Gauge mediation Messengers of SUSY breaking We will first consider a - PowerPoint PPT Presentation

Gauge mediation

Messengers of SUSY breaking We will first consider a model with N f messengers φ i , ¯ φ i Goldstino multiplet X with an expectation value: � X � = M + θ 2 F W = X ¯ φ i φ i for gauge unification, φ i and ¯ φ i should form complete GUT multiplets. The existence of the messengers shifts the coupling at the GUT scale � µ GUT δα − 1 GUT = − N m � 2 π ln M where N m = � N f i =1 2 T ( r i ) For the unification to remain perturbative we need 150 N m < ln( µ GUT /M )

Soft Masses � X � = M + θ 2 F ⇒ messenger fermion mass = M ⇒ messenger scalars mass 2 = M 2 ± F λ λ one-loop gaugino mass: M λi ∼ α i 4 π N m F M

Scalar Soft Masses two-loops squared masses for squarks and sleptons � 2 ∼ M 2 � α i M 2 F s ∼ � i λ 4 π M inserting messenger loop corrections in the one-loop sfermion mass di- agrams spoils the cancellation by destroying the relation between the couplings

RG calculation of soft masses effective Lagrangian below the messenger mass: i d 2 θ τ ( X, µ ) W α W α � L G = − 16 π Taylor expanding in the F � � i ∂τ X = M F = i ∂ ln τ F M λ = � � 2 τ ∂X 2 ∂ ln X M � � X = M � µ � � τ ( X, µ ) = τ ( µ 0 ) + i b ′ X + i b � 2 π ln 2 π ln µ 0 X b ′ is the β function coefficient including the messengers b is the β function coefficient in the effective theory (i.e. the MSSM) b ′ = b − N m So the gaugino mass is simply given by M λ = α ( µ ) 4 π N m F M

Gaugino Masses M λ = α ( µ ) 4 π N m F M the ratio of the gaugino mass to the gauge coupling is universal: M λ 1 M λ 2 M λ 3 = N m F = = α 1 α 2 α 3 M this was once thought to be a signature of gravity mediation models

Sfermion Masses consider wavefunction renormalization for the matter fields of the MSSM: d 4 θ Z ( X, X † ) Q ′† Q ′ , � L = Z is real and the superscript ′ indicates not yet canonically normalized Taylor expanding in the superspace coordinate θ 2 + � 2 � � ∂X F θ 2 + ∂ 2 Z Z + ∂Z ∂Z d 4 θ ∂X∂X † F θ 2 F † θ � ∂X † F † θ X = M Q ′† Q ′ L = � � Canonically normalizing: ∂X F θ 2 � � Q = Z 1 / 2 � 1 + ∂ ln Z X = M Q ′ � � so 2 � � � � � ∂ 2 Z ∂ ln Z ∂ ln Z ∂X † − 1 d 4 θ F θ 2 F † θ � X = M Q † Q L = 1 − � ∂X∂X † ∂X Z �

Sfermion Masses ∂ 2 ln Z � FF † m 2 Q = − � ∂ ln X∂ ln X † MM † � X = M Rescaling also introduces an A term in the effective potential from Taylor expanding the superpotential: � ∂X F θ 2 � � � 1 − ∂ ln Z QZ − 1 / 2 � W ( Q ′ ) = W � � X = M so the A term is � Z − 1 / 2 ∂ ln Z ∂W X = M F Q � ∂X ∂ ( Z − 1 / 2 Q ) � which is suppressed by a Yukawa coupling

RG Calculation √ XX † , so that Z ( X, X † ) is invariant calculate Z and replace M by under X → e iβ X . At l loops RG analysis gives ln Z = α ( µ 0 ) l − 1 f ( α ( µ 0 ) L 0 , α ( µ 0 ) L X ) where � � � � µ 2 µ 2 L 0 = ln , L X = ln µ 2 XX † 0 so ∂ 2 ln Z ∂ ln X∂ ln X † = α ( µ ) l +1 h ( α ( µ ) L X ) two-loop scalar masses are determined by a one-loop RG equation

RG Calculation at one-loop d ln µ = C 2 ( r ) d ln Z α ( µ ) π � 2 C 2 ( r ) /b ′ � � 2 C 2 ( r ) /b � α ( µ 0 ) α ( X ) Z ( µ ) = Z 0 α ( X ) α ( µ ) where � � α − 1 ( X ) = α − 1 ( µ 0 ) + b ′ XX † 4 π ln µ 2 0 � � µ 2 α − 1 ( µ ) = α − 1 ( X ) + b 4 π ln XX † So we obtain � � F Q = 2 C 2 ( r ) α ( µ ) 2 � 2 ξ 2 + N m m 2 b (1 − ξ 2 ) � 16 π 2 N m M where 1 ξ = 1+ b 2 π α ( µ ) ln( M/µ )

Gauge mediation and the µ problem electroweak sector in the MSSM needed two types of mass terms: a supersymmetric µ term: W = µH u H d and a soft SUSY-breaking b term: V = bH u H d with a peculiar relation between them b ∼ µ 2 In gauge mediated models we need 1 16 π 2 F µ ∼ m soft ∼ M

Gauge mediation and the µ problem If we introduce a coupling of the Higgses to the SUSY breaking field X , W = λXH u H d we get µ = λM , b = λ F ∼ 16 π 2 µ 2 so b is much too large A more indirect coupling W = X ( λ 1 φ 1 ¯ φ 1 + λ 2 φ 2 ¯ φ 2 ) + λH u φ 1 φ 2 + ¯ λH d ¯ φ 1 ¯ φ 2 yields a one-loop correction to the effective Lagrangian: d 4 θ λ ¯ 16 π 2 f ( λ 1 /λ 2 ) H u H d X λ � ∆ L = X † This unfortunately still gives the same, nonviable, ratio for b/µ 2

Gauge mediation and the µ problem The correct ratio can be arranged with two additional singlet fields: W = S ( λ 1 H u H d + λ 2 N 2 + λφ ¯ N ) + Xφ ¯ φ − M 2 φ then µ = λ 1 � S � , b = λ 1 F S A VEV for S is generated at one-loop F 2 1 � S � ∼ X 16 π 2 MM 2 N but F S is only generated at two-loops: F 2 16 π 2 µ M 2 1 1 F S ∼ M 2 ∼ X N (16 π 2 ) 2 M Thus, b ∼ µ 2 provided that M 2 N ∼ F X