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

Sebastian Paßehra in collaboration with Helvecio Fargnolib, Christoph Gnendigerc , Dominik Stöckingerc and Hyejung Stöckinger-Kimc , arXiv:1309.0980 [hep-ph], arXiv:1311.1775 [hep-ph].

aMax Planck Institute for Physics, Munich, bUniversidade Federal de Minas Gerais, Belo Horizonte, c Institute for Nuclear and Particle Physics, TU Dresden.

DPG Spring Meeting, Mainz, 26th of March 2014 content

1 Introduction

2 Standard Model Contributions to aµ

3 MSSM Contributions to aµ

4 Numerical Results

5 Conclusions and Outlook

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 2 / 15 e g − 2 experimentally accesible: ω := ω − ω = − B · , a s c m 2 measurement: ωa 6= 0,

g − 2 anomalous magnetic moment: ⇒ a := , , µ 2

[H. N. Brown, et. al., P. R. L. 86 (2001)]

What is aµ? experimentally:

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 3 / 15 What is aµ? experimentally:

low energetic muon in an external homogeneous magnetic ﬁeld: • circular motion due to the electrical charge, e frequency: ωc = − m B, • precession due to the spin, e g frequency: ωs = − m B · 2 , e g − 2 experimentally accesible: ω := ω − ω = − B · , a s c m 2 measurement: ωa 6= 0,

g − 2 anomalous magnetic moment: ⇒ a := , , µ 2

[H. N. Brown, et. al., P. R. L. 86 (2001)] Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 3 / 15 What is aµ? theoretically:

γ(q = p2 − p1) = µ(p1) µ(p2)

" µν # µ 2 σ qν 2 ieu¯(p2) γ FE(q ) + i FM(q ) u(p1), 2mµ

FM(0) ≡ aµ,

tree-level: FM(0) = 0,

⇒ aµ induced by higher order corrections

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 4 / 15 contributions to aµ in the Standard Model QED (11 658 471.896 ± 0.008) · 10−10 [T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, arXiv:1205.5370 (2012)] hadronic vac. pol., leading order (687.53 ± 4.34) · 10−10 [M. Benayoun, P. David, L. DelBuono, F. Jegerlehner (2012)] hadronic vac. pol., higher order (−9.97 ± 0.09) · 10−10 [F. Jegerlehner and R. Szafron (2011)] hadronic light by light (10.5 ± 2.6) · 10−10 [J. Prades, E. de Rafael, A. Vainshtein, Advanced Series on Directions in High Energy Physics – Vol. 20 (2009)] electroweak (15.36 ± 0.10) · 10−10 [C. Gnendiger, D. Stöckinger and H. Stöckinger-Kim, arXiv:1306.5546 [hep-ph]]

SM −10 theory: aµ = (11 659 175.3 ± 5.1) · 10 BNL −10 experiment: aµ = (11 659 208.9 ± 6.3) · 10

[G. W. Bennett et al., Physical Review D 73 (2006)],[A. Hoecker, W. Marciano, Journal of Physics G 37 (2010)],

BNL SM −10 ⇒ aµ − aµ = (33.6 ± 8.1) · 10 (4.2σ)

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 5 / 15 MSSM contributions to aµ: one-loop corrections loops with Chargino χ˜+ or Neutralino χ˜0 and corresponding slepton χ˜− γ µ µ˜ γ µ , , µ ν˜ µ µ χ˜0

e2 m2 aχ˜i = µ µ 16π2s2 m2 w `˜

h χ˜i χ˜i i × F1(xi )C1 + mχ˜i F2(xi )C2 , [arXiv:1208.2884v2]

χ˜i with Cj : containing all couplings, some parts ∝ sgn(µ) tan β, m2 F : formfactors, depending on x = χ˜i , i i m2 `˜

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 6 / 15 MSSM contributions to aµ: 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) aµ in the MSSM Mainz, 26.03.14 7 / 15 two-loop corrections with closed fermion–sfermion–loop

γ 0 0 χ˜j χ˜i µ˜m 0 0 µ µ χ˜j χ˜i µ˜m µ µ γ

χ˜j− χ˜− χ˜j− χ˜− γ i + i γ χ˜ j− χ˜i− ν˜µ ν˜µ ν˜µ µ µ µ µ µ 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) aµ in the MSSM Mainz, 26.03.14 8 / 15 renormalization and counterterm contributions to aµ

γ χ˜0 χ˜0 j × i

× µ˜m 0 0 µ µ χ˜j χ˜i µ˜m µ µ γ

γ χ˜ χ˜ γ j− χ˜i− j− χ˜i− χ˜− χ˜ × + × γ j × i− ν˜µ ν˜µ ν˜µ µ µ µ µ µ µ

0 0 γ γ χ˜i χ˜i

+ + χ˜i− χ˜i− µ˜m µ˜m µ µ µ µ × × µ µ µ µ × × γ γ ν˜µ ν˜µ

ﬁrst two lines: absorbing divergencies, last line: ﬁnite.

full analytic results: [arXiv:1311.1775 [hep-ph]]

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 9 / 15 renormalization constants

required renormalization constants: 2 2 • mass corrections to W and Z bosons: δMW , δMZ , • mass corrections to µ, M1 and M2: δµ, δM1, δM2,

• ﬁeld corrections to γ and γ–Z–mixing: δZγ, δZZγ,

• ﬁeld corrections to Higgs ﬁelds H1 and H2: δZH1 , δZH2 , 1 necessary for δtβ = 2 tβ (δZH2 − δZH1 ),

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) aµ in the MSSM Mainz, 26.03.14 10 / 15 numerical results

benchmark scenarios:

BM1: all one-loop masses similar, BM4: M2, ML heavy, M1, ME light, µ negative,

BM1 BM4

tβ 40 50 µ [GeV] 350 -160 M1 [GeV] 150 140 M2 [GeV] 300 2000 ME [GeV] 400 200 ML [GeV] 400 2000 1L,SUSY −11 aµ [10 ] 440.2 159.8

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 11 / 15 comparison with known 2L contributions

æ MU3 ,D3 ,Q3 ,E3 ,L3 à MU ,D ,Q three diﬀerent scenarios for ò M Q3 ;MU3 = 1 TeV H tan Β L 2 running sfermion masses, photonic 2 LH a L standard value: M = 1.5 TeV.

BM1

0.10 à à ò à æ à 0.05 æ à ò æ æ æ à æ æ à ò

r 0.00 ò ò -0.05 ò ò

-0.10 103 104 105 106 M @GeVD

2L 1L,SUSY r = aµ aµ

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 12 / 15 sfermion masses

BM1 BM4 0.08 0.07 ç ç ò ò ò ò ç ç ç ç ò ò ò ç ò æ 0.06 ò æ 0.06 ç M U ò ç M U ò ç ç à M ò ô à M 0.05 D 0.04 à ìà ìà ìíà áìíàæ ìáçíàôæ áíàôæ áíàæ D ç áìíà áìíà áìíà áìíà áìíà áìíàæ áìíàæ áìíæ áíæ áôíæ áôæ ô ò ì ì ì M æ æ æ æ æ ô ô ô ç ì M ç Q ô ô ô ô ò Q 0.04 ì 0.02ô ô ç ò r ò ò r ò ç ìò ô M U3 ç M U3 àæ àæ àæ àæ àæ àæ àæ àæ áàæ áàæ áàæ áíàæ áìíàôòæ áàíôæ áàíæ á á á á á á á áí í ôí ôí ìòô ç ç 0.03 í í í ô ô ò ìò ô M D3 0.00 ô M D3 í í í í ô ô ô ò ò ì ç ç ô ô ô ô ò ì ç ò ò ì ç ç M Q3 ç M Q3 0.02ò ò ò ò ì -0.02 ç á ç á ì M E3 ç M E3 0.01 ì ç ì - ç ì ì ì í M L3 0.04ç í M L3 102 103 102 103 M @GeVD M @GeVD

2L,f f˜ 1L,SUSY r = aµ aµ standard value for sfermion masses: M = 1.5 TeV, mixing parameters: Af = 0,

logarithmic behavior for large masses, threshold around 500 GeV.

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 13 / 15 stop mixing

BM1 BM4 r 0.030 0.035 0.040 0.045 r -0.03 -0.01 0.01 0.03 0.05 6 6 024 024 -642 -642

mh0 GeV mh0 GeV

128.6@ D 128.6@ D 127.6 127.6

125.6 125.6

123.6 123.6

122.6 122.6

---640246 ---640246

2L,f f˜ 1L,SUSY ∗ r = aµ aµ , M = MQ3 = MU3 = MD3, Xt = At − µ tβ ,

dashed lines: m˜t1 < 1000 GeV (500 GeV), other sfermion masses: M = 1.5 TeV,

Higgs mass symmetric for Xt ↔ −Xt , aµ ≈ linear in Xt .

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 14 / 15 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,

MSSM −10 ⇒ error on aµ below 1 · 10 ⇒ new experiment/more precise Standard Model prediction become more valuable (stronger restrictions on MSSM parameter space)

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 15 / 15 shift by ∆α

aµ 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,

α(0) choose α(MZ ) = , ∆α is measured, 1−∆α(MZ )

1 h f 2 i h others 2i renormalization: δZγ = − 2 < Σγ MZ − < ∂p2 Σγ p . MZ

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 I / III MSSM content

extension of the SM by Supersymmetry

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 II / III input sectors squarks:

2 2 2 3 2 ∗ ! mq˜ + mq + MZ c2β(Tq − Qqsw) mq Aq − µκq Mq˜ = L , m (A − µ∗κ ) m2 + m2 + M2 c Q s2 q q q q˜R q Z 2β q w 1 κt = , κb = tβ, tβ neutralinos:   M1 0 −MZ swcβ MZ swsβ    0 M2 MZ cwcβ MZ cwsβ Y =   , −MZ swcβ MZ cwcβ 0 −µ  MZ swsβ MZ cwsβ −µ 0 charginos: √ ! M 2M s X = √ 2 W β . 2MW cβ µ

Sebastian Paßehr (MPP Munich) aµ in the MSSM Mainz, 26.03.14 III / III