Anomaly and gaugino mediation
“Supergravity” mediation X is in the hidden sector, M P l suppressed couplings W = W hid ( X ) + W vis ( ψ ) � � c i ψ j † e V ψ i + . . . δ i Pl X † X j − j f = M 2 θ YM 2 π + i 4 π k τ = g 2 + i M Pl X + . . . SUSY breaking VEV � X � = M + F X θ 2 , induced squark and gluino masses: F 2 Pl , M λ = k F X ( M 2 q ) i j = c i X j M 2 M Pl no reason for the c i j to respect flavor symmetries ⇒ FCNCs
Naive Expectation K¨ ahler function might have flavor-blind form: K = X † X + ψ i † e V ψ i ⇒ � � � � ψ i † e V ψ i + . . . 1 + X † X 1 X † X + f = − 3 + M 2 M 2 Pl 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
Extra Dimensions SUSY breaking sector separated by a distance r from the MSSM two sectors on different 3-branes embedded in the higher dimensional theory Interactions suppressed by e − Mr where M is the higher dimensional Planck/string scale If only supergravity states propagate in bulk then setting e a µ = 0 and Σ = M must decouple the two sectors Lagrangian must have the form W = W hid + W vis f = c + f hid + f vis τW 2 τ hid W 2 α 2 hid + τ vis W 2 = α α vis all interactions between the two sectors due to supergravity form of f implies a K¨ ahler function of the form � � 1 − f hid + f vis K = − 3 M 2 Pl ln 3 M 2 Pl
Integrate out Hidden Sector dropping Planck suppressed interactions in effective theory: � � d 4 θψ † e V ψ Σ † Σ M 3 ( m 0 ψ 2 + yψ 3 ) d 2 θ Σ 3 L eff = M 2 + � i d 2 θτW α W α + h.c. − 16 π 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 Σ ψ M → ψ , R [ ψ ] = 2 3 � � d 4 θψ † e V ψ + M m 0 ψ 2 + yψ 3 ) d 2 θ ( Σ L eff = � i d 2 θτW α W α + h.c. − 16 π If m 0 = 0, then the theory is classically scaleand conformally invariant Σ decouples classically
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: � � p 2 M 2 1 G = p 2 h Λ 2 Σ † Σ h can only depend on the combination ΛΣ /M and conjugate because of the classical conformal invariance since Λ is real, only the combination Λ 2 Σ † Σ /M 2 appears effects of the scaling anomaly determined by β functions and γ cutoff dependence only occurs in the K¨ ahler function and τ if we renormalize effective theory down to scale µ we must have: � � � ψ † e V ψ µM ΛΣ , µM d 4 θ Z L eff = ΛΣ † � � d 2 θ yψ 3 − i d 2 θ τW α W α + h.c. + 16 π
Compensator Dependence Z is real and R -symmetry-invariant, must have � � µM Z = Z Λ | Σ | where � � 1 / 2 Σ † Σ | Σ | = for global SUSY, with Σ = M Pl , axial symmetry is anomalous θ YM shifts when the ψ s are re-phased due to the chiral anomaly in superconformal gravity, scale and axial anomalies vanish Σ dynamical, re-phased ⇒ shift in θ YM is canceled since τ is holomorphic we have � � τ = i � µM b 2 π ln ΛΣ µ dependence determines that � b = b
SUSY breaking: gaugino mass SUSY breaking will be communicated to auxiliary supergravity fields: � Σ � = M + F Σ θ 2 induces a θ 2 term in τ ⇒ gaugino mass: � � bg 2 i ∂τ 16 π 2 F Σ Σ= M F Σ = M λ = M . � 2 τ ∂ Σ this SUSY breaking mass arises through the one-loop anomaly this mech- anism is known as anomaly mediation
SUSY breaking We can also Taylor expand Z in superspace: � � � � � � F † |F Σ | 2 M θ 2 + ∂ 2 Z M ¯ M 2 θ 2 ¯ F Σ Z − 1 ∂Z θ 2 + 1 θ 2 Z = Σ � ∂ (ln µ ) 2 2 ∂ ln µ 4 Σ= M canonically normalize kinetic terms by rescaling: M θ 2 � � Z 1 / 2 � � F Σ ψ ′ 1 − 1 ∂ ln Z = Σ= M ψ � 2 ∂ ln µ Using ∂g ∂y γ ≡ ∂ ln Z ∂ ln µ , β g ≡ ∂ ln µ , β y ≡ ∂ ln µ we find � θ 2 � |F Σ | 2 Zψ † e V ψ ψ ′† e V ψ ′ M 2 θ 2 ¯ ∂γ 1 + 1 = 4 ∂ ln µ � � � θ 2 � |F Σ | 2 ψ ′† e V ψ ′ M 2 θ 2 ¯ ∂γ ∂g β g + ∂γ 1 + 1 = ∂y β y 4
Squark and Slepton Masses � � |F Σ | 2 ∂γ ∂g β g + ∂γ ψ = − 1 M 2 ∂y β y � 4 M 2 to leading order 16 π 2 (4 C 2 ( r ) g 2 − ay 2 ) , β g = − b g 3 16 π 2 ( ey 2 − fg 2 ) y 1 γ = 16 π 2 , β y = so � � |F Σ | 2 4 C 2 ( r ) b g 4 + ay 2 ( ey 2 − fg 2 ) M 2 1 ψ = 512 π 4 M 2 � first term is positive for asymptotically free gauge theories negative mass squared for sleptons since in the MSSM the U (1) Y and SU (2) L gauge couplings are not asymptotically free W ( ψ ) after rescaling gives trilinear interactions with coefficient A ijk = 1 2 ( γ i + γ j + γ k ) y ijk F Σ M
Trilinear Terms gauge mediation: messengers have masses � M X θ 2 � 1 + F X � X � = M X anomaly mediation: the cutoff is � M θ 2 � 1 + F Σ Λ Σ M = Λ mass of the regulator fields with anomaly mediation the regulator is the messenger
Heavy SUSY Thresholds after rescaling, the mass m of a SUSY threshold becomes m Σ /M low-energy Z and τ have the following dependence: � � � � Λ | Σ | , | m || Σ | µM µM ΛΣ , m Σ Z , τ Λ | Σ | 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, M W
The µ problem in order to get EWSB in the MSSM need µ and b terms: W = µ H u H d , V = b H u H d with b ∼ µ 2 need µ ∼ soft masses, so in anomaly-mediation require α F Σ µ ∼ 4 π M including a coupling to spurion field Σ that directly gives a µ term: W = µ Σ 3 M 3 H u H d we also gives a tree-level b term 3 F Σ M µ ∼ 12 π α µ 2 b = which is much too large
The µ problem more complicated possibility: � d 4 θ δ X + X † H u H d ΣΣ † L int = ( ∗ ) M 2 + h.c. M where X is a SUSY breaking field, rescale Σ H i → H i M � d 4 θ δ X + X † H u H d Σ † L eff int = Σ + h.c. M θ 2 and θ 2 ¯ θ 2 terms ⇒ assuming � X � = θ 2 F X , picking out the ¯ � � F † F † µ = δ M + X Σ M � � F † F † F X F Σ b = δ M − Σ X M M M b vanishes at tree-level if F Σ ∝ F X
The µ problem b term is generated at one-loop, canonically normalize the Higgs: M θ 2 � � � � Z 1 / 2 1 − 1 F Σ H ′ = 2 γ i Σ= M H i � i i if δ ∼ α/ 4 π , we find: � � F † γ u F Σ M + γ d F Σ = O ( µ 2 ) b = δ X 2 M M 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
5D Gravity for r ≪ r c the gravitational potential is 1 r 2 M 3 rather than the 4D Newton potential 1 rM 2 Pl static potential given by the spatial Fourier transform of the graviton propagator with zero energy exchange: � d D − 1 p e i� p.� r 1 V ( r ) ∼ ∼ r D − 3 p 2 � Matching the potentials at r = r c we have M 2 Pl = r c M 3
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: � d 4 θ r c m 2 V 2 + aV ( X + X † ) M 1 / 2 + M 1 / 2 H u H d ΣΣ † bV L = M 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 � r c m 2 ( X + X † ) H u H d ΣΣ † d 4 θ ab L int ∼ M 2 + h.c. + . . . with � α � r c m ∼ O (1) , ab ∼ O 4 π → required interaction existence proof that µ problem can be solved in anomaly-mediation
Slepton masses squark and slepton masses: � � |F Σ | 2 4 C 2 ( r ) b g 4 + ay 2 ( ey 2 − fg 2 ) M 2 1 ψ = � 512 π 4 M 2 b is negative for SU (2) L and U (1) Y ⇒ sleptons are tachyonic possible solutions: • 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”
Anti-Gauge Mediation 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 K¨ ahler term � � � XX † M 2 d 4 θ Z X † X Λ 2 ΣΣ † scalar potential m 2 X ( X ) | X | 2 V ( X ) = � � |F Σ | 2 N m 16 π 2 λ 2 ( X ) Aλ 2 ( X ) − C a g 2 M 2 | X | 2 = a ( X )
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