − n h / 2) / (12 π ) U ( 1 ) : β 1 , 0 = ( − 4 n g − 3 n h / 10) / (12 π ) where n g = 3 is the number of quark and lepton generations and n h = 1 is the number of Higgs doublet fields in the Standard Model. 34 / 65
Gauge coupling unification Loop contributions of superpartners change the beta-functions. In the MSSM one finds: β SUSY SU ( 3 ) : = (27 − 6 n g ) / (12 π ) 3 , 0 β SUSY SU ( 2 ) : = (18 − 6 n g − 3 n h / 2) / (12 π ) 2 , 0 β SUSY U ( 1 ) : = ( − 6 n g − 9 n h / 10) / (12 π ) 1 , 0 35 / 65
Gauge coupling unification Loop contributions of superpartners change the beta-functions. In the MSSM one finds: β SUSY SU ( 3 ) : = (27 − 6 n g ) / (12 π ) 3 , 0 β SUSY SU ( 2 ) : = (18 − 6 n g − 3 n h / 2) / (12 π ) 2 , 0 β SUSY U ( 1 ) : = ( − 6 n g − 9 n h / 10) / (12 π ) 1 , 0 35 / 65
R -parity ◮ In the SM baryon and lepton number are accidental symmetries ◮ The most general superpotential of the SUSY-SM contains baryon and lepton number violating terms: W ∈ λ ijk L i L j E k + λ ′ λ ′′ ijk L i Q j D k + κ i L i H 2 + ijk U i D j D k � �� � � �� � lepton number violating baryon number violating 36 / 65
R -parity ◮ In the SM baryon and lepton number are accidental symmetries ◮ The most general superpotential of the SUSY-SM contains baryon and lepton number violating terms: W ∈ λ ijk L i L j E k + λ ′ λ ′′ ijk L i Q j D k + κ i L i H 2 + ijk U i D j D k � �� � � �� � lepton number violating baryon number violating LQD and UDD couplings lead to rapid proton decay → impose discrete symmetry: R -parity R = ( − 1) 3 B + L +2 S → R SM = + and R SUSY = − 36 / 65
R -parity R -parity conservation has dramatic phenomenological consequences: ◮ lightest SUSY particle (LSP) is absolutely stable → dark matter candidate if also electrically neutral ◮ in collider experiments SUSY particles can only be produced in pairs ◮ in many models SUSY collider events contain missing E T 37 / 65
SUSY breaking Supersymmetry: mass( e − ) = mass(˜ e − L , R ) → SUSY must be broken No agreed model of supersymmetry breaking → phenomenological ansatz Must preserve solution to hierarchy problem → “soft” SUSY breaking 38 / 65
SUSY breaking Supersymmetry: mass( e − ) = mass(˜ e − L , R ) → SUSY must be broken No agreed model of supersymmetry breaking → phenomenological ansatz Must preserve solution to hierarchy problem → “soft” SUSY breaking Introduce ◮ gaugino masses M 1 / 2 χχ : M 1 ˜ B ˜ B , M 2 ˜ W ˜ W , M 3 ˜ g ˜ g ◮ squark and slepton masses M 2 0 φ † φ : e † e † u † u † m 2 e L , m 2 e R , m 2 u L , m 2 e L ˜ L ˜ e R ˜ R ˜ u L ˜ L ˜ u R ˜ R ˜ u R etc. ˜ ˜ ˜ ˜ � ˜ � ν i ◮ trilinear couplings A ijk φ i φ j φ k : A e h 1 ˜ e jR etc. ij ˜ e j L ◮ Higgs mass terms B ij φ i φ j : Bh 1 h 2 etc. 38 / 65
SUSY breaking MSSM w/o breaking: two additional parameters from Higgs sector Soft SUSY breaking ◮ A e ij , A d ij , A u → 27 real + 27 phases ij ◮ M 2 Q , M 2 U , M 2 D , M 2 L , M 2 E → 30 real + 15 phases ˜ ˜ ˜ ˜ ˜ ◮ M 1 , M 2 , M 3 → 3 real + 1 phase → 124 parameters in the MSSM! (but strong constraints from FCNS’s, flavour mixing and CP violation) 39 / 65
SUSY breaking MSSM w/o breaking: two additional parameters from Higgs sector Soft SUSY breaking ◮ A e ij , A d ij , A u → 27 real + 27 phases ij ◮ M 2 Q , M 2 U , M 2 D , M 2 L , M 2 E → 30 real + 15 phases ˜ ˜ ˜ ˜ ˜ ◮ M 1 , M 2 , M 3 → 3 real + 1 phase → 124 parameters in the MSSM! (but strong constraints from FCNS’s, flavour mixing and CP violation) Simple framework constrained MSSM: breaking is universal at GUT scale ◮ universal scalar masses: M 2 Q , M 2 U , M 2 D , M 2 L , M 2 E → M 2 0 at M GUT ˜ ˜ ˜ ˜ ˜ ◮ universal gaugino masses: M 1 , M 2 , M 3 → M 1 / 2 at M GUT ◮ universal trilinear couplings A e ij , A d ij , A u ij → A · h e ij , A · h d ij , A · h u ij at M GUT → 6 additional parameters: M 0 , M 1 / 2 , A , B , µ , tan( β ) 39 / 65
SUSY mass spectrum In QFT the (s)particle masses “run”: dM i ( µ ) d ln µ 2 = γ i M i 40 / 65
SUSY mass spectrum In QFT the (s)particle masses “run”: dM i ( µ ) d ln µ 2 = γ i M i SPS1a 600 m Ql ( µ 2 +m Hd 2 ) 1/2 400 M 3 GeV M 2 200 M 1 m Er 0 ( µ 2 +m Hu 2 ) 1/2 SOFTSUSY3.0.5 -200 2 4 6 8 10 12 14 16 log 10 ( µ /GeV) 40 / 65
SUSY mass spectrum In QFT the (s)particle masses “run”: dM i ( µ ) d ln µ 2 = γ i M i SPS1a 600 m Ql ( µ 2 +m Hd 2 ) 1/2 typical mass pattern e.g. from 400 M 3 M 1 ( µ ) α 1 ( µ ) = M 2 ( µ ) α 2 ( µ ) = M 3 ( µ ) GeV α 3 ( µ ) M 2 200 M 1 m Er → M 3 ( M Z ) : M 2 ( M Z ) : M 1 ( M Z ) ≃ 7 : 2 : 1 0 ( µ 2 +m Hu 2 ) 1/2 SOFTSUSY3.0.5 -200 2 4 6 8 10 12 14 16 log 10 ( µ /GeV) 40 / 65
Radiative EWK symmetry breaking ◮ RGE drives ( µ 2 + m H 2 u ) negative → EWK symmetry breaking ◮ Masses of W and Z bosons fix B and | µ | ◮ cMSSM has 4 1/2 parameters: M 0 , M 1 / 2 , A , tan( β ) and sign( µ ) 41 / 65
Mixing After SU(2) L × U(1) Y breaking, mixing will occur between any two or more fields which have the same color, charge and spin ◮ ( ˜ W ± , ˜ χ ± H ± ) → ˜ i =1 , 2 : charginos ◮ (˜ B , ˜ W 3 , ˜ H 0 χ 0 1 , 2 ) → ˜ i =1 , 2 , 3 , 4 : neutralinos ◮ (˜ t L , ˜ t R ) → ˜ t 1 , 2 etc.: sfermion mass eigenstates 42 / 65
Mixing After SU(2) L × U(1) Y breaking, mixing will occur between any two or more fields which have the same color, charge and spin ◮ ( ˜ W ± , ˜ χ ± H ± ) → ˜ i =1 , 2 : charginos ◮ (˜ B , ˜ W 3 , ˜ H 0 χ 0 1 , 2 ) → ˜ i =1 , 2 , 3 , 4 : neutralinos ◮ (˜ t L , ˜ t R ) → ˜ t 1 , 2 etc.: sfermion mass eigenstates Note: ◮ mixing involves various SUSY parameters → cross sections and branching ratios become model dependent ◮ sfermion mixing ∝ m f → large only for 3rd generation (˜ t 1 , 2 , ˜ τ 1 , 2 ) 42 / 65
Outline ◮ The supersymmetric harmonic oscillator ◮ Motivation for SUSY: Symmetry & the hierarchy problem ◮ The MSSM ◮ SUSY searches 43 / 65
Summary of SUSY searches so far. . . 44 / 65
Summary of SUSY searches so far. . . . . . but let’s see what to expect in 2011 & 2012. . . 44 / 65
Outline ◮ The supersymmetric harmonic oscillator ◮ Motivation for SUSY: Symmetry & the hierarchy problem ◮ The MSSM ◮ SUSY searches ◮ indirect searches through quantum fluctuations ◮ direct searches at colliders 45 / 65
Indirect SUSY searches Wealth of precision measurements from B / K physics, ( g − 2), astrophysics (DM) and collider limits → constraints on certain SUSY masses e.g. through anomalous magnetic moment ( g − 2) 46 / 65
Indirect SUSY searches: ( g − 2) µ Hamiltonian for interaction of µ -spin with external magnetic field e � S µ · � H = g µ B 2 m µ with g µ = 2 in leading order Loop-corrections modify the interaction of the µ with the electromagnetic field � g − 2 � = α 2 π = 0 . 00116114 ⇒ 2 QED 23 / 51
Indirect SUSY searches: ( g − 2) µ There are additional diagrams in supersymmetric QED, e.g. which is given by √ √ d 4 k � 1 i = (2 π ) 4 ( ie 2) P R P L ( ie 2) I ( p ′ − k ) 2 − m 2 � k − M ˜ γ ˜ µ L i × ( ie )( p ′ + p − 2 k ) ν ( p − k ) 2 − m 2 ˜ µ L After a short calculation (using standard QED techniques) one finds � 1 = − m 2 µ e 2 x 2 (1 − x ) � g − 2 � dx µ x 2 + ( m 2 µ L − M 2 µ ) x + M 2 2 8 π 2 m 2 γ − m 2 0 SQED ˜ ˜ ˜ γ 24 / 51
Indirect SUSY searches: ( g − 2) µ In the limit m ˜ γ , m µ we find µ L ≫ M ˜ m 2 � g − 2 � = − α µ m 2 2 6 π SQED ˜ µ L ◮ SUSY contribution decouples rapidly for m ˜ µ L ≫ m µ ◮ SUSY contribution ∝ m f → effects in ( g − 2) e suppressed Including mixing: → dependence on further SUSY parameters ( A and tan β ) 25 / 51
Indirect SUSY searches → CMSSM fit to B , K and EWK observables, ( g − 2) µ and Ω DM 47 / 65
Indirect SUSY searches → CMSSM fit to B , K and EWK observables, ( g − 2) µ and Ω DM Mass Spectrum of SUSY Particles no LHC Mass Spectrum of SUSY Particles no LHC 1 Environment σ 1400 2 σ Environment Best Fit Value 1200 Particle Mass [GeV] 1000 800 600 400 200 0 ~ ~ ∼ ∼ ~ ~ ~ ~ ~ ~ ~ + 0 0 0 H 0 0 0 0 χ + χ + h A H χ χ χ χ τ τ q q t t g l l b b 1 2 1 2 3 4 1 2 R L R L 1 2 1 2 47 / 65
Indirect SUSY searches → CMSSM fit to B , K and EWK observables, ( g − 2) µ and Ω DM Mass Spectrum of SUSY Particles no LHC Mass Spectrum of SUSY Particles no LHC 1 Environment σ 1400 2 σ Environment Best Fit Value 1200 Particle Mass [GeV] 1000 800 600 400 200 0 ~ ~ ∼ ∼ ~ ~ ~ ~ ~ ~ ~ + 0 0 0 H 0 0 0 0 χ + χ + h A H χ χ χ χ τ τ q q t t g l l b b 1 2 1 2 3 4 1 2 R L R L 1 2 1 2 ◮ global fits point to light sparticle spectrum with ˜ m < 1 TeV ◮ current data cannot constrain more general SUSY models 47 / 65
Indirect SUSY searches → CMSSM fit without ( g − 2) µ and Ω DM ◮ prediction of light SUSY spectrum rests on ( g − 2) µ and Ω DM 48 / 65
SUSY particle production at the LHC SUSY particles would be produced at the LHC via QCD processes 49 / 65
SUSY particle production at the LHC SUSY particles would be produced at the LHC via QCD processes 1000 q¯ σ (pp → ˜ g˜ g / ˜ q / ˜ ˜ q˜ q / ˜ q˜ g + X) [pb] √ s = 7 TeV; 100 q = m ˜ m ˜ g NLO+NLL 10 1 0 . 1 ˜ g ˜ g q ¯ ˜ ˜ q 0 . 01 q ˜ ˜ q q ˜ ˜ g 0 . 001 200 400 600 800 1000 1200 m [GeV] √ → σ ≈ 100 fb for m ≈ 1000 GeV at S = 7 TeV 49 / 65
SUSY particle production at the LHC SUSY particles would be produced at the LHC via QCD processes 10 3 q¯ σ (pp → ˜ g˜ g / ˜ ˜ q / ˜ q˜ q / ˜ q˜ g + X) [pb] √ s = 14 TeV; 10 1 q = m ˜ m ˜ g 10 NLO+NLL 1 10 − 1 10 − 2 g ˜ ˜ g 10 − 3 q ¯ ˜ ˜ q 10 − 4 q ˜ ˜ q ˜ q ˜ 10 − 5 g 500 1000 1500 2000 2500 3000 m [GeV] √ → σ ≈ 2 . 5 pb for m ≈ 1000 GeV at S = 14 TeV 50 / 65
SUSY searches at hadron colliders → Powerful MSSM signature at the LHC: cascade decays with E T , miss 51 / 65
SUSY searches at hadron colliders → Powerful MSSM signature at the LHC: cascade decays with E T , miss Generic signature for many new physics models which address – the hierarchy problem – the origin of dark matter → predict spectrum of new particles at the TeV-scale with weakly interacting & stable particle ( ← discrete parity) 51 / 65
Squark and gluino searches at the LHC Atlas limits (165 pb − 1 ) β µ MSUGRA/CMSSM: tan = 10, A = 0, >0 0 600 [GeV] ATLAS Preliminary Observed 95% C.L. limit Median expected limit 550 0 lepton 2011 combined 0 lepton 2011 combined 1/2 int -1 L = 165 pb , s =7 TeV CL Observed 95 % C.L. limit s m 500 CL Median expected limit s ~ q (1400) Reference point 450 2010 data PCL 95% C.L. limit CMS 2010 Razor,Jets/MHT ~ 400 q ∼ ( ~ ± 1 χ g (1000) LEP 2 0 0 1 0 ) ~ ~ β µ -1 D0 g , q , tan =3, <0, 2.1 fb 350 ~ ~ β µ -1 CDF g , q , tan =5, <0, 2 fb 300 ~ g (800) ~ q ( 6 0 0 250 ) ~ 200 g (600) 150 500 1000 1500 2000 2500 m [GeV] 0 > 950 GeV → m ˜ q ≈ m ˜ g ∼ 52 / 65
Direct SUSY searches at the LHC: expected limits The LHC is probing the preferred region of SUSY parameter space 2D 95% CL no LHC 1D 68% CL no LHC -1 95% CL exclusion 35pb 800 800 -1 95% CL exclusion 1fb -1 95% CL exclusion 2fb 95% CL exclusion 7fb -1 700 700 600 600 [GeV] [GeV] 500 500 1/2 1/2 M M 400 400 300 300 200 200 0 0 100 100 200 200 300 300 400 400 500 500 600 600 700 700 800 800 M M [GeV] [GeV] 0 0 53 / 65
Direct SUSY searches at the LHC: expected limits But what if we do not see any SUSY signal at the LHC? 2D 95% CL no LHC 1D 68% CL no LHC -1 95% CL exclusion 35pb 800 800 -1 95% CL exclusion 1fb -1 95% CL exclusion 2fb 95% CL exclusion 7fb -1 700 700 600 600 [GeV] [GeV] 500 500 1/2 1/2 M M 400 400 300 300 200 200 0 0 100 100 200 200 300 300 400 400 500 500 600 600 700 700 800 800 M M [GeV] [GeV] 0 0 54 / 65
Direct SUSY searches at the LHC: expected limits We have considered the SUSY search in the 4 jets + E T , miss signature with M eff = � i p T , i + E T , miss 55 / 65
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