The Anomalous Magnetic Moment of the Muon in the Minimal Supersymmetric Standard Model for tan β =

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1 The Anomalous Magnetic Moment of the Muon in the Minimal Supersymmetric Standard Model for tan β = Markus Bach Institut für Kern- und Teilchenphysik Technische Universität Dresden IKTP Institute Seminar February 6, / 40

2 Outline 1 Anomalous Magnetic Moment of the Muon Minimal Supersymmetric Standard Model for tan β = 3 Calculation and Approximation 4 Identification of Viable Parameter Regions / 40

3 Outline 1 Anomalous Magnetic Moment of the Muon Minimal Supersymmetric Standard Model for tan β = 3 Calculation and Approximation 4 Identification of Viable Parameter Regions 3 / 40

4 Definition magn. moment of a fermion: µ = g Q e s (g - gyromagn. ratio) m c interaction with an external photon: A ρ (q) decomposition: f (p) f (p ) = i Q e ū(p ) Λ ρ u(p) ū(p ) Λ ρ u(p) = ū(p ) [ γ ρ ( FE (q ) m F M (q ) ) + (p + p ) ρ F M (q ) +... ] u(p) classical limit: g = ( 1 m F M (0) ) anomalous magnetic moment: a := g = m F M (0) 4 / 40

5 Experimental Determination muons from pion decay in a storage ring spin precession in magnetic field B relative to the momentum with ω = e ( [a µ B a µ 1 ) ] β E γ 1 c m µ vanishes for magic momentum p = 3,094 GeV detection of electrons from muon decay result from E81 experiment at Brookhaven National Laboratory: a exp µ = ( ,9 ± 6,3) / 40

6 Standard Model Prediction different kinds of contributions QED: massless class, massive class hadronic: vacuum polarization, light-by-light scattering electroweak combined result [Davier et al., arxiv: ]: a SM µ = ( , ± 4,9) discrepancy between experiment and theory: a µ = a exp µ a SM µ = (8,7 ± 8,0) new physics? 6 / 40

7 Contributions from New Physics a µ not only points to BSM physics but also restricts it close connection between Feynman diagrams for a µ and self energy: aµ NP mµ = O(1) M NP mµ (NP) m µ [Czarnecki, Marciano, arxiv:hep-ph/0101] m µ (NP) m µ is promising MSSM one-loop result for common sparticle mass M S : a MSSM,1L µ 1, sgn(µm ) tan β What happens for tan β? ( 100 GeV M S ) 7 / 40

8 Outline 1 Anomalous Magnetic Moment of the Muon Minimal Supersymmetric Standard Model for tan β = 3 Calculation and Approximation 4 Identification of Viable Parameter Regions 8 / 40

9 Vector Superfields keep Standard Model gauge group SU (3) c SU () L U (1) Y Group Generators Coupling Superfields Component Fields Spin 1 Spin 1 U (1) Y Ŷ g 1 V λ B µ SU () L T a g V a λ a W a µ SU (3) c T d s g 3 V d s λ d s G d µ total gauge multiplet: V g := g 1 Ŷ V + g T a V a + g 3 T d s V d s 9 / 40

10 Chiral Superfields Superfield L i = ( Lνi L ei ) Component Fields Spin 0 Spin 1 ) ( νil νil l il = l ẽ il = il e il SU (3) c I 3 Y 1 1/ 1/ 1/ E i ẽ ir e ir ) Qui (ũil uil 1/ Q i = q Q il = q il = 3 1/6 di d il d il 1/ U i ũ ir u ir 3 0 /3 D i H u = ( H + u H 0 u H d = ( H 0 d H d ) ) d ir d ir 3 0 1/3 h + h u = u ψ + hu 0 H ψ Hu = u 1/ ψh 0 1 1/ u 1/ h d = ( h 0 d h d ) ψ 0 ψ Hd = Hd ψ H d 1 1/ 1/ 1/ 10 / 40

11 Supersymmetric Lagrangian R-parity conserving superpotential (no generation mixing): W MSSM = 3 (y ei H d L i E i + y ui H u Q i U i + y di H d Q i D i) µ H d H u i=1 supersymmetric, gauge invariant and renormalizable Lagrangian: L susy = d 4 θ ( L i e Vg L i + E i e Vg E i + Q i e Vg Q i + U i e Vg U i + D i e Vg D i ) + H u e Vg H u + H d evg H d [ ( + d 1 θ W α W 16g1 α g [ ] + d θ W MSSM θ=0 + h. c. W aα W a α g 3 ) ] Ws dα Wsα d + h. c. θ=0 11 / 40

12 Soft Supersymmetry Breaking up to now: sparticles have same mass as corresponding particles supersymmetry breaking realized through additional terms: L soft = 3 ( m l li il + mẽ i ẽ ir + m q i q il + mũ i ũ ir + m di d ir ) i=1 mh u h u mh d h d + 1 ( M1 λ λ + M λ a λ a + M 3 λ d s λ d s + h. c. ) [ 3 ( Aei y ei h d l il ẽ ir + A u i y ui h u q il ũ ir + A ) d i y di h d q il d ir i=1 ] Bµ h d h u + h. c. 1 / 40

13 Electroweak Symmetry Breaking vacuum expectation values for the Higgs doublets: 0 0 h u 0 =, 0 h d 0 = important parameter: gauge boson masses: v u tan β := v u v d M W = g v u + v d, M Z = vd 0 g1 + g vu + vd = M W c W 13 / 40

14 Electroweak Symmetry Breaking vacuum expectation values for the Higgs doublets: 0 0 h u 0 =, 0 h d 0 = important parameter: gauge boson masses: v u tan β := v u v d M W = g v u + v d, M Z = vd 0 g1 + g vu + vd = M W c W tan β = implies: v u = MW g, v d = 0 13 / 40

15 Fermion Masses at tree level: m 0 e i = y ei v d, m 0 u i = y ui v u, m 0 d i = y di v d usual constraint: tan β 50 such that the Yukawa couplings remain perturbative if mass is created at tree-level tan β 50 is possible if one considers loop corrections which are tan β-enhanced [Dobrescu, Fox, arxiv: ] e.g. muon mass: m µ = m 0 µ Σ µ = m 0 µ (1 + µ ) = y µ v d (1 + µ ) 14 / 40

16 Fermion Masses at tree level: m 0 e i = y ei v d, m 0 u i = y ui v u, m 0 d i = y di v d usual constraint: tan β 50 such that the Yukawa couplings remain perturbative if mass is created at tree-level tan β 50 is possible if one considers loop corrections which are tan β-enhanced [Dobrescu, Fox, arxiv: ] e.g. muon mass: m µ = m 0 µ Σ µ = m 0 µ (1 + µ ) = y µ v d (1 + µ ) tan β = implies: m 0 µ = 0 m µ = Σ µ 14 / 40

17 Muon Sneutrino and Smuon Masses muon sneutrino mass: m ν µ = m L + 1 M Z cos β smuon mass matrix: ( (m 0 M µ µ ) + ml + MZ (sw 1 = ) cos β ) m0 µ (A µ µ tan β) mµ 0 (A µ µ tan β) (mµ) 0 + mr MZ sw cos β diagonalization: U µ M µ U µ = diag(m µ 1, m µ ) 15 / 40

18 Muon Sneutrino and Smuon Masses muon sneutrino mass: m ν µ = m L 1 M Z smuon mass matrix: ( M µ ml MZ (sw 1 = ) m0 µ (A µ µ tan β) mµ 0 (A µ µ tan β) mr + MZ sw ) diagonalization: U µ M µ U µ = diag(m µ 1, m µ ) 15 / 40

19 Muon Sneutrino and Smuon Masses muon sneutrino mass: m ν µ = m L 1 M Z smuon mass matrix: ( M µ ml MZ (sw 1 = ) y µ µ M W /g y µ µ M W /g mr + MZ sw ) diagonalization: U µ M µ U µ = diag(m µ 1, m µ ) smuon masses: ( m µ 1/ = 1 m L + m R + 1 M Z [ ] 1 ± ml m R + M Z s W + 8 yµ µ M W g ) 15 / 40

20 Chargino Masses chargino mass matrix: X = diagonalization: ( M U X V 1 = diag(m χ 1 MW cos β µ, m χ ) Hermitian form: V X X V 1 = diag(m χ, m 1 χ ) ) MW sin β 16 / 40

21 Chargino Masses chargino mass matrix: X = ( M MW 0 µ ) diagonalization: U X V 1 = diag(m χ 1, m χ ) Hermitian form: V X X V 1 = diag(m χ, m 1 χ ) chargino masses: m χ 1/ = 1 ( µ + M + M W ) ( µ ± + M + MW ) 4 µ M 16 / 40

22 Neutralino Masses neutralino mass matrix: M 1 0 M Z s W cos β M Z s W sin β 0 M M Z c W cos β M Z c W sin β Y = M Z s W cos β M Z c W cos β 0 µ M Z s W sin β M Z c W sin β µ 0 diagonalization: N Y N 1 = diag(m χ 0 1, m χ 0, m χ 0 3, m χ 0 4 ) Hermitian form: N Y Y N 1 = diag(m χ 0, m 1 χ 0, m χ 0, m 3 χ 0 4) 17 / 40

23 Neutralino Masses neutralino mass matrix: M M Z s W 0 M 0 M Z c W Y = µ M Z s W M Z c W µ 0 diagonalization: N Y N 1 = diag(m χ 0 1, m χ 0, m χ 0 3, m χ 0 4 ) Hermitian form: N Y Y N 1 = diag(m χ 0, m 1 χ 0, m χ 0, m 3 χ 0 4) 17 / 40

24 Outline 1 Anomalous Magnetic Moment of the Muon Minimal Supersymmetric Standard Model for tan β = 3 Calculation and Approximation 4 Identification of Viable Parameter Regions 18 / 40

25 Parameters and Feynman Diagrams relevant free MSSM parameters: y µ, µ, M 1, M, m L, m R approximation: m µ µ, M 1, M, m L, m R one-loop calculation sufficient calculation with mass eigenstates diagonal propagators Feynman Diagrams for muon self energy: χ l µ k µ µ µ µ ν µ χ 0 m 19 / 40

26 Parameters and Feynman Diagrams relevant free MSSM parameters: y µ, µ, M 1, M, m L, m R approximation: m µ µ, M 1, M, m L, m R one-loop calculation sufficient calculation with mass eigenstates diagonal propagators Feynman Diagrams for muon anomalous magnetic moment: χ l A ρ µ k A ρ µ µ ν µ µ µ χ 0 m 19 / 40

27 Auxiliary and Loop Functions expressions with sparticle mixing matrices occur: m χ U l l V l1 (U µ k1 ) U µ k1 m χ 0 m N m1 N m3 substitution introduces different functions auxiliary functions: x 1,J = A J q q 3 q 4 + B J (q q 3 + q q 4 + q 3 q 4 ) C J (q + q 3 + q 4 ) + D J (q 1 q ) (q 1 q 3 ) (q 1 q 4 ) y 1,J = q 3 q 4 (B J q 3 q 4 C J (q 3 + q 4 ) + D J ) (q 1 q 3 ) (q 1 q 4 ) (q q 3 ) (q q 4 ) loop function for muon self energy: I (q a, q b, q c ) = q a q b ln q a + q q b q c ln q b + q b q c q a ln q c c q a (q a q b ) (q b q c ) (q a q c ) 0 / 40

28 Auxiliary and Loop Functions expressions with sparticle mixing matrices occur: m χ U l l V l1 (U µ k1 ) U µ k1 m χ 0 m N m1 N m3 substitution introduces different functions auxiliary functions: x 1,J = A J q q 3 q 4 + B J (q q 3 + q q 4 + q 3 q 4 ) C J (q + q 3 + q 4 ) + D J (q 1 q ) (q 1 q 3 ) (q 1 q 4 ) y 1,J = q 3 q 4 (B J q 3 q 4 C J (q 3 + q 4 ) + D J ) (q 1 q 3 ) (q 1 q 4 ) (q q 3 ) (q q 4 ) loop functions for muon anomalous magnetic moment: K C/N (q a, q b, q c ) = G C/N(q a, q c ) G C/N (q a, q b ) q b q c with: G C (q a, q b ) = 3 q a + 4 q a q b q b + q a ln q a q b (q a q b ) 3 G N (q a, q b ) = q a + q b + q a q b ln q a q b (q a q b ) 3 0 / 40

29 Contribution to the Muon Self Energy Σ MSSM µ = 16π yµ { g Re(µ M ) M W I (m ν µ, m χ, m χ ) I (m χ 0, m µ m 1, m µ ) m=1 [ g Re µ M 1 W x g m,11 + g 1 x m,1 + 4 n,p=1 n p + y µ µ M g W x m,33 + g 1 x m,13 m µ mr MZ sw 1 ( ) ] g1 x m,13 + g x m,3 m µ 1 ml MZ 1 s W [ 1 Re(g 4 1 y np,13 + g y np,3 ) I (m µ 1, m χ 0, m n χ 0) p ] } g 1 Re(y np,13 ) I (m µ, m χ 0, m n χ 0) p 1 / 40

30 Contribution to the Muon Anomalous Magnetic Moment a MSSM µ = yµ mµ 16π { g Re(µ M ) M W K C (m ν µ, m χ, m χ ) K N (m χ 0, m µ m 1, m µ ) m=1 [ g Re µ M 1 W x g m,11 + g 1 x m,1 + 4 n,p=1 n p + y µ µ M g W x m,33 + g 1 x m,13 m µ mr MZ sw 1 ( ) ] g1 x m,13 + g x m,3 m µ 1 ml MZ 1 s W [ 1 Re(g 4 1 y np,13 + g y np,3 ) K N (m µ 1, m χ 0, m n χ 0) p ] } g 1 Re(y np,13 ) K N (m µ, m χ 0, m n χ 0) p / 40

31 Properties of the Results direct dependency on parameters y µ, µ, M 1, M, m L, m R symmetries: Σ MSSM µ Σ MSSM µ and aµ MSSM aµ MSSM if y µ y µ or µ µ or (M 1, M ) ( M 1, M ) m µ = Σ MSSM µ can be used to determine y µ numerically a MSSM µ only depends on µ, M 1, M, m L, m R if y µ is eliminated from the formula via the self energy: a MSSM µ invariant under µ µ and (M 1, M ) ( M 1, M ) Which sign is usual for a MSSM µ? 3 / 40

32 Start of the Approximation relevant MSSM mass parameters assumed to be real w.l.o.g.: µ, M 1, m L and m R positive; M positive or negative approximation: M Z µ, M 1, M, m L, m R sparticle masses: m ν µ = m µ 1 = m L m µ = m R m χ 1 = µ m χ = M m χ 0 1 = m χ 0 = µ m χ 0 3 = M 1 m χ 0 4 = M 4 / 40

33 Approximation for the Self Energy I dimensionless loop function: ( ma Î, m ) b := m m c m a m b I (ma, mb, mc) c v a, v b : 0 < Î (v a, v b ) < 1 5 / 40

34 Approximation for the Self Energy II [ ( Σ MSSM y µ µ M 3π g W sgn(m ) 3 g µ Î, M ) m L m L g1 µ Î, M 1 m L m L + g1 µ Î, M 1 m R m R ] g1 Î M1, m L µ m R m R m L [ ( µ y µ sgn(m ) 0,705 GeV Î, M ) m L m L µ 0,067 GeV Î, M 1 m L m L µ + 0,135 GeV Î, M 1 m R m R ] 0,135 GeV Î M1, m L µ m R m R m L 6 / 40

35 Approximation for the Self Energy II [ ( Σ MSSM y µ µ M 3π g W sgn(m ) 3 g µ Î, M ) m L m L g1 µ Î, M 1 m L m L + g1 µ Î, M 1 m R m R g 1 Î ( M1 m R, m L m R ) µ m L ] W H u HW1: W H d µ L µ R ν µ 6 / 40

36 Approximation for the Self Energy II [ ( Σ MSSM y µ µ M 3π g W sgn(m ) 3 g µ Î, M ) m L m L g1 µ Î, M 1 m L m L + g1 µ Î, M 1 m R m R g 1 Î ( M1 m R, m L m R ) µ m L ] W 0 H 0 u HW: W 0 H 0 d µ L µ R µ L 6 / 40

37 Approximation for the Self Energy II [ ( Σ MSSM y µ µ M 3π g W sgn(m ) 3 g µ Î, M ) m L m L g1 µ Î, M 1 m L m L + g1 µ Î, M 1 m R m R g 1 Î ( M1 m R, m L m R ) µ m L ] B 0 H 0 u HBL: B 0 H 0 d µ L µ R µ L 6 / 40

38 Approximation for the Self Energy II [ ( Σ MSSM y µ µ M 3π g W sgn(m ) 3 g µ Î, M ) m L m L g1 µ Î, M 1 m L m L + g1 µ Î, M 1 m R m R g 1 Î ( M1 m R, m L m R ) µ m L ] H 0 u B 0 HBR: H 0 d B 0 µ L µ R µ R 6 / 40

39 Approximation for the Self Energy II [ ( Σ MSSM y µ µ M 3π g W sgn(m ) 3 g µ Î, M ) m L m L g1 µ Î, M 1 m L m L + g1 µ Î, M 1 m R m R g 1 Î ( M1 m R, m L m R ) µ m L ] BLR: B 0 B 0 µ L µ R µ L µ R 6 / 40

40 Approximation for the Anomalous Magnetic Moment I dimensionless loop function: ( qa ˆK N, q ) b := q q c q a q b K N (q c, q a, q b ) c w a, w b : 0 < ˆK N (w a, w b ) < 1 7 / 40

41 Approximation for the Anomalous Magnetic Moment I dimensionless loop function: ( qa ˆK W, q ) b := q q c q a q b [ K C (q c, q a, q b ) + K N (q c, q a, q b ) ] c w a, w b : 0 < ˆK W (w a, w b ) < 1,155 7 / 40

42 Approximation for the Anomalous Magnetic Moment II [ aµ MSSM y µ m 3π µ M g W sgn(m ) g 1 µ ˆK W, M µ M ml ml + g1 1 µ ˆK N, M 1 µ M 1 ml ml g1 1 µ ˆK N, M 1 µ M 1 mr mr ] + g1 µ M 1 m ˆK L ml N, m R m R M1 M1 [ y µ sgn(m ), (1000 GeV) µ ˆK W, M µ M ml ml + 0, (1000 GeV) µ ˆK N, M 1 µ M 1 ml ml 1, (1000 GeV) µ M 1 ˆK N ( µ + 1, µ M 1 (1000 GeV) m L m R m R ), M 1 m R ( m ˆK L N, m R M1 M1 ) ] 8 / 40

43 Result for Common Sparticle Mass special case: µ = M 1 = M = m L = m R = M S M Z approximation for Σ MSSM µ is linear in y µ y µ 3π g g 1 sgn(m ) 3 g m µ M W insertion yields: a MSSM µ 1 3 m µ M S g1 + sgn(m ) 5 g g1 sgn(m ) 3 g (1000 GeV) M S { 7, für M > 0 5, für M < 0 How can a positive sign be achieved? 9 / 40

44 Outline 1 Anomalous Magnetic Moment of the Muon Minimal Supersymmetric Standard Model for tan β = 3 Calculation and Approximation 4 Identification of Viable Parameter Regions 30 / 40

45 Scaling Behavior rescaling of the mass parameters by common factor k > 0 leads to: y µ (k µ, k M 1, k M, k m L, k m R ) y µ (µ, M 1, M, m L, m R ) a MSSM µ (k µ, k M 1, k M, k m L, k m R ) 1 k amssm µ (µ, M 1, M, m L, m R ) main task: generate a positive a MSSM µ 31 / 40

46 Possible Parameter Scenarios reminder: a MSSM µ 3π y µ g m µ M W [ sgn(m ) g 1 µ ˆK W, M µ M ml ml + g1 1 µ ˆK N, M 1 µ M 1 ml ml g1 1 µ ˆK N, M 1 µ M 1 mr mr ) ] + g1 µ M 1 m L m R ( m ˆK L N, m R M1 M1 y µ > 0 y µ < 0 M < 0 scenario 1 scenario M > 0 scenario 4 scenario 3 3 / 40

47 Benchmark Point for Scenario 1 µ M 1, m L, m R for dominance of BLR contribution m L = m R for symmetry reasons P1b : (µ, M 1, M, m L, m R ) = (3000, 600, 3000, 600, 600) GeV with y µ (P1b) 0,11, a MSSM µ (P1b) 3, a Μ MSSM BLR HW HBL & HBR total / 40

48 Restrictions from the Tau Sector requirement for non tachyonic staus: yτ < g µ MW ( 0 < 0,397 Î ( m L,τ + M Z ( 1 s W ) µ, M + 0,038 m L,τ m Î L,τ )) (m R,τ + M Z s W µ M, 1 m L,τ m L,τ + µ m L,τ [ 0,076 Î ( ) ml,τ m R,τ (174,1 GeV) µ ( 0,076 Î µ, m R,τ ) M 1, m L,τ m R,τ m R,τ ) M 1 m R,τ ] 174,1 GeV m R,τ 34 / 40

49 Restrictions from the Tau Sector requirement for perturbation theory: 0 < 1,407 Î ( yτ < 4 π ( ) µ, M + 0,134 m L,τ m Î L,τ µ m L,τ, ) ( M 1 0,69 m Î L,τ ( µ + 0,69 Î m L,τ µ m R,τ, ) M 1, m L,τ 1 m R,τ m R,τ ) M 1 m R,τ 34 / 40

50 Parameter Dependency: µ-m 1 -Plane a MSSM µ regions for M = 3000 GeV, m L = m R m L = 600 GeV : 35 / 40

51 Parameter Dependency: µ-m L -Plane a MSSM µ regions for M = 3000 GeV, m L = m R M 1 = 600 GeV : 36 / 40

52 Parameter Dependency: M 1 -m L -Plane a MSSM µ regions for M = 3000 GeV, m L = m R µ = 3000 GeV : 37 / 40

53 Maximization of Sparticle Masses limit case: µ while M 1 = m L = m R benchmark point: P1c : a MSSM µ 1 3 m µ m L 9 (1000 GeV) 3,7 10 ml (µ, M 1, M, m L, m R ) = (40 000, 1000, , 1000, 1000) GeV 38 / 40

54 Benchmark Points for All Scenarios µ/gev M 1 /GeV M /GeV m L /GeV m R /GeV y µ P1a ,18 P1b ,11 P1c ,03 P ,86 P ,75 P4a ,3 large mass splittings large masses compared to the normal MSSM huge y µ values compared to the Standard Model 39 / 40

55 Conclusion and Outlook MSSM for tan β = is capable of explaining the a µ discrepancy large Yukawa couplings diagrams with Higgs boson loops relevant? analysis of other observables to further constrain or exclude this model 40 / 40

56 Conclusion and Outlook MSSM for tan β = is capable of explaining the a µ discrepancy large Yukawa couplings diagrams with Higgs boson loops relevant? analysis of other observables to further constrain or exclude this model Thanks for your attention! 40 / 40