Dynamical Term in Gauge Mediation µ Pietro Slavich CERN & LAPTH Annecy ILC Physics in Florence - 12/09/2007 Based on: A. Delgado, G.F. Giudice and P.S., arXiv:0706.3873
The problem in SUSY theories µ • To give mass to both up- and down-type quarks In SUSY extensions of the SM we • To allow for a higgsino mass term must introduce two Higgs doublets with opposite hypercharge: • To cancel anomalies Higgs/higgsino mass term in the superpotential � d 2 θ H d H u L ⊃ µ There are also soft SUSY-breaking mass terms for the Higgses in the scalar potential H d | H d | 2 + m 2 H u | H u | 2 − B µ ( H d H u + h . c . ) V soft ⊃ m 2 In the MSSM, is the only superpotential term with the dimension of a mass µ The problem: if is allowed in the SUSY limit, why is it not of ? O ( M P ) µ µ
The Giudice-Masiero mechanism: is forbidden in the SUSY limit, and is generated in µ (1988) the low-energy theory by SUSY-breaking effects Parametrize the SUSY-breaking sector with a chiral superfield that acquires a vev X � X � = M + θ 2 F The SUSY-breaking spurion couples to the Higgses in a non-minimal Kahler potential � X † M + X † X � � d 4 θ H d H u M 2 + . . . L ⊃ � F 2 � � F d 2 θ H d H u + ( H d H u + h . c . ) + . . . ∼ M M � F 2 B µ F � µ ∼ F M , B µ ∼ Therefore, ∼ µ M M F In gravity-mediated SUSY-breaking is the typical soft mass m ∼ ˜ ∼ TeV M P
When the soft terms are loop-induced (GMSB, AMSB) the GM mechanism has a problem In gauge mediation the SUSY-breaking sector couples only to heavy messenger fields � φ = | κ M | 2 ± | κ F | Φ = | κ M | 2 , d 2 θ X Φ ¯ m 2 m 2 L ⊃ κ Φ , The soft masses for the MSSM fields are generated at loop level by the gauge interactions φ F f ∼ α M λ ∼ m ˜ φ 4 π M λ λ λ λ Φ f ∼ O ( α 2 ) A ˜ ˜ ˜ f f Φ f F α µ ∼ ˜ m ∼ ∼ TeV We also want 4 π M B µ F B µ ∼ (10 − 100 TeV) 2 But !!! ∼ µ M Such a huge would require an unacceptable fine tuning in the Higgs sector B µ
NMSSM alternative: generate and at the weak scale through the vev of a light singlet B µ µ � d 2 θ L ⊃ λ µ = λ � N � , B µ = λ � F N � N H d H u Is it worth the pain? a light singlet requires the introduction of several new soft terms, and it can even pick up a tadpole from the SUSY-breaking sector, destabilizing the hierarchy • Neither of these issues is too problematic in gauge mediation, where the soft terms are calculable and the SUSY-breaking scale is relatively low • Also, the singlet-doublet interaction can give a positive contribution to the lightest Higgs boson mass and help lifting it above the LEP bound Does NMSSM+GMSB result in an acceptable EWSB ?
The Higgs sector of the NMSSM Superpotential and soft SUSY-breaking interactions for the Higgses and the singlet W ⊃ λ N H d H u − k 3 N 3 � � λ A λ NH d H u − k H u | H u | 2 + ˜ H d | H d | 2 + ˜ N | N | 2 + 3 A k N 3 + h . c . m 2 m 2 m 2 V soft ⊃ ˜ tan β ≡ � H u � v 2 ≡ � H d � 2 + � H u � 2 , � H d � , Define MSSM-like parameters: B µ ≡ λ k � N � 2 − λ 2 v 2 µ ≡ λ � N � , sin 2 β − λ A λ � N � 2 | ˜ m N | , A λ , A k ≪ v � N � ≪ v In gauge mediation we have . Therefore, we get . This results in a very light scalar+pseudoscalar pair, ruled out by LEP searches.
We need some mechanism to generate sizeable soft SUSY-breaking terms for the singlet | ˜ m N | , A λ , A k ∼ O ( ˜ m ) � N � ≫ v � N � ≫ v In the limit the singlet and doublet sectors decouple from each other, and the h i tree-level masses for the two CP-odd ( ) and three CP-even ( ) neutral scalars are a i a 2 = 3 k 2 2 B µ w � N � 2 + O ( v 2 ) , m 2 sin 2 β + O ( v 2 ) , m 2 a 1 = � 2 � � � λ A λ k + 2 wA k − 1 sin 2 β Z cos 2 2 β + λ 2 v 2 sin 2 2 β − m 2 h 1 = M 2 + O ( v 4 ) , 1 1 − 4 w h 3 = 4 w − 1 m 2 h 2 = m 2 a 1 + O ( v 2 ) , m 2 m 2 a 2 + O ( v 2 ) 3 � � � m 2 1 − 8 ˜ > 1 N w ≡ where 1 + A 2 3 k
We must include radiative corrections. Defining the effective potential V e ff = V 0 + ∆ V φ i = ( H d , H u , N ) the mass matrices for CP-even and CP-odd parts of become S,P ) e ff = S,P ) 0 + ∆ M 2 √ � √ ( M 2 � ( M 2 Z Z S,P ∂ 2 ∆ V ∂ 2 ∆ V � � ij = 1 ij = 1 ∆ M 2 � ∆ M 2 � � � � � , � � S P ∂ Re φ i ∂ Re φ j ∂ Im φ i ∂ Im φ j 2 2 � � min min ∆ M 2 O ( h 4 O ( h 2 t ) t ) We keep the terms in and the terms in . We also include some Z S,P leading-logarithmic two-loop corrections controlled by the top Yukawa and strong couplings. For we agree with the code NMHDECAY ( Ellwanger, Hugonie & Gunion ) within 5 GeV m h 1 O ( h 4 In the limit of heavy singlet the dominant corrections to are just as in the MSSM: t ) m h 1 3 m 4 ln M 2 + X 2 X 4 � � � � h 1 ) 1 − loop ≃ t t t ( ∆ m 2 S − , X t = A t + λ � N � cot β 4 π 2 v 2 m 2 M 2 12 M 4 t S S A t ( M ) ≃ 0 In GMSB , and only a moderate weak-scale value is generated by RG evolution We will need a largish (~ TeV) to evade the LEP bounds on the Higgs mass M S
NMSSM+GMSB with singlet-messenger interactions: N-GMSB Φ V V The soft masses for the Higgs doublets Φ are mediated by the gauge interactions: H u,d H u,d H u,d m 2 ˜ To generate a mass for the singlet we can couple it directly to the messengers N ¯ V N Φ ¯ ¯ ¯ ¯ N N Φ Φ Φ Φ ¯ Φ Φ Φ N N N N N N Φ Φ N A λ , A k This will also generate trilinear interactions (but no mass term) at one loop ¯ ¯ Φ Φ H d N N N N N H u N Φ Φ
This model was first proposed (without a detailed study) by Giudice & Rattazzi in 1997 _ We must introduce two pairs of messenger fields in the 5 and 5 representations of SU(5) + λ N H d H u − k � ¯ Φ 1 Φ 1 + ¯ 3 N 3 � ξ N ¯ W ⊃ X Φ 2 Φ 2 + Φ 1 Φ 2 X = M + θ 2 F ( parametrizes the SUSY-breaking sector) ( Φ , ¯ A single messenger pair coupling to both and would destabilize the weak scale Φ ) X N 16 π 2 N F 2 V e ff = ξ d Φ W ⊃ X ¯ Φ Φ + ξ N ¯ Φ Φ M We must also distinguish between the doublet and triplet components of the messengers
This model was first proposed (without a detailed study) by Giudice & Rattazzi in 1997 _ We must introduce two pairs of messenger fields in the 5 and 5 representations of SU(5) 2 + λ NH d H u − k � i ¯ i ¯ κ D Φ D i Φ D i + κ T Φ T i Φ T 3 N 3 � � � ξ D ¯ Φ D 1 Φ D 2 + ξ T ¯ Φ T 1 Φ T � W ⊃ X + N i 2 i =1 X = M + θ 2 F ( parametrizes the SUSY-breaking sector) ( Φ , ¯ A single messenger pair coupling to both and would destabilize the weak scale Φ ) X N 16 π 2 N F 2 V e ff = ξ d Φ W ⊃ X ¯ Φ Φ + ξ N ¯ Φ Φ M We must also distinguish between the doublet and triplet components of the messengers
We use analytical continuation in superspace to extract the soft SUSY-breaking terms at the messenger scale from the wave function renormalization of the observable fields ( we don’ t need to explicitly compute two-loop diagrams ) The gaugino and sfermion soft masses are the same as in the usual GMSB α 2 F 2 F α i ˜ f � i m 2 M i = n c i M , f = 2 n c i C M 2 , ( n = 2) ˜ i 4 π (4 π ) 2 i The singlet-messenger interactions generate A-terms at 1-loop and scalar masses at 2-loop � F A λ = A k 1 � 2 ξ 2 D + 3 ξ 2 M , = − T 16 π 2 3 � � � 1 D + 2 m 2 8 ξ 4 D + 15 ξ 4 T + 12 ξ 2 D ξ 2 T − 16 g 2 s ξ 2 T − 6 g 2 ξ 2 D − 2 g ′ 2 ξ 2 3 ξ 2 ˜ = N T (16 π 2 ) 2 �� F 2 − 4 k 2 � 2 ξ 2 D + 3 ξ 2 T M 2 �� F 2 � 3 g 4 + 5 g ′ 4 1 � � m 2 m 2 − λ 2 � 2 ξ 2 D + 3 ξ 2 ˜ H u = ˜ H d = n T (16 π 2 ) 2 2 6 M 2
Phenomenology of the N-GMSB Three new parameters w.r.t. the usual GMSB: ξ U ≡ ξ D,T ( M GUT ) , λ , k (but no ) µ , B µ The size of the soft SUSY-breaking parameters is determined by and . We M F choose them such as to maximize the radiative correction to the light Higgs mass � • Large generates a sizeable stop mass scale F/M M S ≡ m ˜ t 1 m ˜ t 2 • Large generates a sizeable through RG evolution A t ( M S ) M M = 10 13 GeV F/M = 1 . 72 × 10 5 GeV Take and (such that ) M S ≈ 2 TeV , A t ≈ − 1 . 4 TeV Conditions on the parameters are imposed at different scales ( ) M t , M S , M , M GUT We solve the RGE of a tower of effective theories and get all the parameters at M S � H d � , � H u � � N � The EWSB conditions imposed at the scale determine and . M S v 2 = � H d � 2 + � H u � 2 Fixing as input, we can use them to determine and tan β , � N � k λ ( M S ) ξ U Two free parameters to play with: and
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