Two-Loop Corrections to the Muon Magnetic Moment from Fermion/Sfermion Loops in the MSSM

Transcription

1 Two-Loop Corrections to the Muon Magnetic Moment from Fermion/Sfermion Loops in the MSSM Sebastian Paßehr a in collaboration with Helvecio Fargnoli b, Christoph Gnendiger c, Dominik Stöckinger c and Hyejung Stöckinger-Kim c, arxiv: [hep-ph], arxiv: [hep-ph]. a Max Planck Institute for Physics, Munich, b Universidade Federal de Minas Gerais, Belo Horizonte, c Institute for Nuclear and Particle Physics, TU Dresden. DPG Spring Meeting, Mainz, 26th of March 2014

2 content 1 Introduction 2 Standard Model Contributions to a µ 3 MSSM Contributions to a µ 4 Numerical Results 5 Conclusions and Outlook Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

3 What is a µ? experimentally: low energetic muon in an external homogeneous magnetic field: circular motion due to the electrical charge, frequency: ω c = e m B, precession due to the spin, frequency: ω s = e m B g 2, Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

4 What is a µ? experimentally: low energetic muon in an external homogeneous magnetic field: circular motion due to the electrical charge, frequency: ω c = e m B, precession due to the spin, frequency: ω s = e m B g 2, experimentally accesible: ω a := ω s ω c = e m B g 2, 2 measurement: ω a 0, anomalous magnetic moment: a µ := g 2, 2 [H. N. Brown, et. al., P. R. L. 86 (2001)] Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

5 What is a µ? theoretically: (q = p 2 p 1 ) µ(p 1 ) µ(p 2 ) = [ ] ieū(p 2 ) µ F E (q 2 ) + i σµν q ν F M (q 2 ) u(p 1 ), 2m µ F M (0) a µ, tree-level: F M (0) = 0, a µ induced by higher order corrections Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

6 contributions to a µ in the Standard Model QED ( ± 0.008) [T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, arxiv: (2012)] hadronic vac. pol., leading order ( ± 4.34) [M. Benayoun, P. David, L. DelBuono, F. Jegerlehner (2012)] hadronic vac. pol., higher order ( 9.97 ± 0.09) [F. Jegerlehner and R. Szafron (2011)] hadronic light by light (10.5 ± 2.6) [J. Prades, E. de Rafael, A. Vainshtein, Advanced Series on Directions in High Energy Physics Vol. 20 (2009)] electroweak (15.36 ± 0.10) [C. Gnendiger, D. Stöckinger and H. Stöckinger-Kim, arxiv: [hep-ph]] theory: experiment: a SM µ = ( ± 5.1) a BNL µ = ( ± 6.3) [G. W. Bennett et al., Physical Review D 73 (2006)],[A. Hoecker, W. Marciano, Journal of Physics G 37 (2010)], a BNL µ a SM µ = (33.6 ± 8.1) (4.2σ) Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

7 MSSM contributions to a µ : one-loop corrections loops with Chargino χ + or Neutralino χ 0 and corresponding slepton χ µ µ µ,, µ ν µ µ χ 0 a χ i µ = e2 mµ 2 16π 2 sw 2 m 2 l [ ] F 1 (x i )C χ i 1 + m χ i F 2 (x i )C χ i 2, [arxiv: v2] with C χ i j : containing all couplings, some parts sgn(µ) tan β, F i : formfactors, depending on x i = m2 χ i m 2 l, Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

8 MSSM contributions to a µ : two-loop corrections 1 MSSM corrections to Standard Model diagrams, [S. Heinemeyer, D. Stöckinger, G. Weiglein, Nuclear Physics B 690 (2004)] [S. Heinemeyer, D. Stöckinger, G. Weiglein, Nuclear Physics B 699 (2004)] 2 corrections to diagrams with supersymmetric particles: photonic corrections, [G. Degrassi, G. F. Giudice, Physical Review D 58 (1998)], [P. von Weitershausen, M. Schäfer, H. Stöckinger-Kim, D. Stöckinger, Phys.Rev. D81 (2010)] corrections with SM fermions and superpartners: this talk, [H. Fargnoli, C. Gnendiger, SP, D. Stöckinger, H. Stöckinger-Kim, Phys. Lett. B726 (2013)], [H. Fargnoli, C. Gnendiger, SP, D. Stöckinger, H. Stöckinger-Kim, JHEP 1402 (2014)], Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

9 two-loop corrections with closed fermion sfermion loop χ 0 j χ 0 i µ m χ 0 j χ 0 i µ m χ j χ i + χ j χ i χ j χ i ν µ ν µ ν µ calculated by using two different methods: usual two-loop evaluation, [S. Heinemeyer, D. Stöckinger and G. Weiglein, Nucl. Phys. B 690 (2004)] special method for Barr Zee type diagrams (first compute inner one-loop; get integral representation) [S. M. Barr and A. Zee, Phys. Rev. Lett. 65 (1990)] Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

10 renormalization and counterterm contributions to a µ χ 0 j χ 0 i χ 0 j χ 0 i µm µm χ j χ i νµ χ j χ i + νµ χ j χ i νµ χ 0 i χ 0 i + µ µm µm µ χ i + µ µ νµ νµ χ i first two lines: last line: absorbing divergencies, finite. full analytic results: [arxiv: [hep-ph]] Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

11 renormalization constants required renormalization constants: mass corrections to W and Z bosons: δm 2 W, δm2 Z, mass corrections to µ, M 1 and M 2 : δµ, δm 1, δm 2, field corrections to and Z mixing: δz, δz Z, field corrections to Higgs fields H 1 and H 2 : δz H1, δz H2, necessary for δt β = 1 2 t β (δz H2 δz H1 ), on-shell renormalization of the MSSM as far as possible: SM particles on-shell both Charginos and the lightest Neutralino on-shell DR definition of the Higgs renormalization constants (i. e. DR definition of tan β) Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

12 numerical results benchmark scenarios: BM1: BM4: all one-loop masses similar, M 2, M L heavy, M 1, M E light, µ negative, BM1 BM4 t β µ [GeV] M 1 [GeV] M 2 [GeV] M E [GeV] M L [GeV] a 1L,SUSY µ [10 11 ] Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

13 comparison with known 2L contributions M U3, D3, Q3, E3, L3 M U, D, Q M Q3 ; M U3 1 TeV tan Β 2 photonic 2 L a three different scenarios for running sfermion masses, standard value: M = 1.5 TeV. r BM1 r BM M GeV / r = aµ 2L aµ 1L,SUSY M GeV Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

14 r r sfermion masses BM MGeV M U M D M Q M U3 M D3 M Q3 M E3 M L3 / r = a 2L,f f µ aµ 1L,SUSY BM MGeV M U M D M Q M U3 M D3 M Q3 M E3 M L3 standard value for sfermion masses: M = 1.5 TeV, mixing parameters: A f = 0, logarithmic behavior for large masses, threshold around 500 GeV. Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

15 stop mixing r BM1 r BM m h 0 GeV m h 0 GeV L,f f r = aµ / / aµ 1L,SUSY, M = M Q3 = M U3 = M D3, X t = A t µ t β, dashed lines: m t1 < 1000 GeV (500 GeV), other sfermion masses: M = 1.5 TeV, Higgs mass symmetric for X t X t, a µ linear in X t. Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

16 summary conclusions: two-loop corrections with a closed fermion sfermion loop have been evaluated analytically in the Feynman-diagrammatic approach logarithmically enhanced contributions by heavy sfermions possibly strongest two-loop contributions in the MSSM outlook: full two-loop calculation in progress, error on a MSSM µ below new experiment/more precise Standard Model prediction become more valuable (stronger restrictions on MSSM parameter space) Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, / 15

17 shift by α a µ proportional to α at one-loop level, considered class of two-loop corrections: renormalization of α from photon vacuum polarization including light quarks, α in the Thomson limit large intrinsic uncertainties, choose α(m Z ) = α(0) 1 α(m Z ), α is measured, [ renormalization: δz = 1 R Σ f ) M ( ] [ M 2 Z 2 Z R p 2Σ others ( p 2 )]. Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, I / III

18 MSSM content extension of the SM by Supersymmetry Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, II / III

19 input sectors squarks: ( ) m + m 2 ql q 2 + MZ 2 c 2β(Tq 3 Q q sw) 2 m q (A q µκ q M q = m q (A q µ κ q ) κ t = 1, κ b = t β, t β neutralinos: M 1 0 M Z s w c β M Z s w s β 0 M Y = 2 M Z c w c β M Z c w s β M Z s w c β M Z c w c β 0 µ, M Z s w s β M Z c w s β µ 0 charginos: ( ) M2 2MW s X = 2MW β. c β µ m 2 q R + m 2 q + M 2 Z c 2βQ q s 2 w ), Sebastian Paßehr (MPP Munich) a µ in the MSSM Mainz, III / III