Anomaly and gaugino mediation

Transcription

1 Anomaly and gaugino mediation

2 Supergravity mediation X is in the hidden sector, P l suppressed couplings SUSY breaking VEV W = ( W hid (X) + W vis ) (ψ) f = δj i ci j X X ψ j e V ψ 2 i +... Pl τ = θ Y 2π + i 4π g + i k 2 Pl X +... induced squark and gluino masses: X = + F X θ 2, ( 2 q ) i j = ci j F 2 X 2 Pl, λ = k F X Pl no reason for the c i j to respect flavor symmetries FCNCs

3 Naive Expectation Kähler function might have flavor-blind form: K = X X + ψ i e V ψ i f = Pl [ X X + ( ) ] 1 + X X ψ i e V ψ 2 i +... Pl interactions are flavor-blind but there are direct interactions induced by Planck scale (string) states which have been integrated out these interactions should not be flavor-blind they must generate Yukawa couplings

4 Extra Dimensions SUSY breaking sector separated by a distance r from the SS two sectors on different 3-branes embedded in the higher dimensional theory Interactions suppressed by e r where is the higher dimensional Planck/string scale If only supergravity states propagate in bulk then setting e a µ = 0 and Σ = must decouple the two sectors Lagrangian must have the form W = W hid + W vis f = c + f hid + f vis τwα 2 = τ hid Wα 2 2 hid + τ viswαvis 2 all interactions between the two sectors due to supergravity form of f implies a Kähler function of the form ) K = 3Pl (1 2 ln f hid+f vis 3 2 Pl

5 Integrate out Hidden Sector dropping Planck suppressed interactions in effective theory: L eff = d 4 θψ e V ψ Σ Σ 2 + d 2 θ Σ3 (m 3 0 ψ 2 + yψ 3 ) d 2 θτw α W α + h.c. i 16π where the conformal weights of the fields determine the R-charges to be R[Σ] = 2 3, R[ψ] = 0 only trace of the hidden sector is in compensator field Σ, rescale Σψ ψ, R[ψ] = 2 3 L eff = d 4 θψ e V ψ + d 2 θ( Σ m 0ψ 2 + yψ 3 ) d 2 θτw α W α + h.c. i 16π If m 0 = 0, then the theory is classically scaleand conformally invariant Σ decouples classically

6 Super-Weyl anomaly quantum corrections break scale-invariance: couplings run e.g. a two-point function has dependence on the cutoff Λ dependence on Σ spurion of conformal symmetry: ( ) G = 1 p h p Λ 2 Σ Σ h can only depend on the combination ΛΣ/ and conjugate because of the classical conformal invariance since Λ is real, only the combination Λ 2 Σ Σ/ 2 appears effects of the scaling anomaly determined by β functions and γ cutoff dependence only occurs in the Kähler function and τ if we renormalize effective theory down to scale µ we must have: L eff = ( ) d 4 θ Z µ ΛΣ, µ ψ e V ψ ΛΣ + d 2 θ yψ 3 i 16π d 2 θ τw α W α + h.c.

7 Compensator Dependence Z is real and R-symmetry-invariant, must have ) Z = Z where ( µ Λ Σ Σ = ( Σ Σ ) 1/2 for global SUSY, with Σ = Pl, axial symmetry is anomalous θ Y shifts when the ψs are re-phased due to the chiral anomaly in superconformal gravity, scale and axial anomalies vanish Σ dynamical, re-phased shift in θ Y is canceled since τ is holomorphic we have ( τ = i b 2π ln µ ΛΣ µ dependence determines that b = b )

8 SUSY breaking: gaugino mass SUSY breaking will be communicated to auxiliary supergravity fields: Σ = + F Σ θ 2 induces a θ 2 term in τ gaugino mass: λ = i τ 2τ Σ F Σ = bg2 Σ= F Σ 16π 2. this SUSY breaking mass arises through the one-loop anomaly this mechanism is known as anomaly mediation

9 SUSY breaking We can also Taylor expand Z in superspace: [ ( ) Z = Z 1 Z F Σ 2 ln µ θ 2 + F Σ θ Z F Σ 2 4 (ln µ) 2 θ θ2] Σ= 2 2 canonically normalize kinetic terms by rescaling: ψ = Z 1/2 (1 1 2 ln Z ln µ F Σ θ 2 ) Σ= ψ Using we find Zψ e V ψ = γ ln Z ln µ, β g g ln µ, β y = [ [ γ ( ln µ γ F Σ 2 2 ] θ 2 θ2 g β g + γ y β y ) FΣ 2 2 y ln µ ψ e V ψ ] θ 2 θ2 ψ e V ψ

10 Squark and Slepton asses 2 ψ = 1 4 ( γ g β g + γ y β y ) FΣ 2 2 to leading order so γ = 1 16π 2 (4 C 2 (r)g 2 ay 2 ), β g = b g3 16π 2, β y = y 16π 2 (ey 2 fg 2 ) 2 ψ = [ 1 512π 4 4 C2 (r) b g 4 + ay 2 (ey 2 fg 2 ) ] F Σ 2 2 first term is positive for asymptotically free gauge theories negative mass squared for sleptons since in the SS the U(1) Y SU(2) L gauge couplings are not asymptotically free and W (ψ) after rescaling gives trilinear interactions with coefficient A ijk = 1 2 (γ i + γ j + γ k ) y ijk F Σ

11 Trilinear Terms gauge mediation: messengers have masses anomaly mediation: the cutoff is X = X (1 + F X X θ 2 ) Λ Σ = Λ ( 1 + F Σ θ 2) mass of the regulator fields with anomaly mediation the regulator is the messenger

12 Heavy SUSY Thresholds after rescaling, the mass m of a SUSY threshold becomes mσ/ low-energy Z and τ have the following dependence: ( ) ( ) Z µ Λ Σ, m Σ Λ Σ, τ µ ΛΣ, mσ ΛΣ gaugino and sfermion masses are independent of m since m/λ has no dependence on the spurion Σ anomaly is insensitive to UV physics, completely determined by the low-energy effective theory threshold and regulator contribute with opposite signs and cancel SUSY breaking in the mass term cancellation would not complete soft masses only depend on β g, β y, γ/ g, γ/ y at weak scale, W

13 with The µ problem in order to get EWSB in the SS need µ and b terms: W = µ H u H d, V = b H u H d b µ 2 need µ soft masses, so in anomaly-mediation require µ α 4π including a coupling to spurion field Σ that directly gives a µ term: we also gives a tree-level b term F Σ W = µ Σ3 3 H u H d which is much too large b = 3 F Σ µ 12π α µ2

14 more complicated possibility: The µ problem L int = d 4 θ δ X+X H uh d ΣΣ 2 + h.c. ( ) where X is a SUSY breaking field, rescale ΣH i H i L eff int = d 4 θ δ X+X H Σ uh d Σ + h.c. assuming X = θ 2 F X, picking out the θ 2 and θ 2 θ2 terms ( ) F µ = δ X + F Σ b = δ b vanishes at tree-level if F Σ F X ( F X F Σ ) F X F Σ

15 The µ problem b term is generated at one-loop, canonically normalize the Higgs: H i = Z 1/2 i ( γ i F Σ θ 2) Σ= H i if δ α/4π, we find: b = δ F X 2 ( γu F Σ + γ d F Σ ) = O(µ 2 ) this relies on the coefficients of X and X in (*) being equal seems fine-tuned generated in 5D toy model without fine-tuning fifth (extra) dimension has a compactification radius r c

16 5D Gravity for r r c the gravitational potential is 1 r 2 3 rather than the 4D Newton potential 1 r 2 Pl static potential given by the spatial Fourier transform of the graviton propagator with zero energy exchange: V (r) d D 1 p ei p. r p 2 1 r D 3 atching the potentials at r = r c we have 2 Pl = r c 3

17 5D Vector Exchange introduce a massive vector superfield V which propagates in the 5D bulk (canonical dimension 3/2) Integrating over fifth dimension, assume the 4D effective theory has form: L = d 4 θ r c m 2 V 2 + av (X + X ) 1/2 + bv ΣΣ H 1/2 u H d 2 + h.c. first term is a mass term and V is normalized to dimension 1 2 Integrating out V and performing the usual rescaling gives with L int d 4 θ ab r c m 2 (X + X )H u H d ΣΣ 2 + h.c r c m O(1), ab O ( ) α 4π required interaction existence proof that µ problem can be solved in anomaly-mediation

18 squark and slepton masses: Slepton masses 2 ψ = [ 1 512π 4 4 C2 (r) b g 4 + ay 2 (ey 2 fg 2 ) ] F Σ 2 2 b is negative for SU(2) L and U(1) Y possible solutions: sleptons are tachyonic new bulk fields which couple leptons and the SUSY breaking fields new Higgs fields with large Yukawa couplings new asymptotically free gauge interactions for sleptons, leptons and sleptons are composite heavy SUSY violating threshold (messengers) with a light singlet consider the last possibility, sometimes known as anti-gauge mediation

19 Anti-Gauge ediation consider a singlet X and N m messengers φ and φ in s and s of SU(5) GUT with a superpotential W = λxφφ X is pseudo-flat: it gets a mass through anomaly mediation when we renormalize down to a scale X we have a Kähler term d 4 θ Z ( XX 2 Λ 2 ΣΣ ) X X scalar potential V (X) = m 2 X (X) X 2 = N m 16π λ 2 (X) [ Aλ 2 (X) C a ga(x) ] 2 F Σ 2 2 X 2 2

20 Anti-Gauge ediation If messengers have asymptotically free gauge interactions then m 2 X (X) can change sign, and X is stabilized nearby (Coleman Weinberg) X = m then F component of X is proportional to mf Σ : F X N mλ 2 16π 2 m F Σ splitting in the messenger masses is a loop effect threshold depends on light VEV extra contribution to soft masses low-energy couplings only depend on X = X Σ, F X = F X X m F Σ because of the loop factor suppression F Σ

21 Taylor Expansion in Superspace gaugino mass: λ = 1 2τ ( = 1 2τ τ F ln Σ Σ Σ= τ ln µ + τ ln X = α(µ) 4π (b N m) F Σ Pl Pl ) FΣ first term is usual anomaly mediation second term is minus the gauge mediation answer Pl hence the name Anti-Gauge ediation

22 Taylor Expansion in Superspace squark or slepton mass squared: ( ) 2 2 ψ = ln µ + ln X ln Z(µ, X ) F Σ 2 4 [ 2 Pl = 2C 2(r)b (4π) α 2 (µ) α 2 (µ) N 2 m b + (α 2 (µ) α 2 (m)) N 2 m b 2 first term is anomaly mediation term second term is minus the gauge mediation term final term is RG running induced by gaugino mass ] FΣ 2 2 Pl

23 Slepton asses 2 ψ ( = ln µ + ) 2 ln X ln Z(µ, X ) F Σ Pl [ = 2C 2(r)b (4π) α 2 (µ) α 2 (µ) N 2 m b + (α 2 (µ) α 2 (m)) N 2 m b 2 ] FΣ 2 2 Pl for the sleptons 2 ψ > 0 RG term dominates N m sufficiently large cannot m too large, higher dimension operators dominate, e.g. d 4 θ X X ψ e V ψ 2 Pl would give for m GUT we need N m 4 2 ψ = F X 2 2 Pl

24 µ Problem adding singlet S cangenerate µ and b terms d 2 θλ SH u H d + k 3 S3 + y 2 S2 X one-loop a kinetic mixing: d 4 θ ZSX + h.c. for X 0, S is massive and can be integrated out: L eff = λ y S λ y H u H d X d 4 θ X X H uh d Z ( X Pl Λ Σ ) + h.c. produces µ term at one-loop, b term at two-loops: µ = λ y 1 2 Z ln X, b = λ y Z (ln X ) 2

25 Gaugino mediation RG running from gaugino mass + mass 2 for squarks and sleptons consider models where only gauginos get masses at leading order squarks have strong gauge coupling heavier than sleptons large top Yukawa coupling and heavy stops radiative EWSB simple set up: compact extra dimension with radius r c 1 GUT gauge fields propagate in bulk SUSY breaking on brane at the other end of the fifth dimension Yukawa couplings only source of flavor violation GI mechanism suppresses FCNCs

26 Gaugino mediation SS SUSY gaugino propagating in bulk mediates SUSY breaking

27 Gaugino mediation 4D gauge coupling related to the 5D coupling by 1 F g µνf a aµν = g5 2 dx 5 F a µνf aµν, g 2 4 = g2 5 r c no chirality in 5D, minimal SUSY theory has N = 2 vector supermultiplet 4D vector + adjoint chiral: (A N, λ L, λ R, φ) (A ν, λ L ) + (φ + ia 5, λ R ) fifth component of gauge field is a scalar choose boundary conditions so: adjoint chiral supermultiplet vanishes on one 3-brane vector multiplet does not only vector supermultiplet has massless mode (independent of x 5 ) breaks SUSY N = 1

28 Gaugino mediation SUSY breaking on one brane communicated by local interactions L dx 5 d 2 θ ( ) 1 + δ(x 5 r c ) X W α W 2 α + h.c. r c λ σ µ D µ λ + F X λ λ gaugino mass generated by auxiliary fields on SUSY breaking brane λ = 1 r c F X

29 Gaugino mediation bulk gluino loops with two mass insertions give the largest contribution to the squark/slepton masses: 2 ψ g2 5 16π 2 ( FX 2 ) 2 1 r 3 c = g2 4 16π 2 2 λ, suppressed relative to gluino mass squared for r c 1 W 4D RG running µ d dµ m2 Q g2 2 λ + cg4 Tr ( ( 1) 2F m 2 i dominates by a large logarithm, ln r c W, over the 5D loop contribution ) all the soft masses are determined by gaugino masses and r c very predictive scenario