Vacuum instabilities around the corner

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Vacuum instabilities around the corner destabilizing the vacuum at the SUSY scale Wolfgang Gregor Hollik Institut für Theoretische Teilchenphysik (TTP) Karlsruher Institut für

1 Vacuum instabilities around the corner destabilizing the vacuum at the SUSY scale Wolfgang Gregor Hollik Institut für Theoretische Teilchenphysik (TTP) Karlsruher Institut für Technologie (KIT) Karlsruhe Center for Particle and Astroparticle Physics Karlsruhe School for Particle and Astroparticle Physics June 26, 2015 CINVESTAV México City W. G. H. MSSM vacuum

2 Motivation and Outline I think, we have it! Rolf Heuer

3 Motivation and Outline Eureka! Archimedes

4 What do we have? [particlezoo]

5 Having pinned down the missing piece... [Hambye, Riesselmann: Phys.Rev. D55 (1997) 7255]

6 The SM phase diagram 200 Instability Top mass Mt in GeV Meta stability Stability Non perturbativity [Degrassi et al. JHEP 1208 (2012) 098] h

7 The SM Higgs sector V (Φ) = m 2 Φ Φ + λ ( Φ Φ ) 2 SU(2) doublet: Φ = (φ +, φ 0 ) Hypercharge Y Φ = 1 2

8 The SM Higgs sector V (Φ) = m 2 Φ Φ + λ ( Φ Φ ) 2 SU(2) doublet: Φ = (φ +, φ 0 ) Hypercharge Y Φ = 1 2 V develops non-trivial minimum if m 2 < 0, v/ 2 = µ/ 2λ

9 The SM Higgs sector V (Φ) = m 2 Φ Φ + λ ( Φ Φ ) 2 SU(2) doublet: Φ = (φ +, φ 0 ) Hypercharge Y Φ = 1 2 V develops non-trivial minimum if m 2 < 0, v/ 2 = µ/ 2λ spontaneous symmetry breaking, Higgs mechanism, coupling to weak bosons,...

10 The SM Higgs sector V (Φ) = m 2 Φ Φ + λ ( Φ Φ ) 2 SU(2) doublet: Φ = (φ +, φ 0 ) Hypercharge Y Φ = 1 2 V develops non-trivial minimum if m 2 < 0, v/ 2 = µ/ 2λ spontaneous symmetry breaking, Higgs mechanism, coupling to weak bosons,...

11 The SM Higgs sector V (Φ) = m 2 Φ Φ + λ ( Φ Φ ) 2 SU(2) doublet: Φ = (φ +, φ 0 ) Hypercharge Y Φ = 1 2 V develops non-trivial minimum if m 2 < 0, v/ 2 = µ/ 2λ spontaneous symmetry breaking, Higgs mechanism, coupling to weak bosons,...

12 Back again [Hambye, Riesselmann: Phys.Rev. D55 (1997) 7255]

13 Constraining the SM Higgs sector Theoretical considerations λ < 4π perturbativity λ > 0 unbounded from below, aka vacuum stability

14 Constraining the SM Higgs sector Theoretical considerations λ < 4π perturbativity λ > 0 unbounded from below, aka vacuum stability trivial at the classical (i.e. tree) level V (Φ) = m 2 Φ Φ + λ ( Φ Φ ) 2 Quantum level [courtesy of Max Zoller] dominant contribution: top quark (y t 1) dependence on the energy scale: β function!

15 Energy scale dependence Scale-independent loop-corrected effective potential Q d d Q V loop(λ i, φ, Q) = 0 Approximation for large field values evaluated at Q = φ V loop (φ) = λ(φ)φ 4, β function for coupling λ i β i (λ i ) = Q d λ i(q) d Q running of λ determines stability of the loop potential upper bound: Landau pole; lower bound: λ > 0

16 Stability, instability or metastability? [Courtesy of Max Zoller]

17 Stability, instability or metastability? [Courtesy of Max Zoller]

18 Precise analysis: up to three loops! [Zoller 2014]

19 Precise analysis: up to three loops! [Zoller 2014]

20 Stability of the Standard Model m h = 125 GeV: metastable electroweak vacuum metastability: decay time of false vacuum large instability scale around GeV SM sufficiently stable neutrino masses missing

21 Stability of the Standard Model m h = 125 GeV: metastable electroweak vacuum metastability: decay time of false vacuum large instability scale around GeV SM sufficiently stable neutrino masses missing Why New Physics?

22 Stability of the Standard Model m h = 125 GeV: metastable electroweak vacuum metastability: decay time of false vacuum large instability scale around GeV SM sufficiently stable neutrino masses missing Why New Physics? Why Supersymmetry? m h = 125 GeV: perfectly within the range metastability absolute stability Task: Do not introduce further instabilities!

23 Stability of the Standard Model m h = 125 GeV: metastable electroweak vacuum metastability: decay time of false vacuum large instability scale around GeV SM sufficiently stable neutrino masses missing Why New Physics? Why Supersymmetry? m h = 125 GeV: perfectly within the range metastability absolute stability Task: Do not introduce further instabilities! (generically difficult)

24 Supersymmetry relates bosonic and fermionic degrees of freedom: absolute zero the only non-trivial extension of Poincaré symmetry

25 Supersymmetry relates bosonic and fermionic degrees of freedom: absolute zero the only non-trivial extension of Poincaré symmetry [Particle Fever]

26 Supersymmetry relates bosonic and fermionic degrees of freedom: absolute zero the only non-trivial extension of Poincaré symmetry [Particle Fever] W MSSM = Y e ijh d L L,i Ē R,j + Y u ij H u Q L,i Ū R,j Y d ijh d Q L,i DR,j

27 Soft SUSY breaking L νmssm SUSY = ( ) q L,i m 2 Q ij q L,j + ũ ( ) R,i m 2 u ij ũr,j + d ( ) R,i m 2 d d ij R,j + l ( ) L,i m 2 l l ij L,j + ẽ ( ) R,i m 2 e ij ẽr,j [ + h d q L,i A d d ij R,j + q L,i h u A u ijũ R,j + h d l L,i A e ijẽ R,j + h. c. ] + m 2 h d h d 2 + m 2 h u h u 2 + (B µ h d h u + h. c. )

28 The Higgs sector of the MSSM and its stability Higgs potential of 2HDM type II V = m 2 11 H d H d + m 2 22 H uh u + ( m 2 12 H u H d + h.c. ) + λ 1 ( H 2 d H ) 2 λ 2 ( ) d + H 2 2 u H u ( )( + λ 3 H u H u H d H ) ( )( d + λ4 H u H d H d H 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 H d, λ tree 1 = λ tree 2 = λ tree 3 = g2 + g 2, 4 m 2 tree 22 = µ 2 + m 2 H u, λ tree 4 = g2 2, m 2 tree 12 = B µ, λ tree 5 = λ tree 6 = λ tree 7 = 0.

29 The Higgs sector of the MSSM and its stability Higgs potential of 2HDM type II V = m 2 11 H d H d + m 2 22 H uh u + ( m 2 12 H u H d + h.c. ) + λ 1 ( H 2 d H ) 2 λ 2 ( ) d + H 2 2 u H u ( )( + λ 3 H u H u H d H ) ( )( d + λ4 H u H d H d H 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 H d, λ tree 1 = λ tree 2 = λ tree 3 = g2 + g 2, 4 m 2 tree 22 = µ 2 + m 2 H u, λ tree 4 = g2 2, m 2 tree 12 = B µ, λ tree 5 = λ tree 6 = λ tree 7 = 0.

30 The Higgs sector of the MSSM and its stability Higgs potential of 2HDM type II V = m 2 11 H d H d + m 2 22 H uh u + ( m 2 12 H u H d + h.c. ) + λ 1 ( H 2 d H ) 2 λ 2 ( ) d + H 2 2 u H u ( )( + λ 3 H u H u H d H ) ( )( d + λ4 H u H d H d H 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, Haber 2003] always fulfilled in the tree Extending the tree loop corrections? [Gorbahn, Jäger, Nierste, Trine 2011]

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

32 Effective 2HDM [Gorbahn, Jäger, Nierste, Trine 2011] integrating out heavy SUSY particles requirement of large SUSY scale M SUSY M A v ew effective theory: generic 2HDM, λ i calculated from SUSY loops H H t R t L H H t L t R H χ 0 H H H collecting all SUSY contributions: λ i = λ i (tan β, µ, M 1, M 2, m 2 Q, m 2 u, m 2 d, m2 L, m 2 e, A u, A d, A e ).

33 Triviality bounds on the effective 2HDM 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,...

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

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

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

37 Calculating the 1-loop effeective potential t L h h t R t L,R h 0 u t L,R h t R,L +... h 0 u t L,R h h 0 u h 0 u dominant contribution from third generation squarks quadrilinear couplings ( Y t 2 ) trilinear coupling to a linear combination (µ Y t h d A th 0 u) series summable to an infinite number of external legs

38 Calculating the 1-loop effeective potential t L h h t R t L,R h 0 u t L,R h t R,L +... h 0 u t L,R h h 0 u h 0 u dominant contribution from third generation squarks quadrilinear couplings ( Y t 2 ) trilinear coupling to a linear combination (µ Y t h d A th 0 u) series summable to an infinite number of external legs Do not stop after renormalizable / dim 4 terms!

39 All roads lead to Rome... 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]

40 The reasoning behind The effective potential V eff (φ): average energy density

41 The reasoning behind The effective potential V eff (φ): average energy density The ground state of the theory V eff minimized: d V eff d φ = 0 φ=v

42 The reasoning behind The effective potential V eff (φ): average energy density The ground state of the theory V eff minimized: d V eff d φ = 0 φ=v Technically: generating function for 1PI n-point Green s functions: V eff (φ) = n=2 G (n) (p i = 0)φ n

43 The reasoning behind The effective potential V eff (φ): average energy density The ground state of the theory V eff minimized: d V eff d φ = 0 φ=v Technically: generating function for 1PI n-point Green s functions: V eff (φ) = n=2 G (n) (p i = 0)φ n conversely: calculate all 1PI n-point functions V eff

44 The reasoning behind The effective potential V eff (φ): average energy density The ground state of the theory V eff minimized: d V eff d φ = 0 φ=v Technically: generating function for 1PI n-point Green s functions: V eff (φ) = n=2 G (n) (p i = 0)φ n conversely: calculate all 1PI n-point functions V eff (with subleties)

45 Summing up external legs H H t R t L H t L H 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

46 Summing up external legs H H t R t L t L H t R H h t L t R 0 h = hd A t µ h 0 u Y t

47 Summing up external legs H H t R t L t L H t R H 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)

48 Summing up external legs H H t R t L t L H t R H 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

49 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

50 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

51 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

52 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)!

53 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

54 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) ]

55 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 branch cut at x y = 1: take real part (analytic continuation) ignore imaginary part: log(1 + y x) = 1 2 log ( (1 + y x) 2) always bounded from below minimum independent of Higgs parameters from tree potential minimum determined by SUSY scale parameters

56 Tree, loop and tree + loop 0.5 (a) V (x) x

57 Tree, loop and tree + loop 8 (b) 6 4 V (x) x

58 Tree, loop and tree + loop 4 (c) 3 2 V (x) x

59 Tree, loop and tree + loop 0.15 (d) V (x) x

60 Cooking up phenomenologically viable parameters 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 FeynHiggs to determine light Higgs 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 H ±, m H m h include sbottom (drives minimum), take A b = 0.,

61 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

62 Discussion of stability tan β = 40 m A = 800 GeV M = 1 TeV µ = 3.83 TeV A t 1.5 TeV

63 Discussion of stability tan β = 40 m A = 800 GeV M = 1 TeV µ = 3.75 TeV A t 1.5 TeV

64 [

65 Naive exclusions: Constraint in µ-tan β tan β µ/tev M SUSY = 1TeV M SUSY = 2TeV 4 5

66 New interpretation Access to Charge and Color breaking minima µ = 3810 GeV µ = 3860 GeV µ = 3910 GeV V eff /GeV v u + ϕ u /GeV

67 New interpretation Access to Charge and Color breaking minima ( m M 2 t (h0 u, h 0 2 d ) = Q + Y t h 0 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 u 2 M 2 b(h ( m 0 u, h 0 2 d ) = Q + Y b h 0 d 2 A b h 0 d µ Y b h 0 ) u A b h0 d µy b h0 u m 2 b + Y bh 0 d 2 non-trivial behaviour of sfermions masses with Higgs vev

68 New interpretation Access to Charge and Color breaking minima ( m M 2 t (h0 u, h 0 2 d ) = Q + Y t h 0 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 u 2 M 2 b(h ( m 0 u, h 0 2 d ) = Q + Y b h 0 d 2 A b h 0 d µ Y b h 0 ) u A b h0 d µy b h0 u m 2 b + Y bh 0 d 2 non-trivial behaviour of sfermions masses with Higgs vev: m 2 b1,2 (h 0 u, h 0 d ) = m2 Q + m2 b + Y b h 0 d 2 2 ± 1 ( m 2 Q 2 m2 b )2 + 4 A b h 0 d µ Y b h 0 u 2 expand theory around new minimum: m 2 b2 < 0

69 New interpretation Access to Charge and Color breaking minima ( m M 2 t (h0 u, h 0 2 d ) = Q + Y t h 0 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 u 2 M 2 b(h ( m 0 u, h 0 2 d ) = Q + Y b h 0 d 2 A b h 0 d µ Y b h 0 ) u A b h0 d µy b h0 u m 2 b + Y bh 0 d 2 non-trivial behaviour of sfermions masses with Higgs vev: m 2 b1,2 (h 0 u, h 0 d ) = m2 Q + m2 b + Y b h 0 d 2 2 ± 1 ( m 2 Q 2 m2 b )2 + 4 A b h 0 d µ Y b h 0 u 2 expand theory around new minimum: m 2 b2 < 0 tachyonic squark mass!

70 [commons.wikimedia.org]

71 Rolling down the hill What does a tachyonic mass mean? mass second derivative: m 2 φ = 2 V/ φ 2 m 2 φ < 0 negative curvature non-convex potential: imaginary part log(1 + y x) log(m 2 φ )

72 Rolling down the hill What does a tachyonic mass mean? mass second derivative: m 2 φ = 2 V/ φ 2 m 2 φ < 0 negative curvature non-convex potential: imaginary part log(1 + y x) log(m 2 φ )

73 Rolling down the hill What does a tachyonic mass mean? mass second derivative: m 2 φ = 2 V/ φ 2 m 2 φ < 0 negative curvature non-convex potential: imaginary part log(1 + y x) log(m 2 φ )

74 A more complete picture Including colored directions V tree b = b L(M 2 Q + Y b v d 2 ) b L + b R(M 2 b + Y b v d 2 ) b R ] [ b L (µ Y b h 0 u A b v d ) b R + h. c. + Y b 2 b L 2 b R 2 + D-terms.

75 A more complete picture Including colored directions V tree b = b L(M 2 Q + Y b v d 2 ) b L + b R(M 2 b + Y b v d 2 ) b R ] [ b L (µ Y b h 0 u A b v d ) b R + h. c. + Y b 2 b L 2 b R 2 + D-terms. D-flat direction D-terms: g 2 φ 4 V D = g2 ( 1 h 0 8 u 2 h 0 d b L b R 2) 2 + g2 2 8 will always take over ( h 0 u 2 h 0 d 2 b L 2) 2 g 2 ( + 3 bl 2 b R 2) 2. 6 take e.g. b L = b R = h 0 u and h 0 d 0

76 CCB minima and the one-loop Higgs potential Preliminary results previously safe false but CCB conserving minima turn into to deep global minima with b L = b R 0 and h 0 u v u b L h u 0

77 Conclusions the Higgs potential in the SM is (un/meta)stable MSSM: multi-scalar theory, has several unwanted minima formation of new CCB conserving 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 global analysis shows more severe bounds (preliminary) squark contribution to the effective Higgs potential: only third generation squarks light A t fixed by m h, A b 0 (for simplicity) sign(a t ) = sign µ for CCB conserving minimum 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 FeynHiggs: determines right light Higgs mass new CCB bounds exist for all µ-a t sign combinations

78 Finally... Greetings from Señor Higgs (courtesy of Jens Hoff)

79 Backup Slides

Unstable vacua in the MSSM

Unstable vacua in the MSSM ongoing work starting from Phys. Rev. D 90, 035025 (2014) Markus Bobrowski, Guillaume Chalons, Wolfgang Gregor ollik, Ulrich Nierste Institut für Theoretische Teilchenphysik