Unstable vacua in the MSSM

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1 Unstable vacua in the MSSM ongoing work starting from Phys. Rev. D 90, (2014) Markus Bobrowski, Guillaume Chalons, Wolfgang Gregor ollik, Ulrich Nierste Institut für Theoretische Teilchenphysik (TTP) Karlsruher Institut für Technologie (KIT) LPSC Grenoble DPG Frühjahrstagung 2015 Wuppertal March 9, 2015 W. G.. MSSM vacuum

2 Motivation and outline discovery of iggs boson [ATLAS, CMS: 4 th July 2012] m h = 126 GeV: light SUSY (MSSM) iggs severly constrains any SUSY parameter space Stability of the scalar potential: electroweak vacuum ground state of the theory (global minimum) further minima vacuum decay or stable vacuum This talk new class of constraints from vacuum stability on SUSY (iggs) parameters using 1-loop effective potential access to charge and color breaking deeper minimum

3 The iggs sector of the MSSM and its stability iggs potential of 2DM type II V = m 2 11 d d + m 2 22 u u + ( m 2 12 u d + h.c. ) + λ 1 ( 2 d ) 2 λ 2 ( ) d u u ( )( + λ 3 u u d ) ( )( d + λ4 u d d u) + {λ5, λ 6, λ 7 } In the MSSM: tree potential calculated from D-terms and L soft m 2 tree 11 = µ 2 + m 2 d, λ tree 1 = λ tree 2 = λ tree 3 = g2 + g 2, 4 m 2 tree 22 = µ 2 + m 2 u, λ tree 4 = g2 2, m 2 tree 12 = B µ, λ tree 5 = λ tree 6 = λ tree 7 = 0.

4 The iggs sector of the MSSM and its stability iggs potential of 2DM type II V = m 2 11 d d + m 2 22 u u + ( m 2 12 u d + h.c. ) + λ 1 ( 2 d ) 2 λ 2 ( ) d u u ( )( + λ 3 u u d ) ( )( d + λ4 u d d u) + {λ5, λ 6, λ 7 } Unbounded from below requirements λ 1 > 0, λ 2 > 0, λ 3 > λ 1 λ 2 and others... [Gunion, aber 2003] always fulfilled in the tree Extending the tree loop corrections? [Gorbahn, Jäger, Nierste, Trine 2011]

5 Recovery from being unbounded from below??? V (φ) = µ 2 φ 2 + λφ 4

6 Recovery from being unbounded from below??? V (φ) = µ 2 φ 2 λφ 4

7 Recovery from being unbounded from below??? V (φ) = µ 2 φ 2 λφ 4 +λ (6) φ 6

8 Calculating the 1-loop effeective potential t L t R t L,R t L,R + t R,L t L,R quadrilinear couplings ( Y t 2 ) trilinear coupling to a linear combination (µ Y t h d A th 0 u)

9 Features of the resummed series V 1 (h 0 u, h 0 d ) = N cm 4 [ 32π 2 (1 + y + x) 2 log(1 + y + x) + (1 + y x) 2 log(1 + y x) ] 3(x 2 + y 2 + 2y) x 2 = µy t 2 h h M 4, h = h 0 d A t µ h 0 Y u, y = Y th 0 u 2 t M 2 M = m 2 Q = m 2 t = M SUSY branch cut at x y = 1: take real part (analytic continuation) always bounded from below minimum independent of iggs parameters from tree potential minimum determined by SUSY scale parameters

10 Features of the resummed series M SUSY = 1 TeV, A t = 1.5 TeV, µ = 3 TeV, tan β = V1(h 0 u, vd)/gev h 0 u/gev

11 Implementing the electrowek vacuum Minimum at the electroweak scale v = 246 GeV m 2 tree 11 = m 2 tree 12 tan β v2 2 cos(2β)λtree 1 1 v cos β m 2 tree 22 = m 2 tree 12 cot β + v2 2 cos(2β)λtree 1 1 v sin β m h = 126 GeV δ V 1 δφ d δ V 1 δφ u φu,d 0 χ u,d 0 φu,d 0 χ u,d 0 using Feyniggs to determine light iggs mass by adjusting A t (several solutions: sign A t = sign µ) connection to potential: m A pseudoscalar mass m A less dependent on higher loops decoupling limit: m A, m ±, m m h include sbottom (drives minimum), take A b = 0.,

12 One-loop effective potential with charge and color conserving second minimum µ = 3810 GeV µ = 3860 GeV µ = 3910 GeV V eff /GeV v u + ϕ u /GeV tan β = 40, m A = 800 GeV, M SUSY = 1 TeV A t 1.8 TeV

13 Constraint in µ-tan β tan β µ/tev M SUSY = 1TeV M SUSY = 2TeV 4 5

14 Conclusions formation of new minima at the 1-loop level stability of the electroweak vacuum: bounds on µ tan β instability of electroweak vacuum by second minimum in standard model direction v u : global CCB minimum squark contribution to the effective potential: only third generation squarks light A t fixed by m h, A b 0 (sign A t = sign µ) free parameters: M SUSY = m t = m b, tan β, µ no new insights if m t L, b L m t R, b R gluino and electroweak gauginos heavy bottom resummation: higgsino contribution! Greetings from Señor iggs (courtesy of Jens off)

15 Backup Slides

16 tan β resummation for bottom yukawa coupling Yukawa coupling not given directly by the mass m b y b = v d (1 + b ) gluino b higgsino = 2α s 3π µm G tan βc 0 ( m b1, m b2, M G), b = Y t 2 16π 2 µa t tan βc 0 ( m t 1, m t 2, µ). b gluino b higgsino Μ MG GeV 0.2

17 Check versus known results 1-loop effective potential [Coleman, Weinberg 1973] V 1 (h u, h d ) = 1 [ ( M 2 ) 64π 2 STr (h u, h d ) M4 (h u, h d ) ln Q 2 3 ] 2 field dependent mass M(h u, h d ) STr accounts for spin degrees of freedom same result can be obtained by the tadpole method T h V 1(h) V 1 (h) dh T (h) [Lee, Sciaccaluga 1975] functional methods: effective potential for arbitrary number of scalars: V 1 (φ 1, φ 2,... φ n ) [Jackiw 1973]

18 Summing up external legs t R t L t L t R most dominant contribution from top Yukawa y t and A t can be easily summed for m t R = m t L M 1-PI potential as generating function for 1-PI Green s functions V 1 PI (φ) = Γ 1 PI (φ) = n 1 n! G n(p ext = 0)φ n classical field value φ 0 φ 0 dv (φ) dφ = 0 determines ground state of the theory

19 Summing up external legs t R t L t L t R h V 1 n t L 0 h = hd A t µ h 0 u Y t t R a ( ) n n n! 2 h a n h, n! 2 = 1 n(n 1)(n 2) V 1 = N cm 4 32π 2 [ (1 + x) 2 log(1 + x) + (1 x) 2 log(1 x) 3x 2] Q 2 = M 2 x 2 = µy t 2 h h/m 4, m 2 t L = m 2 t R = M 2

20 Summing up (ctd.) Field dependent stop mass ( m 2 t M 2 t (h0 u, h 0 d ) = + Y t h 0 L u 2 A t h 0 u µ Y t h 0 d A t h 0 u µyt h 0 d m 2 t + Y t h 0 R u 2 ) trilinear h(h 0 d, h0 u), quadrilinear h 0 u 2 diagrams with mixed contributions h 0 u h 0 u t L,R t L,R

21 Summing up (ctd.) Field dependent stop mass ( m 2 t M 2 t (h0 u, h 0 d ) = + Y t h 0 L u 2 A t h 0 u µ Y t h 0 d A t h 0 u µyt h 0 d m 2 t + Y t h 0 R u 2 ) trilinear h(h 0 d, h0 u), quadrilinear h 0 u 2 diagrams with mixed contributions h 0 u h 0 u t L,R t L,R h t R,L h

22 Summing up (ctd.) Field dependent stop mass ( m 2 t M 2 t (h0 u, h 0 d ) = + Y t h 0 L u 2 A t h 0 u µ Y t h 0 d A t h 0 u µyt h 0 d m 2 t + Y t h 0 R u 2 ) trilinear h(h 0 d, h0 u), quadrilinear h 0 u 2 diagrams with mixed contributions h 0 u h 0 u t L,R t L,R h t R,L h

23 Summing up (ctd.) Field dependent stop mass ( m 2 t M 2 t (h0 u, h 0 d ) = + Y t h 0 L u 2 A t h 0 u µ Y t h 0 d A t h 0 u µyt h 0 d m 2 t + Y t h 0 R u 2 ) trilinear h(h 0 d, h0 u), quadrilinear h 0 u 2 diagrams with mixed contributions h 0 u h 0 u t L,R t L,R h t R,L h gummi bear factor (2n + k 1)! k!(2n 1)!

24 Summing up (ctd.) Field dependent stop mass ( m 2 t M 2 t (h0 u, h 0 d ) = + Y t h 0 L u 2 A t h 0 u µ Y t h 0 d A t h 0 u µyt h 0 d m 2 t + Y t h 0 R u 2 ) t L h h t R + t L,R h 0 u t L,R h t R,L +... h 0 u h 0 u h 0 u t L,R h

25 Summing up (ctd.) Field dependent stop mass ( m 2 t M 2 t (h0 u, h 0 d ) = + Y t h 0 L u 2 A t h 0 u µ Y t h 0 d A t h 0 u µyt h 0 d m 2 t + Y t h 0 R u 2 ) V 1 = n=0 k=0 n=0 k=0 a kn x 2n y k, x 2 = µy t 2 h h M 4, y = Y th 0 u 2 M 2 1 (2n + k 1)! x 2n y k n(2n + k 1)(2n + k 2) k!(2n 1)! = [ (1 + y + x) 2 log(1 + y + x) +(1 + y x) 2 log(1 + y x) 3(x 2 + y 2 + 2y) ]

26 Effective 2DM [Gorbahn, Jäger, Nierste, Trine 2011] integrating out heavy SUSY particles requirement of large SUSY scale M SUSY M A v ew effective theory: generic 2DM, λ i calculated from SUSY loops

27 Effective 2DM [Gorbahn, Jäger, Nierste, Trine 2011] integrating out heavy SUSY particles requirement of large SUSY scale M SUSY M A v ew effective theory: generic 2DM, λ i calculated from SUSY loops t R t L t L t R χ 0 collecting all SUSY contributions: λ i = λ i (tan β, µ, M 1, M 2, M 2 Q, M 2 ũ, M 2 d, M 2 L, M 2 ẽ, A u, A d, A e ).

28 Triviality bounds on the effective 2DM simple check: λ 1 > 0, λ 2 > 0, λ 3 > λ 1 λ 2, where now λ i = λ tree i + λino i + λ sferm i 16π 2. Severe UFB limits Bounds on λ 1,2,3 transfer into bounds on m 0, A t, µ,...

29 Discussion of stability V eff /GeV tan β = 40 m A = 800 GeV M = 1 TeV A t 1.5 TeV h 0 u /GeV µ = 2.55 TeV

30 Discussion of stability V eff /GeV tan β = 40 m A = 800 GeV M = 1 TeV A t 1.5 TeV h 0 u /GeV µ = 2.51 TeV