SUSY breaking and the MSSM
Spontaneous SUSY breaking at tree-level O’Raifeartaigh, Fayet, Iliopoulos
Spontaneous SUSY Breaking � 0 | H | 0 � > 0 implies that SUSY is broken. 2 D a D a , V = F i ∗ F i + g 2 find models where F i = 0 or D a = 0 cannot be simultaneously solved then use this SUSY breaking sector to generate the soft SUSY breaking
O’Raifeartaigh model have nonzero F -terms W O ′ R = − k 2 Φ 1 + m Φ 2 Φ 3 + y 2 Φ 1 Φ 2 3 . scalar potential is |F 1 | 2 + |F 2 | 2 + |F 3 | 2 V = | k 2 − y 3 | 2 + | mφ ∗ 3 | 2 + | mφ ∗ 2 φ ∗ 2 2 + yφ ∗ 1 φ ∗ 3 | 2 . = no solution where both F 1 = 0 and F 2 = 0 For large m , minimum is at φ 2 = φ 3 = 0 with φ 1 undetermined vacuum energy density is V = |F 1 | 2 = k 4 .
O’Raifeartaigh model Around φ 1 = 0, the mass spectrum of scalars is 0 , 0 , m 2 , m 2 , m 2 − yk 2 , m 2 + yk 2 . There are also three fermions with masses 0 , m, m. Note that these masses satisfy a sum rule for tree-level breaking Tr[ M 2 scalars ] = 2Tr[ M 2 fermions ]
O’Raifeartaigh model: One Loop For k 2 � = 0, loop corrections will give a mass to φ 1 ψ φ 3 φ 3 3 φ 2 ψ 3 φ 3 φ 3 φ 2 Figure 1: Crosses mark an insertion of yk 2 . yk 2 insertions must appear with an even number in order to preserve the orientation of the arrows flowing into the vertices correction to the φ 1 mass from the top three graphs vanishes by SUSY
O’Raifeartaigh model: One Loop bottom two graphs give � d 4 p iy 2 k 4 iy 2 k 4 − im 2 2 π 4 ( − iy 2 ) ( p 2 − m 2 ) 3 + ( iym ) 2 i 1 = ( p 2 − m 2 ) 3 , p 2 − m 2 yields a finite, positive, result |F 1 | 2 y 4 k 4 y 4 m 2 1 = 48 π 2 m 2 = . 48 π 2 m 2 the classical flat direction is lifted by quantum corrections, the potential is stable around φ 1 = 0 the massless fermion ψ 1 stays massless since it is the Nambu–Goldstone particle for the broken SUSY generator, a goldstino ψ 1 is the fermion in the multiplet with the nonzero F component.
Fayet–Iliopoulos mechanism uses a nonzero D -term for a U (1) gauge group add a term linear in the auxiliary field to the theory: L FI = κ 2 D , where κ is a constant parameter with dimensions of mass scalar potential is 2 D 2 − κ 2 D + gD � V = 1 i q i φ i ∗ φ i , and the D equation of motion gives D = κ 2 − g � i q i φ i ∗ φ i . If the φ i s have large positive mass squared terms, � φ � = 0 and D = κ 2 in the MSSM, however, squarks and sleptons cannot have superpotential mass terms
Problems Fayet–Iliopoulos and O’Raifeartaigh models set the scale of SUSY breaking by hand. To get a SUSY breaking scale that is naturally small compared to the Planck scale, M P l , we need an asymptotically free gauge theory that gets strong through RG evolution at some much smaller scale Λ ∼ e − 8 π 2 / ( bg 2 0 ) M P l , and breaks SUSY nonperturbatively can’t use renormalizable tree-level couplings to transmit SUSY breaking, since SUSY does not allow scalar–gaugino–gaugino couplings we expect that SUSY breaking occurs dynamically in a “hidden sec- tor” and is communicated by non-renormalizable interactions or through loop effects. If the interactions that communicate SUSY breaking to the MSSM (“visible”) sector are flavor-blind it is possible to suppress FCNCs
Gauge-Mediated Scenario add “messenger” chiral supermultiplets where the fermions and bosons are split and which couple to the SM gauge groups MSSM superpartners get masses through loops: �F� m soft ∼ α i 4 π M mess � If M mess ∼ �F� , then the SUSY breaking scale can be as low as � �F� ∼ 10 4 –10 5 GeV.
Gravity-Mediated Scenario interactions with the SUSY breaking sector are suppressed by powers of M P l hidden sector field X with a nonzero �F X � , then MSSM soft terms of the order m soft ∼ �F X � M P l . � �F X � ∼ 10 10 – To get m soft to come out around the weak scale we need 10 11 GeV. Alternatively, if SUSY is broken by a gaugino condensate � 0 | λ a λ b | 0 � = δ ab Λ 3 � = 0, then Λ 3 m soft ∼ P l , M 2 which requires Λ ∼ 10 13 GeV. This can, of course, be rewritten as: �F X � = Λ 3 /M P l .
Effective Lagrangian Below the M P l is: � � � j ψ i ψ j ∗ + ˆ d 4 θ X ∗ M P l ˆ b ′ ij ψ i ψ j + XX ∗ m i b ij ψ i ψ j L eff = − ˆ + h.c. M 2 � � � P l M 3 G α G α + ˆ ˆ M 2 W α W α + ˆ d 2 θ X M 1 B α B α − + h.c. 2 M P l � X d 2 θ a ijk ψ i ψ j ψ k + h.c. − M P l ˆ where G α , W α , B α , and ψ i are the chiral superfields of the MSSM, and the hatted symbols are dimensionless If � X � = �F� then � � − �F X � M 3 � ˆ G � G + ˆ M 2 � W � W + ˆ M 1 � B � L eff = B + h.c. 2 M P l � � − �F X ��F ∗ ψ j ∗ + ˆ b ij � X � j � ψ i � ψ i � m i ˆ ψ j + h.c. M 2 � P l a ijk � ψ k − �F ∗ X � − �F X � ψ i � ψ j � d 2 θ ˆ b ′ ij ψ i ψ j + h.c. M P l ˆ M P l
Assumptions assume ˆ M i = ˆ m i mδ i M , ˆ j = ˆ j we have generated a µ -term with µ ij = ˆ H u δ j b ′ δ i H d �F ∗ X � /M P l assuming a ijk = ˆ aY ijk and ˆ b ij = ˆ H u δ j bδ i ˆ H d , then soft parameters have a universal form (when renormalized at M P l ) gaugino masses are equal M �F X � M i = m 1 / 2 = ˆ M P l , the scalar masses are universal m |�F X �| 2 f = m 2 H u = m 2 H d = m 2 m 2 0 = ˆ , M 2 P l A and b terms are given by ˆ a �F X � �F X � b = A Y f = ˆ M P l Y f , b = Bµ = M P l µ . A f ˆ b ′ µ 2 and b are naturally of the same order of magnitude if ˆ b ′ are of b and ˆ the same order of magnitude
Justified Assumptions? the assumptions avoid problems with FCNCs. Since gravity is flavor- blind, it might seem that this a natural result of gravity mediation. However, the equivalence principle does not guarantee these universal terms, since nothing forbids a K¨ ahler function of the form K bad = f ( X † , X ) i j ψ † j ψ i , m i which leads directly to off-diagonal terms in the matrix ˆ j Taking µ and the four SUSY breaking parameters and running them down from the unification scale (rather than the Planck scale as one would expect) is referred to as the minimal supergravity scenario
MSugra scalar mass m 2 , gaugino mass M , A = 0 Giudice, Rattazzi, hep-ph/0606105
The goldstino Consider the fermions in a general SUSY gauge theory. Take a basis Ψ = ( λ a , ψ i ). The mass matrix is � √ � 2 g a ( � φ ∗ � T a ) i 0 √ M fermion = 2 g a ( � φ ∗ � T a ) j � W ij � eigenvector with eigenvalue zero: √ � � � D a � / 2 �F i � eigenvector is only nontrivial if SUSY is broken. The corresponding canonically normalized massless fermion field is the goldstino: � � � D a � 2 λ a + �F i � ψ i 1 Π = √ F Π where Π = � + � � D a � 2 F 2 i �F i � 2 a 2
The Goldstino masslessness of the goldstino follows from two facts. First the superpo- tential is gauge invariant, ( φ ∗ T a ) i W ∗ i = − ( φ ∗ T a ) i F i = 0 second, the first derivative of the scalar potential ∂W i ∂V ∂φ i = − W ∗ ∂φ i − g a ( φ ∗ T a ) j D a i vanishes at its minimum � ∂V ∂φ i � = �F i �� W ij � − g a ( � φ ∗ � T a ) j � D a � = 0
The Supercurrent iF Π ( σ µ ¯ Π) α + ( σ ν σ µ ψ i ) α D ν φ ∗ i − 2 ( σ ν σ ρ σ µ ¯ 1 J µ λ a ) α F a = νρ , √ α 2 iF Π ( σ µ ¯ Π) α + j µ ≡ α . terms included in j µ α contain two or more fields. supercurrent conservation: α = iF Π ( σ µ ∂ µ ¯ ∂ µ J µ Π) α + ∂ µ j µ α = 0 ( ∗ ) effective Lagrangian for the goldstino F Π (Π ∂ µ j µ + h.c. ) . L goldstino = i ¯ Π σ µ ∂ µ Π + 1 The EQOM for Π is just eqn (*) goldstino–scalar–fermion and goldstino–gaugino–gauge boson interactions allow the heavier superpartner to decay interaction terms have two deriva- tives, coupling is proportional to the difference of mass squared
Eat the Goldstino Nambu–Goldstone boson can be eaten by a gauge boson for gravity, Poincar´ e symmetry, and hence SUSY, must be a local SUSY spinor ǫ α → ǫ α ( x ): supergravity spin-2 graviton has spin-3/2 fermionic superpartner, gravitino , � Ψ α µ , which transforms inhomogeneously under local SUSY transformations: µ = − ∂ µ ǫ α + . . . . δ � Ψ α gravitino is the “gauge” particle of local SUSY transformations when SUSY is spontaneously broken, the gravitino acquires a mass by “eating” the goldstino: the other super Higgs mechanism gravitino mass: m 3 / 2 ∼ �F X � M P l
Gravitino Mass In gravity-mediated SUSY breaking, the gravitino mass ∼ m soft In gauge-mediated SUSY breaking the gravitino is much lighter than the MSSM sparticles if M mess ≪ M P l , so the gravitino is the LSP. � �F X � < 10 6 GeV For a superpartner of mass m � ψ ≈ 100 GeV, and m 3 / 2 < 1 keV the decay � ψ → ψ Π can be observed inside a collider detector
Recommend
More recommend
