Higher dimensional operators. in supersymmetry

Transcription

1 I. Antoniadis CERN Higher dimensional operators in supersymmetry with E. Dudas, D. Ghilencea, P. Tziveloglou Planck 2008, Barcelona

2 Outline Effective operators from new physics integrating out heavy states - higher-dimensional - higher-derivative operators (hdo) SUSY hdo - 2-derivative description - SUSY breaking Higher-dim + hdo in MSSM - generation from heavy fields - classification of dim 5 - physical consequences

3 Effective operators from new physics - Known low-energy physics: local interactions of dimension 4 renormalizable QFT predictive power - However at energies < mass of heavy particles some interactions may look non-renormalizable e.g. four-fermion Fermi interactions - Unknown new physics in the multi-tev range: local effective operators O i of dim 4 + δ i L SM + i c i M δ O(4+δ i) i i - M not far from the electroweak scale: lowest-dim operators O i can affect significantly the low energy physics

4 Integrating out heavy fields two types of higher-dim effective operators: - with two (or less) derivatives from tree-level exchanges of massive states ( µ Z µ)h 2 M 2 i ψγ µ D µ ψ M 2 2 Z µz µ 1 M 2(H µ H) 2 2 Z µz µ 1 M 2( ψγ µ ψ) 2 - higher-derivative operators (hdo) generated by: mixing with heavy states string theory DBI action, α /loop corrections

5 L = 1 2 ( φ)2 λ 1φ ( χ)2 + c( φ)( χ) M 2 χ 2 2 λ 2φ 2 χ 2 2 Integrate out the massive field χ L = 1 2 ( φ)2 λ 1φ c2 2 φ 1 M λ 2 φ 2 φ c2 M 2 ( φ)2

6 SUSY hdo General two-derivative SUSY lagrangian: 3 functions of chiral superfields φ i 1 real: Kähler potential K 2 analytic: superpotential W, gauge kinetic function f L susy = + d 4 θ K(φ i ev, φ i ) d 2 θ [ W (φ i ) + f ab (φ i )W a W b] + h.c. chiral gauge superfield W D 2 DV Higher-dim operators: encoded in power exps K = φ i ev φ i + ci jk M φ i ev φ j φ k + h.c. + W = λ ijk φ i φ j φ k + c ijkl M φi φ j φ k φ l + f ab (φ i ) = δ ab + f abi M φi + the first terms in the rhs are renormalizable

7 hdo operators : - hdo in the superpotential (a) λ ij M d 2 θ Φ i Φ j λ ij M D 2 D 2 d 4 θ Φ i D 2 Φ j - hdo in the Kähler potential (b) (c) k ij M 2 k ijk M 2... d 4 θ Φ i Φ j d 4 θ Φ i Φ jd 2 Φ k In components: Φ = z + 2θψ + θ 2 F (a) contains ψ ψ, F z (b) contains z 2, ψ ψ, F F higher-derivative kinetic terms propagating auxiliary fields

8 Example : L = d 4 θ ( Φ Φ + χ χ ) + d 2 θ (mφχ + M2 χ2 ) + h.c. Integrate out the heavy field χ L = d 4 θ [( 1 + m2 M 2 ) ] Φ Φ + m2 M 4Φ Φ + d 2 θ ( m 2 2M Φ2 + ) m2 2M 3Φ Φ + h.c.

9 Reformulate SUSY theories with hdo in terms of two-derivatives: - coupling to gravity much simpler SUSY breaking via gravity easier to study - coupling to a SUSY breaking sector can be studied by standard methods - theories with hdo ghost (super)fields Is the theory sick? No, if treated as effective at energies E M low-energy truncation of sensible high-energy theory : no ghosts in UV lowest hdo encode leading low-energy effects of new physics: integration out of heavy states

10 L = + hdo in the superpotential d 4 θ Φ Φ d 2 θ ( ± ξ Φ Φ + m 2 Φ2 + λ ) 3 Φ3 + h.c. one particle + one ghost with masses: m 2 1 m2, m ξ of order the cutoff L = d 4 θ [ Φ 1 Φ 1 Φ 2 Φ ] 2 + d 2 θ ( [ 1 2 m kpφ k Φ p λ kplφ k Φ p Φ l Φ ξ D 2 Φ ) = ( a1 a 2 b 1 b 2 unitary matrix ) ( ) Φ1 Φ 2 ] + h.c.

11 Physics of SUSY hdo: SUSY breaking can hdo trigger SUSY breaking? scalar potential not positive definite V = particles F i 2 ghosts F j 2 no if SUSY is unbroken in the absence of hdo SUSY minima are stable but V could vanish with SUSY broken However SUSY may be trivial without hdo decoupled in a non-interacting sector

12 L = d 4 θs S + Φ Φ ( ) d 2 θ m 2 S + h.c. + 1 M 2Φ Φ + λ 4M 2 ( SΦ D 2 Φ + h.c. ) equivalent to Y = 1 M D 2 Φ L = d 4 θ ( S S Φ Φ Y Y ) + d 2 θ { S ( λy 2 m 2) MΦY } + h.c. O R SUSY: ( F S = m 2 +λy 2, F Φ = My) (0, 0) vacuum: y = φ = 0 s = arbitrary F S = m 2

13 Higher-dim + hdo in MSSM Generation from heavy fields Higher-dim operators: via interactions with heavy (super)fields Example: singlet coupled to higsses in MSSM Strumia 99 ; Brignole-Casas-Espinosa-Navarro 03 Dine-Seiberg-Thomas 07 W = λσh 1 H 2 +Mσ 2 W eff = λ2 M (H 1H 2 ) 2 can raise the Higgs mass in MSSM? hdo operators: via mixing with heavy fields

14 MSSM: Higgs mixing with heavy doublets + d 4 θ 3,4 i=1,2 H i H i + ( c 1 H 1 H 3 + c 2 H 2 H 4 + h.c. ) d 2 θ (µh 1 H 2 + MH 3 H 4 ) + h.c. µ << M neglecting gauge interactions d 4 θ ( H 1 H 1 + H 2 H 2 + c2 1 M 2H 1 H 1 + c2 2 M 2H 2 H 2 + d 2 θ (µh 1 H 2 + c 1c 2 M H 1 H 2 ) + h.c. dominant at low energy 1 M gauge interactions d 4 θ ( H 2 e V D 2 e V H 1 + h.c. ) )

15 Low-energy supersymmetry: main LHC target Quantum corrections in MSSM: important to reconcile the tree-level relation m h m Z with the experimental limit m h > 114 GeV Higher-dim / hdo operators: can also affect low-energy predictions masses, couplings, interactions,... classification of dim-5 (R-parity conserving) L = L MSSM + L (5)

16 L MSSM = + d 4 θ ( ) Z 1 H 1 ev H 1 + Z 2 H 2 e V H 2 + d 2 θ ( Q λ U U H 2 Q λ D D H 1 L λ E E H 1 + µ H 1 H 2 ) + h.c. soft terms: Z i (S, S ), λ U,D,E (S), µ(s) spurion S m S θ 2 L (5) = 1 M d 2 θ [ Q U T Q Q D + Q U T L L E + λ H (H 1 H 2 ) 2 ] + h.c. + 1 M d 4 θ [ H 1 ev Q Y U U + Q Y D D e V H 2 + L Y E E e V H 2 +A D α ( B H 2 e V ) D α ( C e V H 1 ) + h.c. ] T Q,L (S), λ H (S), Y U,D,E (S, S ), A(S, S ), B(S, S ), C(S, S ) T, Y dangerous FCNC simple ansatz for their absence: T Q = t Q (S) λ U λ D T L = t L (S) λ U λ E λ F (S) = (1 + A F S)λ F Y F = y F (S, S ) λ F F : U, D, E

17 Field redefinitions: H 1 H 1 1 [ ] M D2 1 e V H M Q ρ U U H 2 H 2 1 M D2 [ 2 H 1 ev ] + 1 M Q ρ D D + 1 M L ρ E E ρ U,D,E (S), i (S, S ) T Q, T L, A, B, C = 0 Y F = y F (S ) λ F F : U, D, E L (5) F (η 1 + η 2 S)(H 1 H 2 ) 2 L (5) D (y U + z U S )H 1 ev Q λ U U + (y D + z D S )Q λ D D e V H 2 +(y E + z E S )L λ E E e V H 2 + h.c.

18 Physical consequences Higgs mass & potential: not important effect perturbativity of L (5) only η 2 can change the tree-level upper bound m h m Z marginally when m A m Z New couplings from L (5) D z F wrong Yukawas H 1 H 2 y F quartic gauge couplings A h f f detailed study under investigation

19 Conclusions/Prospects Effective actions with higher-dim/hdo: appropriate tools to parametrize our ignorance about new physics Hdo can be rewritten as standard two-derivatives adding extra (ghost) (super)fields artifact of the truncation in deriv. expansion General analysis of their effects in MSSM classification of dim 5 no significant effects to the Higgs mass but additional couplings