Patterns of supersymmetry breaking in moduli-mixing racetrack model

Transcription

1 13 June, SUSY 06, UC Irvine Patterns of supersymmetry breaking in moduli-mixing racetrack model Based on articles Hiroyuki Abe (YITP, Kyoto) Phys.Rev.D73 (2006) (hep-th/ ) Nucl.Phys.B742 (2006) 187 (hep-th/ ) hep-th/ with Tetsutaro Higaki and Tatsuo Kobayashi 1

2 I. Introduction Candidate of unified theory of gravity, gauge and matter fields: (higher-dimensional) SUGRA, superstring/m-theory The 4D effective theory has many moduli superfields Φ I = (φ, χ, F φ ) I. M P l : Planck scale, g φ a : gauge couplings, Y ijk : Yukawa couplings,. F φ M a : gaugino masses, m i : squark, slepton masses, A ijk. : A-terms, Thus, moduli stabilization and its effect on SUSY breaking is quite relevant to particle phenomenology/cosmology. 2

3 Sources of moduli potential (in Type IIB O3/O7 on CY 3 with h 1,1 = 1) Gaugino condensation: SU(N a ) SYM, 1/g 2 a = Re f a W GC = N a e 8π2 f a /N a { fd7 = T : Kähler modulus f D3 = S : dilaton Three-form flux: G 3 = F 3 2πiSH 3 W flux = G 3 Ω CY 3 = f RR (U α ) + S f NS (U α ) (Ω : holomolphic three-form) { S : dilaton U α f RR,NS (U α ) : functions determined by flux : complex structure moduli Gukov, Vafa, Witten, Nucl.Phys.B584 (2000) 69 Similarly, moduli potential can arise also in other models (e.g. IIA, hetero) 3

4 Two stage stabilization (KKLT-type) Kachru, Kallosh, Linde, Trivedi, Phys.Rev.D68 (2003) Choi, Falkowski, Nilles, Olechowski, Pokorski, JHEP 0411 (2004) 076,... H.A., Higaki, Kobayashi, hep-th/ By fine-tuning flux such that w 0 = W flux 1 (M P l = 1 unit) while keeping S U αw flux, U α U βw flux 1 we find... Moduli S and U α can be stabilized by flux and receive heavy mass 1 Remaining T is stabilized by GC with mass m 3/2 w 0 W GC 1 We first integrate out S, U α, and derive the effective action of T. 4

5 II. 4D effective SUGRA for single modulus Stabilization at SUSY AdS minimum: F T = e K/2 K T T (W T + K T W ) = 0 V = K I JF I F J 3e K W 2 3m 2 3/2 < 0 Flux + GC: K = 3 ln(t + T ), W = w 0 Ae at at ln(a/w 0 ) 4π 2, σ t = 0, (T = t + iσ t ) A 1, m 3/2 w (fine-tuning) Racetrack : K = 3 ln(t + T ), W = Ae at Be bt (a b)t ln(aa/bb) at bt 4π 2 m 3/2 Ae at 10 14, a b a + b ln M P l m 3/2 (fine-tuning) 5

6 Uplifting AdS to ds (Minkowski) (KKLT model) Kachru, Kallosh, Linde and Trivedi, Phys.Rev.D68 (2003) V lift = D(T + T ) n P 2 V + V lift = V ds 0 At the linear order of δ t, t = t SUSY (1 + δ t ), F T t=tsusy = 0 SUSY breaking order parameters F C flux+gc : m 3/2 w 0, C 0 Racetrack : α 1 F C T + T 4π 2 C 0 F T δ t Choi, Falkowski, Nilles, Olechowski, Nucl.Phys.B718 (2005) n P 2(at) 2 : (n P = 0 for D3) F C C 0 m 3/2 Ae at, F T T + T 2 n P 2 1 at m 3/2 F T T + T 3(2 n P ) 4 1 at bt m 3/2 at 2 1 : flux+gc ln(m P l /m 3/2 ) 2 n P at bt 4 ln(m P l /m 3/2 ) 3(2 n P ) 4π2 : racetrack 6

7 III. Moduli-mixing racetrack model Moduli-mixing gauge coupling Re f a = 1/g 2 a, S = s + iσ s, T = t + iσ t Type IIB O3/O7, magnetized D-branes: Lüst, Mayr, Richter, Stieberger, Nucl.Phys. B696 (2004) 205 f md7 = m D7 S + w D7 T f md9 = m D9 S w D9 T { md7,d9 : magnetic flux number w D7,D9 : winding number Heterotic (M-)theory: Choi, Kim, Phys.Lett.B165 (1985) 71,... Lukas, Ovrut, Waldram, Nucl.Phys. B532 (1998) 43 f H = S βt + f M5, β 1 16π 4 SU(N a ) gaugino condensation: CY J (tr F tr R2 ) W GC = N a e 8π2 f a /N a moduli-mixing superpotential 7

8 Single light modulus H.A., Higaki, Kobayashi, Phys.Rev.D73 (2006) Assuming S U αw flux = 0, W flux = 0 (S is stabilized at high scale) K = ln S + S 3 ln(t + T ) W = Ae a(m a S ±w a T ) ± Be b(m b S ±w b T ) Typical models: A = Ae am a S, B = Be bm b S W D3+mD7 = A B e bw bt : 0 < α 1 W D3+mD9 = A + B e +bw bt : 1 α < 0 W md7+md7 = A e aw at B e bw bt : 1 < α 4π 2 W md7+md9 = A e aw at + B e +bw bt : 4π 2 α < 1 α = 1 F C T + T 4π 2 C 0 F T 8

9 Implications on SUSY phenomenology/cosmology Fine-tuning w 0 1 is not necessary if W flux = 0 and Ae as exists in W GC. Destabilization/overshooting problem may be avoided in some case. V ln V t = t Modulus/anomaly ratio of SUSY breaking mediation, α can take various value depending on m, w and S = U f RR (U)/ U f NS (U). 9

10 Gaugino and sfermion masses at TeV scale Choi, Jeong, Okumura, JHEP09 (2005)

11 Two light moduli H.A., Higaki, Kobayashi, Nucl.Phys.B742 (2006) 187 Assuming S U αw flux = 0, W flux = 0 (S also remains light) K = n S ln(s + S) n T ln(t + T ) W = Ae a(s αt ) Be b(s+βt ) a, b, α, β, A, B, n S, n T : parameters of effective SUGRA SUSY stationary point: F S, F T = 0 for s SUSY n S /2a, n S /2b s SUSY 1 b a ln bb aa, t SUSY n T (b a) 2ab(α + β), (W SUSY 0) Negative vacuum energy: V SUSY = 3(m SUSY 3/2 ) 2 < 0 m SUSY 3/2 b a b ( aa bb ) a b a A (2s SUSY ) n S /2 (2t SUSY ) n T /2 11

12 For s SUSY, t SUSY > 1 with a, b 1 and A, B O(1) b a O(1), B/A > a/b, α, β O(1/ab), Stability of SUSY point: mass eigenvalues (m 2, m2 ) m ( n 2 ln bb ) 4 (ab) 2 S aa (b a) 4(mSUSY 3/2 ) 2 > 0 m 2 (m SUSY 3/2 ) 2 < 0 saddle point m 2 m 2 sharp racetrack s t s t

13 SUSY breaking local minimum t SB = t SUSY (1 + δ t SB ), s SB = s SUSY (1 + δ s SB ) δ t SB = nt O(1), δ s aα + bβ SB = n T ln(bb/aa) δt SB Order parameters m 3/2 e a(δs SB αδt SB ) (1+δ s SB )n S /2 (1+δ t SB )n T /2 m SUSY 3/2, F T T + T δt SB m 3/2, F S S + S n S as Vacuum energy F S V SB = n S S + S 2 F T + n T T + T AdS minimum with modulus-dominated SUSY breaking F T T + T F T T + T 2 3m 2 3/2 < 2m2 3/2 V lift = D e 2K/3 (T + T ) n P (S + S) m P Minkowski minimum with modulus-dominated SUSY breaking α 1 4π

14 Numerical results s SUSY t SUSY V SUSY (m SUSY /m SUSY 3/2 ) 2 m SUSY /m SUSY 3/ s SB t SB V SB (m SB /m 3/2) 2 (m SB /m 3/2) s ds t ds V ds (m ds /mds 3/2 )2 (m ds /mds 3/2 ) AdS F S /(S + S) F T /(T + T ) m 3/2 m SUSY 3/ ds F S /(S + S) F T /(T + T ) m ds 3/2 D

15 IV. Summary Moduli-mixing racetrack model W = Ae af a Be bf b, f a,b = m a,b S + w a,b T Single light modulus S S, T Modulus/anomaly ratio of SUSY breaking mediation α can take various values depending on m, w and S (c.f. α = 1 in KKLT model without moduli mixing) Two light moduli S, T SUSY point is saddle point SUSY breaking AdS local minimum Modulus-dominated SUSY breaking α 1 (c.f. α 1 in racetrack model without moduli mixing) 15