D-term Dynamical SUSY Breaking Nobuhito Maru (Keio University) with H. Itoyama (Osaka City University) arxiv: 1109.76 [hep-ph] 7/6/01 @Y ITP workshop on Field Theory & String Theory
Introduction
SUPERSYMMETRY is one of the attractive scenarios solving the hierarchy problem, but it must be broken at low energy
Dynamical SUSY breaking(dsb) is most desirable to solve the hierarchy problem F-term DSB is induced by non-perturbative effects due to nonrenormalization theorem and well studied so far D-term SUSY breaking @tree level is well known, but no known explicit model of D-term DSB as far as we know
Tree 1-loop
Plan! Introduction! Basic Idea! V eff @1-loop! Gap equation & nontrivial solution F 0 0 D0 0! induced by! Comments on model building! Summary
Basic Idea
N=1 SUSY U(N) gauge theory with an adjoint chiral multiplet L = d 4 θk ( Φ a,φ a,v ) + d θ Im 1 τ ab Φa W a a α = iλ α ( y) + δ β α D a y Φ a = φ a ( y) + θψ a y ( )W aα W b α + d θw Φ a ( ) α β Fµν a ( ) i σ µ σ ν ( y) θ β ( ) +θθf a ( y) y µ = x µ + iθσ µ θ ( ) + h.c. L U N N= N=1 partial breaking models naturally applicable ( ) F Φ ( ) = Im d 4 θtrφe V Φ + d θ 1 ( ) = Tr eφ + m F ( Φ) W Φ Φ Antoniadis, Partrouche & Taylor (1996) Fujiwara, Itoyama & Sakaguchi (005) F ( Φ) Φ a Φ W aα b W b α + ( d θw ( Φ) + h.c. ) Electric & Magnetic FI terms
N=1 SUSY U(N) gauge theory with an adjoint chiral multiplet L = d 4 θk ( Φ a,φ a,v ) + d θ Im 1 τ ab Φa W a a α = iλ α ( y) + δ β α D a y Φ a = φ a ( y) + θψ a y ( )W aα W b α + d θw Φ a ( ) α β Fµν a ( ) i σ µ σ ν ( ) +θθf a ( y) ( y) θ β y µ = x µ + iθσ µ θ ( ) + h.c. Fermion masses Important D=5 operator d θτ ( ab Φ)W aα b W α τ ( abc Φ)ψ c λ a D b + τ ( abc Φ)F c λ a λ b d θw ( Φ) Dirac mass term 1 W ( Φ a b )ψ a ψ b τ abc τ ab ( Φ) φ c
( ψ a ) 1 λ a Fermion mass terms Mixed Majorana-Dirac type masses (<F>=0 assumed) 0 4 τ abc Db 4 τ abc Db ac c W λ c ψ c + h.c. Mass matrix M F 0 4 τ 0aaD 0 4 τ 0aa D0 ac a W <D 0 >: U(1) part of D-term
if D 0 & a a W 0 m = 1 ± a W 1± 1+ a D a a W Gaugino becomes massive by nonzero <D> SUSY is broken D 4 τ 0aa D0
D-term equation of motion: D 0 = 1 ( ) g00 τ 0cd ψ d λ c + τ 0cd ψ d λ c Dirac bilinears condensation The value of <D> will be determined by the gap equation
V eff @1-loop
1-loop effective potential for D-term V 1 loop = a = d 4 k ( ) 4 ln 1 3π π a m a ( m a λ ( + ) k iε )( m a λ ( ) k iε ) ( m a k iε )( k iε ) 4 A d ( ) Δ + 1 8 Δ4 SUSY counterterm added A( d) = 3 4 γ + 1 d λ ( + )4 logλ ( + ) λ ( )4 logλ ( ) m a g aa a a W,λ ( ± ) 1 d=4 1± 1+ Δ,Δ τ D 0aa m a V c.t. = Im Λ d θw α 0 W α 0 = Im Λ ( ) D0
1-loop effective potential for D-term ( D) V 1 loop a Tree level D-term pot. + 1-loop CW pot. + counter term 1 m a 4 4 = m a c + 1 a 64π 1 3π λ + ( D) V 1 loop = c ( Δ) Δ=0 Δ 4 Δ + Λ res 8 ( )4 ( logλ + ) ( + λ )4 ( logλ ) Λ res c + β + Λ res + 1 64π = 1 3π A( d), β g 00 a a W a m a 4 τ 0aa, Λ res Im Λ ( ) a a W a m a 4 τ 0aa
Gap equation & Nontrivial solution
0 = V ( D) 1 loop Δ = Δ c + 1 64π + Gap equation Trivial solution Δ=0 is NOT lifted unlike NJL Λ res 4 Δ 1 λ ( + )3 logλ ( + ) 64π 1+ Δ +1 λ ( ± ) 1 ( 1± 1+ ) Δ { ( ) λ ( )3 ( logλ ( ) +1 )} Approximate form Δ 1: Δ ' Λ res 4c Δ 1:c + 1 64π + Λ ' res ( ) 5 64π Δ ( c < 0) 1 3π Δ log Δ ( ' Λ res > 0)
Gap equation 0 = V ( D) 1 loop Δ = Δ c + 1 64π + Λ res 4 Δ 1 λ ( + )3 logλ ( + ) 64π 1+ Δ +1 1! V 1-loop (D)! 1.0 0.5 { ( ) λ ( )3 ( logλ ( ) +1 )} 0.5 1.0 5 10 15 0 5 30 Nontrivial solution!! c +1 64π = 1, Λ res 8 = 0.001 Δ
E 0 in SUSY Trivial solution Δ=0 is NOT lifted Our SUSY breaking vac. is a local min. V(φ) 15 10 5! 0 1 3 4 5 Δ=0 φ
Metastability of our false vacuum <D> = 0 tree vacuum is not lifted check if our vacuum <D> 0 is sufficiently long-lived 15 V(φ) 10 Long-lived for m a << ΔV (m a Λ) Decay rate of the false vacuum 5 our vac. 1 3 4 5 Δφ Λ φ 4 Δφ exp ΔV exp Λ Coleman & De Luccia(1980) m a 1 m: mass of Φ, Λ: cutoff scale
F 0 0 induced by D 0 0 $ Fermion mass
D 0 0 induces nonvanishing F 0 Effective potenital up to 1-loop V = g ab a W b W 1 g abd a D b + V 1 loop + V c.t. Stationary condition δv = 0 F 0 + m 0 g 00 0 g 00 F 0 + 1 D0 + g 00 V 1 loop = 0 with δ V = 0 we further obtain m 0 g 00 0 g 00 F 0 = m 0 g 00 0 g 00 F 0 These determine the value of nonvanishing F-term
SU(N) part: Fermion masses ( holo L ) mass = 1 g 0a,a F 0 ψ a ψ a + i 4 F 0aa F 0 λ a λ a 1 a a W ψ a ψ a + 4 F 0aa D 0 ψ a λ a U(1) part: NG fermion: admixture of λ 0 and ψ 0
Comments on Model Building
Following the model of Fox, Nelson & Weiner (00), consider a N= gauge sector & N=1 matter sector in MSSM Chirality, Asymptotic freedom Take the gauge group G G SM (G =U(1):hidden sector) D=5 gauge kinetic term provides Dirac gaugino mass term d θτ ( c abc Φ)Φ W SM αa W b τ αsm abc Φ ( ) D a c b ψ SM λ SM Scalar gluon: distinct from the other proposals Gaugino masses are generated at tree level
Once gaugino masses are generated at tree level, sfermion masses are generated by RGE effects Sfermion masses @1-loop M sf C ( i R)α i π M λi log m a M λi (i = SU(3)C, SU()L, U(1)Y) Fox, Nelson & Weiner, JHEP08 (00) 035 Flavor blind No SUSY flavor & CP problems
Summary! New mechanism of DDSB proposed! Shown a nontrivial solution of the gap eq. with nonzero <D> in a self-consistent H-F approx. Our vacuum is metastable & can be made long-lived Phenomenological Application briefly discussed