Dynamical SUSY Breaking with Anomalous U(1) and the SUSY Flavor Problem

Transcription

1 Dynamical SUSY Breaking with Anomalous U(1) and the SUSY Flavor Problem Wang Kai DEPARTMENT OF PHYSICS OKLAHOMA STATE UNIVERSITY In Collaboration with Dr. K.S. Babu and Ts. Enkhbat November 25, 2003

2 1 - Outline Supersymmetry Breaking SUSY QCD & Dynamical SUSY Breaking SUSY Flavor Problem SUSY Breaking Models Involving Anomalous U(1) a Model

3 Supersymmetry(SUSY) 2 - Supersymmetry Breaking Gauge Hierarchy: Stabilize Higgs mass from Quantum Correction Non-renormalization Theorem m 2 H = λ2 t 8π 2Λ2 + λ2 t 8π 2(m2 t Λ2 ) Gauge Unification: sin 2 θ W (M Z ) from MSSM (α S (M Z )?) Consistency of String Theory No SUSY has been discovered at low energy. Spontaneously Breaking of Supersymmetry

4 How to Break SUSY Hamiltonian : H = ω 2 {Q a,q a} 0 H 0 = a 0 Q2 a 0 = 0 = a Q a 0 2 SUSY Limit : Q a 0 = 0 E = 0 UNLIKE GAUGE SYMMETRY, NON-ZERO VEV MAY NOT BREAKING SUSY Sufficient: Positive Vacuum Energy Witten Index Tr( 1) F = n 0 B n0 F, n: # of zero energy state

5 V = a F a 2 + g2 2 D2 V 0 : F a 0 or D 0 F a = W/ φ a ; D = ( a q a φ a 2 ) Moduli Space & Flatness F = 0 or D = 0: a field space solution: Flat Direction Classical Potential: Flat direction Classical Moduli Space D-Flat: Parametrizing (Luty-Taylor, 1996) F-Flat: usual W linear in fields vanish at the origin & infinity at φ Run-away vacuum Non-renormalization Theorem: Flat at the first order Vanish up to All order perturbation Classical Flat directions can only be lifted by Non-perturbative effect

6 SUSY Breaking at Tree Level O Raifeartaigh Mechanism (F -term) ψ x : Goldstino W = m 2 X F = m 2 ; V = m 4 > 0 Fayet-Illiopoulos (D-term) d 4 θ[φ +e gv Φ + 2kV ] a Shift of k in D-term

7 How to Mediate Breaking into Observable Sector SUGRA Mediated Gauge Mediated: extra matters Anomalous U(1) Mediated: both SM and hidden sector fields are charged Anomaly Mediated (suppressing the tree SUGRA contribution) Gaugino Mediated

8 Soft Breaking of SUSY No quadratic divergence L soft = m 2 Q Q 2 + m 2 ũ ũ 2 + m 2 d d 2 + m 2 L L 2 + m 2 ẽ ẽ 2 +m 2 ν ν 2 + m 2 H u H u 2 + m 2 H d H d 2 + A u Qũ c H u + A d Q dc H d + A l Lẽ c H d + A ν L ν c H u + h.c. + B µ H u H d + h.c M 1 B B M 2 Wα W α M 3 Ga Ga M SUSY O(TeV)

9 3 - Dynamical Breaking & the New Physics Scale M Pl O(10 18 GeV) M St O(10 17 GeV) M GUT O(10 16 GeV) M PQ O(10 11 GeV) M SUSY O(10 3 GeV) M EW O(10 2 GeV)

10 Higgs-type Breaking Induced Scale: Require two scales SUSY QCD & Confinement SU(N C ) quark confinement M EW M SUSY M2 PQ M Pl α N = 1 Λ M Pl e α st/2b ; ǫ αβγ Q α i Q β j Qγ k Λ2 B; Q Q ΛM b as β function coef. of SU(N c ) Applications: Technicolor & Electroweak Gauge Symmetry Breaking Dynamical SUSY Breaking

11 DSB of SUSY: How to lift the flat directions (Witten, 1980; Affleck, Dine and Seiberg, 1984,1985) Examples of Working Models SU(3) SU(2) ((3,2) Model) W tree = ǫ αβ Q α d i j L β SU(4) U(1) Model SU(2) I=3/2 representation Φ αβγ SUSY QCD with a Singlet non-chiral Usual Non-chiral theory: MQ Q may decouple and becomes Pure Super Yang-Mills theory SYM Witten Index Tr( 1) F = 0

12 Non-perturbative Correction Lift Flat Directions As a result of the Non-renormalization Theorem: Flat at the first order Vanish up to All order perturbation Classical Flat directions can only be lifted by Non-perturbative effect(instanton Generated) t Hooft Anomaly Matching Condition Superpotential: consist of linear forms of the fields Dimensional analysis plus gauge invariance is sufficient to guess the form of superpotential.

13 SUSY QCD with Singlet (Intriligator-Thomas, 1996; Izaw-Yanagida, 1996) W = SQ i Q i + S N f/nc Λ (3N C N f )/N c F M = S V = Λ 2(3N c N f )/N c S 2(N f N c )/N c For N f < N c S, V = 0 Flat Direction along F S : Run-away Vacuum

14 For N f = N c = N Classical Potential W Cl = λsm + bb + b B λ S < Λ so that Q, Non-perturbative Correction Superpotential U Lagrangian Multiplier Q won t decouple from the theory detm B B Λ 2N f = 0 W = W Cl + U (detm B B Λ 2N )

15 Conditions to be Satisfied for DSB Lift All Classical Directions Non Run-away Vacuum Broken Global R-symmetry t Hooft Anomaly Matching

16 4 - Flavor Symmetry & SUSY Flavor Problem Absence of FCNC in SM Weak Eigen-basis Mass Eigen-basis (GIM Mechanism) Unitary Transformation SUSY: Chiral Multiplets Q = Q + θq + θ 2 F D-term Splitting Problem or SUGRA Generated Mixing Solutions to Flavor Violation(FCNC) Universal soft squared mass & Identical A-term Structure to Yukawa U 1U = 1 Gauge Mediate or String Dilaton Domination Decouple the first two generations

17 String Dilaton Wislon Coupling in SYM: 1 g 2 W d 2 θ a k a TrW a αw aα + h.c. 1 g 2 W 1 = S ıθ or in string : = S + S gst 2 2 d 2 θ S k a TrW a 4 αw aα + h.c. a Chiral field S as a background field and has zero mass dimension Universal sfermions mass (gauge coupling) Gaugino Mass Question: How to Stabilize the Dilaton potential

18 Constraint from K 0 K 0 Mixing (In mass eigenbasis) sin 2 θ d ( m 2 d m 2 d ) 2 ( ) 2 30 TeV 1 m 2 d: averaged squark mass, m 2 d: mass-squared difference; sin 2 θ d: mixing angle between d 1,2 m 2 d 0 universal sin θ d 0 alignment m d 30 TeV decoupling m d

19 Low Energy Gauged Flavor Symmetry Induced D-term U(1) Froggatt-Nelson Mechanism SU(2) H U(1) : controllable D-term & B φk S SU(3) H Problems: GMSB, No Low Energy Flavor Gauge Symmetry: No Understanding of Flavor Problem Additional Discrete Flavor Symmetry Gauged or Global (from Dimensional Reduction & T Duality?)

20 existence of SUGRA: d 4 θq i Q Z Z j MPl 2 m 2 Q F Z M Pl 2 Decoupled Model (Low Energy Flavor Symmetry): with first two generations too heavy at low energy, negative masses of third generation squarks (two loop β function) which implies a VEV of q 3 breaking SU(3) C

21 5 - Anomalous U(1) Symmetry 4D Effective Theory arising from String L = d 4 θln(s + S δ GS V X ) [ ] + d 2 S θ 4 ( k a TrWa α W aα + k X TrW X W X ) + h.c. a S is the string dilaton which is a massless particle carries coupling. g 2 X = 2/k X (S + S ) k a : Kac Moody levels where a: SU(3) C, SU(2) L or U(1) Y Kac-Moody level k i Z + for non-abelian Group, k 1 positive rational number

22 [T a m,t b n] = ıf abc T c m+n + d ab mnj k j CFT s Unification: String Theory so central charges of Virasoro algebra add up implies k i should not be large. A µ A µ + µ α(x) S S + ı 2 δ GSα(x) Anomalous Symmetry TrQ i 0 Anomaly Induced Fayet-Illiopoulos term D = ( i q i Q i 2 + ξ)

23 where ξ = g2 TrQ i 192π 2 M2 Pl Green-Schwarz Anomaly Cancellation Mechanism Cancellation of anomalies by the modified transformation law of p-form potential in Chern-Simons terms. A 3 = A 2 = A 1 = A Gravity k 3 k 2 k 1 24 = δ GS A n : mixed anomalies of [SU(n)] 2 U(1) A

24 6 - msugra + Anomalous U(1) Flavor Symmetry msugra: SUSY Breaking at M Pl (For example, Polonyi) W = µ 2 (Z + β) needs to add soft-breaking when minizing Anomalous U(1) Potential Froggatt-Nelson Type: D-term splitting is proportional to m S. Solution Anomalous U(1) SUSY Breaking Extra SU(2) H symmetry: SU(2) H U(1) A (Babu & Mohapatra, 1999)

25 7 - Anomalous U(1) A Mediated SUSY Breaking (Dvali & Pomarol, 1996; Mohapatra & Riotto, 1997) Hidden sector and observable sector are both charged under Anomalous U(1). Only φ is negatively charged. W = µ φ + φ + W MSSM F φ+ = µ φ ; F φ = µ φ + ; D = ( i q i Q i 2 + φ + 2 φ 2 + ξ) where ξ = g2 TrQ i 192π 2 M2 Pl

26 V = i F i 2 + g2 2 D2 = (µ 2 ξg 2 ) φ 2 + g2 2 φ 4 + (µ 2 + ξg 2 ) φ g2 2 φ + 4 φ = ξ µ 2 /g 2 ǫm Pl φ + = 0 Fφ+ = µ φ = µ ξ µ 2 /g 2 = µ ǫm Pl Fφ = µ φ + = 0 D = µ 2 /g 2 ǫ 0.2

27 SUGRA mediated Contribution (F -term induced in Kähler potential) Soft scalar mass from F φ+ : d 4 θ φ +φ + Q Q M 2 Pl F 2 φ + M 2 Pl Q Q m 2 Q ǫ 2 µ 2 Gaugino d 2 θw α W αφ +φ M 2 Pl ǫ 2 µ String DilatonF S How to stabilize dilaton potential

28 Anomalous U(1) mediated D-term contribution: sfermion mass splitting m 2 Q i = q i µ 2 m 2 Q: qµ 2 ǫ 2 µ 2 D-term Dominate (F -term of the dilaton S depends on stabilization of dilaton potential.)

29 Flavor Violation Source in Soft Sfermion Masses D-term Splitting between Different Families Flavor Mixing from SUGRA mediated ( d 4 θq i Q j φ M Pl ) nij ( φ ) mij φ +φ + M Pl MPl 2

30 Binétruy, Dudas Model SU(N c ) U(1) A Origin of µ scale? N f < N c No Baryon A [SU(Nc )] 2 U(1) A = N f(q + q) 2 = K N δ GS Anomalous U(1) Breaking Scale ξ = k X δ GS g 2 X M Pl/2 ǫm Pl Confinement Scale Λ = M Pl e 8π2 k N S/(3N c N f )

31 (b 0 = (3N c N f )) S Λ S + ıδ GS α(x)/2 Λe ı N f(q+ q) 3Nc N f α(x) U(1) charge of Λ q Λ = N f (q + q)/(3n c N f ) and U(1) charge of detm q det = N f (q + q) gauge aq Λ b q det = 0; Dimensional Analysis a 2bN f = 3 Non-perturbative Potential W (3Nc Nf) Λ (Nc N f ) (detm) 1 Nc N f ( ) (q+ q) φ + Tr(ΛM) M Pl where M i j = Q i Q j

32 q + q = n µ ǫ n 1 Λ; F M = ǫ n ΛM Pl ǫµ M Pl Gaugino Mass from F M or F φ ǫ n F M /M Pl or ǫ n F φ /M Pl NOT ENOUGH If use Dilaton F S, F S to the scalar mass dominated.

33 8 - Summary of SUSY Flavor Problem Source of Gaugino masses? If SUGRA Dominates λλ F Z M Pl SUGRA Mediate Generated m f ( d 4 θq i Q j φ M Pl ) nij ( φ ) mij Z Z M Pl MPl 2 m 2 ij = ǫ n ij+m ij F Z M Pl 2 May Imply Gauge Mediate Domination

34 9 - Anomalous U(1) Breaking & Gauge Mediated (Dvali-Pomarol, 1998; Hisano-Kurosawa-Nomura, 2000) SM particles are neutral under U(1), to Cancel the anomalies Extra Matter Needed! W = Tr(f φ M)(Φ 1 Φ ) + WNP F M 0 φ 1 0 & F φ1 0; M 0 m λ F M / M GSW A 3 /k 3 = A 2 /k 2 : 5 = (3, 1) + (1, 2) Even if SM is charged under U(1), 5 5, No Mixed Non-Abelian Anomaly SUGRA Mediated (cannot dominate) d 4 θf f φ φ M 2 Pl m f F φ M Pl

35 Gaugino Mass & Soft Scalar Masses in GMSB ψ g g ψ M λi (t) Λ G = k iα i (t) 4π Λ G = N F X M [1 + O(F 2 /M 2 )]

36 ψ f g g f f ψ ex f g g f

37 m 2 f(t) = 2 i C f i k (αi(t)) 2 i (4π) 2 {Λ2 S + h i Λ 2 G} where Λ 2 S = N F 2 M 2[1 + O(F 2 /M 4 )] h i = k i b i [1 α2 i(t) α 2 i (0)] α i (t) = α i (0)[1 + α i(0) 4π b it] 1

38 Some Modification Hisano-Kurosawa-Nomura s Model: Local Minimal 5 0 under U(1) VEVs of 5 or 5 break SU(3) SU(2) to nothing W = Tr(f φ M)Φ m Φ n 1 + W NP W = Tr(f φ M)(Φ 1 Φ 2 ) + 5 5Φ1 + W NP

39 QUESTIONS NEED TO BE ADDRESSED SUSY Breaking? Enough Gaugino Mass Source Natural Explanation to the smallness of µ and B µ Acceptable FCNC & Right Flavor Structure in Yukawa Sector A Complete Model: Anomaly-free Symmetry

40 SU(5) Compatible Flavor Model with Lopsided Neutrino Mass Matrix U ij = L ij = ǫ 6 ǫ 5 ǫ 3 ǫ 4 ǫ 3 ǫ 3 ǫ 5 ǫ 4 ǫ 2 H u, D ij = ǫ 3 ǫ 2 ǫ 2 ǫ 3 ǫ 2 1 ǫ 1 1 ǫ 4 ǫ 3 ǫ ǫ 2 ǫ ǫ ǫ 3 ǫ 2 1 ǫp H d, νij D = ǫ 1 1 ǫ 3 ǫ 2 1 ǫ 1 1 ǫp H d, ǫa 1 H u, Yukawa Couplings 10 10H u H d + 5 1H u

41 The U(1) Charges(Froggatt-Nelson): 10 i = (3,2,0); 5 i = (1,0,0); 1 i = (1,0,0); Lepton Flavor Violation from SUGRA Mediate SUSY Breaking Kähler Potential term f i f j(s /M Pl ) n ij (S/M Pl ) m ij Z Z/MPl 2 1 ǫ ǫ 1 ǫ ǫ 3 m 2 L L = i j ǫ 1 1, m2 = ẽ ǫ 1 ǫ iẽj 2 ǫ 1 1 ǫ 3 ǫ 2 1 where ǫ = ǫ. After Charged Lepton Yukawa Matrix is diagonalized (Weak Basis to Mass Basis). Strictly Constrained from µ eγ & K 0 K 0 Mixing

42 10 - Model: SU(N c ) U(1) A W = bqqq + b q q q + q qx + (q q) 2 S + X 3 MPl 2 +qqqxs + H u H d (qqq + q q q + X 3 ) + W α W α XS +U(detM B B Λ 6 ) A 3 = A 2 = A 1 = A [SU(N c )] 2 U(1) A k 3 k 2 k 1 k N = A gravity 24 = δ GS