Little Randall- Sundrum (RS) Models or Tale of Logarithms & - PowerPoint PPT Presentation

Little Randall- Sundrum (RS) Models or Tale of Logarithms & Exponentials

Custodial RS: Gauge Sector SU (2) L × SU (2) R → SU (2) V ) 1 ( U Y → ) 1 ( U X × ) 2 ( U R S A M , Z M , L ± A ± Z M , ˜ A M , ˜ M M M , V ± M , R ± Z H Z ′ M M IR brane UV brane

Custodial RS: Quark Sector u (++) λ ( − +) � � L L 2 / 3 5 / 3 ( 2 , 2 ) 2 / 3 ∋ Q L ≡ , d (++) u ′ ( − +) L L − 1 / 3 2 / 3 2 / 3 � � u c (++) ( 1 , 1 ) 2 / 3 ∋ u c 2 / 3 , R ≡ R 2 / 3 T     Λ ′ ( − +) D (++) R R 5 / 3 − 1 / 3 U ′ ( − +) U ( − +)     ( 3 , 1 ) 2 / 3 ⊗ ( 1 , 3 ) 2 / 3 ∋ T R ≡ ⊗     R R 2 / 3 2 / 3     D ′ ( − +) Λ ( − +) R R − 1 / 3 5 / 3 2 / 3 2 / 3

Question Gauge & in particular structure of quark sector needed to protect T & Z → bb in custodial RS (RSc) model baroque Is there another, possibly more simple way to tame corrections to both oblique corrections (T) b & Zb L b L (g L ) ? To answer question, first have to understand problem better

Prelude In RS model there is only one moderately large parameter, namely L = ln Λ UV Λ IR where Λ UV ( Λ IR ) is cutoff scale on UV (IR) brane Solving gauge-hierarchy problem between weak M W & Planck scale M Pl , requires 10 16 � � L RS ≈ ln ≈ 37

Problem Unfortunately, in SU(2) L × U(1) Y RS variant many observables are L-enhanced: 4 π Π W W (0) − c 2 � � T = w Π ZZ (0) w M 2 e 2 c 2 Z π v 2 L , ≈ w M 2 2 c 2 KK � � M 2 2 − s 2 F 2 ( c Q 3 ) � 1 � ∆ g b w Z L L ≈ 2 M 2 3 3 + 2 c Q 3 KK

Solution! Let’ s curb our ambitions & address hierarchy problem only up to Λ UV = 10 3 TeV , which means 10 3 � � L LRS ≈ ln ≈ 7 It is readily seen, that in such a little RS (LRS) model, one has: T LRS ≈ L LRS T RS ≈ 1 5 T RS L RS

Solution! cont’ d Relative to usual RS 10 68 � CL 95 � CL model constraint from 99 � CL L � ln � 10 16 � 8 T relaxed by factor 68 � CL 95 � CL M KK � TeV � 99 � CL 6 of > 2 in LRS setup: L � ln � 10 3 � 4 M KK � 1 . 5 TeV , 2 M Z (1) ,W (1) ≈ 2 . 5 M KK 0 0 200 400 600 800 1000 � 4 TeV m h � GeV �

Solution! Really? In RS model, flavor non-universal observables, like Z → bb , feature both logarithms, i.e., terms enhanced by volume of extra dimension (XD), & exponentials, i.e., wave functions that describe localization of fermions in XD Simple rescaling of effects by factor L LRS L RS as done in case of T, might thus be incorrect if one considers Z → bb , ε K , ...

Quark Localization Instead of usual bulk mass parameters c Q i = M Q i c q i = − M q i , k k where M Ai denotes 5D masses & k curvature, it turns out to be more useful to work with d A i = max ( − c A i − 1 / 2 , 0) , A = Q, q which parametrize distance from critical point c Ai = -1/2 where F(c Ai ) switch from exponential to square root behavior

Quark Localization cont’ d UV IR u R d u > 0 d t = 0 O (1) t R e − Ld u ≈ 0 ǫ = e − L t 1

Froggatt-Nielsen Quark masses & mixings are related to d Ai via √ 2 m q i ∼ | Y | e − L ( d Qi + d qi ) , v λ ∼ e − L ( d Q 1 − d Q 2 ) , A ∼ e − L (3 d Q 2 − 2 d Q 1 − d Q 3 ) where |Y| = O(1) Yukawa couplings. Wolfenstein parameters ρ , η = O(1), but exact amount of CP not explained

Froggatt-Nielsen cont’ d To satisfy constraints due to masses & mixing of quarks for different L, d Ai obviously have to scale like = L RS d LRS d RS A i A i L LRS which implies that d Ai are larger in LRS model than in native RS setup, resulting in stronger IR localization of light quark wave functions

Aside: d Ai Parameters Assuming that d t = 0, needed to explain large top-quark mass with |Y| = O(1), it is easy to show that in right-handed (RH) down sector � � A λ 3 m t L d d ∼ ln ≈ 6 . 1 , m d � � A λ 2 m t L d s ∼ ln ≈ 4 . 8 , m s � m t � L d b ∼ ln ≈ 4 . 2 m b

Aside: d Ai Parameters cont’ d In case of left-handed (LH) quark bulk mass parameters one obtains instead � 1 � | Y | v L d Q 1 ∼ ln ≈ 4 . 9 , √ A λ 3 2 m t � 1 � | Y | v L d Q 2 ∼ ln ≈ 3 . 4 , √ A λ 2 2 m t � | Y | v � L d Q 3 ∼ ln ≈ 0 . 2 √ 2 m t

Aside: RH vs. LH FCNCs Latter relations imply that for c t > -1/2, RH ∼ couplings are in general strongly suppressed relative to LH counterparts: � � ( g d � R ) ij � ≈ F ( c d i ) F ( c d j ) F ( c Q i ) F ( c Q j ) ≈ e L ( d di + d dj − d Qi − d Qj ) � � � � ( g d L ) ij 7 · 10 − 2 ,  s → dZ   6 · 10 − 3 , b → dZ ≈ 5 · 10 − 3 ,  b → sZ 

Aside: RH vs. LH FCNCs cont’ d In consequence, to 1 obtain RH FCNCs in 0.1 RSc model comparable original � d � ij in magnitude to LH 0.01 custodial �� g L ones in SU(2) L × U(1) Y 0.001 d � ij variant requires bulk s → dZ �� g R 10 � 4 b → dZ mass c t for RH top of b → sZ 10 � 5 O(1) or larger � 0.5 0.0 0.5 1.0 1.5 2.0 c t

Aside: RH vs. LH FCNCs cont’ d Notice that c t > 1 1 means M t > k , which 0.1 original � raises question why d � ij 0.01 custodial �� g L RH top quark should 0.001 be treated as brane- d � ij s → dZ �� g R localized & not bulk 10 � 4 b → dZ fermion b → sZ 10 � 5 � 0.5 0.0 0.5 1.0 1.5 2.0 c t

K-K Mixing In RS model, leading contributions to Δ S = 2 interactions arise from Kaluza-Klein (KK) gluon exchange s d g ( k ) ∞ � k =1 d s & can be described by effective Lagrangian L ∆ S =2 ∋ 8 πα s L ( ˜ ∆ D ) 12 ⊗ ( ˜ ∆ d ) 12 ( ¯ d R s L )( ¯ d L s R ) M 2 KK

Mixing Matrices In terms of LH & RH rotations U d & W d , mixing matrices entering Δ S = 2 interactions can be written as ( ˜ ∆ D ) 12 ⊗ ( ˜ ∆ d ) 12 d ) 1 i ( U d ) i 2 ( ˜ ≈ ( U † ∆ Dd ) ij ( W † d ) 1 j ( W d ) j 2 with ∆ Dd ) ij = 1 ( ˜ 2 F 2 ( c Q i ) F 2 ( c q j ) � 1 � 1 dt ′ min t 2 c Qi ( t ′ ) 2 c qj t 2 , t ′ 2 � � dt × ǫ ǫ

Mixing Matrices cont’ d Evaluating double integral, one finds ( ˜ ∆ D ) 12 ⊗ ( ˜ ∆ d ) 12  F ( c Q 1 ) F ( c Q 2 ) F ( c d ) F ( c s ) , c Q 2 + c s > − 2  ∼ � � ǫ 2 / F 2 ( c Q 2 ) F 2 ( c s ) , c Q 2 + c s < − 2  which implies that in 2 nd case, Δ S = 2 FCNCs are enhanced by e 2 L | 2+ c Q 2 + c s | ≫ 1 with respect to usual RS-GIM result

UV Dominance UV IR � L π s R g (2) e − Ld d ≈ 0 g (1) � L − π t 1 ǫ

UV Dominance cont’ d UV If c Q2 + c s < -2 weight factor min (t 2 ,t’ 2 ) in s R overlap integral does not fall off sufficiently g (1) fast near UV brane to compensate for strong g (2) increase of quark profiles t ǫ

Values of c Ai : RS vs. LRS RS model (L = 37) LRS model (L = 7) c Q1 -0.63 ± 0.03 -1.34 ± 0.16 c Q2 -0.57 ± 0.05 -1.04 ± 0.18 c Q3 -0.34 ± 0.32 -0.49 ± 0.34 c u -0.68 ± 0.04 -1.58 ± 0.18 c c -0.51 ± 0.12 -0.79 ± 0.26 c t ]-1/2 , 2] ]-1/2 , 5/2] c d -0.65 ± 0.03 -1.44 ± 0.17 c s -0.62 ± 0.03 -1.28 ± 0.17 c b -0.58 ± 0.03 -1.05 ± 0.13

Bounds on UV Cutoff To avoid UV dominance in Δ S = 2 processes, one must require that d Q2 + d s < 1, which translates into bound Λ UV L LRS = 8 . 2 > 3600 , ( ∆ S = 2) ⇒ Λ IR For Δ S = 1 FCNCs it turns out that weaker condition d s < 1 is enough to avoid enhancement: Λ UV L LRS = 4 . 8 > 120 , ( ∆ S = 1) ⇒ Λ IR

ε K : LRS vs. RS Under assumption that mixed-chirality operator dominates Δ S = 2 transition, it is easy to derive that ratio of new-physics contribution to ε K in RS & LRS scenario is given by � | Y | v � � 2 � | ∆ ǫ K | LRS ≈ L LRS 1 , e − 2 L LRS max √ | ∆ ǫ K | RS L RS 2 m s ≈ L LRS � 1 , e 2(8 . 2 − L LRS ) � max 37

ε K : LRS vs. RS cont’ d � UV � � IR For generic RS 10 2 10 4 10 6 10 8 10 10 10 12 10 14 10 16 parameter points, 5 approx. featuring values of 4 exact ε K of O(100) larger � �Ε K LRS � �Ε K RS than SM prediction, 3 L dependence of 2 exact results nicely 1 follows approximate formula 0 5 10 15 20 25 30 35 L

ε K : LRS vs. RS cont’ d � UV � � IR L dependence of 10 2 10 4 10 6 10 8 10 10 10 12 10 14 10 16 curves corresponding 5 approx. to points consistent 4 exact with measured value � �Ε K LRS � �Ε K RS of ε K , can look more 3 complicated, but 2 characteristic feature 1 of UV dominance stays intact 0 5 10 15 20 25 30 35 L

Big Picture: LRS vs. RS 100 � of consistent points 50 20 10 5 Z → b ¯ b LRS 2 ǫ K RS 1 ǫ ′ / ǫ 2 4 6 8 10 M KK � TeV �

Summary Considering volume-truncated versions of RS setup with UV cutoff Λ UV << M Pl allows to mitigate constraints from both T & Z → bb ε K provides bound on Λ UV of few 10 3 TeV . Even if bound is satisfied no improvement in ε K can be achieved in LRS compared to native RS model Effect arises since for c Q2 + c s < -2, overlap integrals of 5D gluon propagator with profiles of 1 st & 2 nd generation quarks are dominated by region near UV brane, which partially evades RS-GIM mechanism

Higgs-Boson FCNCs or Fun with δ & Θ distributions

Higgs Localization UV IR bulk u R Higgs V( β ) t R 1 t ǫ