Elementary/Composite Mixing in Randall-Sundrum Models

Transcription

1 Elementary/Composite Mixing in Randall-Sundrum Models Brian Batell University of Minnesota with Tony Gherghetta - arxiv: arxiv: Cornell 1/30/08

2 5D Warped Dimension = 4D Strong Dynamics AdS/CFT duality: Extra dimension is a calculational tool Randall-Sundrum models Standard Model partial compositeness How to quantify elementary/composite mixing? Understand structure and phenomenology of 4D dual theory Answer: The Holographic Basis: Φ(x, y) = ϕ s (x)g s (y) + ϕ n CF T (x)g n (y) n=1 Cornell 1/30/08 1

3 Outline Randall-Sundrum models and geometrical hierachies AdS/CFT and Holography The Kaluza-Klein Basis The Holographic Basis Elementary/composite content of SM fields Explain warped phenomenology in a 4D language (e.g. RS GIM mechanism) Cornell 1/30/08 2

4 A slice of AdS Randall, Sundrum 99 ds 2 = e 2ky η µν dx µ dx ν + dy 2 Warped geometry = Energy scales depend on location Planck/Weak scale hierarchy: Λ weak M P e πkr k O(M P ), πkr O(30) R can be naturally stabilized Goldberger, Wise 99 UV Brane Higgs IR Brane 0 y π R Cornell 1/30/08 3

5 Standard Model in the bulk Davoudiasl, Hewett, Rizzo 99; Pomarol 99 Grossman, Neubert 99 Chang et al. 99; Gherghetta, Pomarol 00 UV IR Theory of Flavor! Natural Yukawa hierarchies FCNC suppressed e Aµ t KK H Cornell 1/30/08 4

6 Bulk fields Scalar field with tuned bulk and boundary masses S = d 5 x [ g 1 2 ( MΦ) 2 1 ] 2 ak2 Φ 2 bkφ 2 (δ(y) δ(y πr)) Tuning: b = 2 ± 4 + a Why consider this toy model? Tuning allows for a localized zero mode: f 0 (y) e (b 1)ky = Holographic interpretation depends on b special values for b mimic bulk graviton and gauge boson Cornell 1/30/08 5

7 is for our purposes... AdS/CFT duality Maldacena 97 Weakly coupled gravity dual Strongly coupled gauge in warped 5D theory (CFT) in 4D Large N c gauge theory ( exp ) ϕ 0 O CFT = exp [ Γ(ϕ 0 ) ] Gubser, Klebanov, Polyakov 97; Witten 97 Cornell 1/30/08 6

8 Dictionary 5D 4D bulk field Φ(x, y) CFT operator O(x) BC Φ(x, y 0 ) = ϕ 0 (x) source: ϕ 0 (x)o(x) bulk mass dimension of O e.g. Bulk gauge field global symmetry current A µ (x, y) J CF T µ m 2 A = 0 J = 3 Cornell 1/30/08 7

9 Holography for RS1 Arkani-Hamed, Porrati, Randall 00 UV Brane IR Brane Rattazzi, Zaffaroni 00 Perez-Victoria 00 y p M p UV cutoff Conformal TeV Scale of CFT Breaking zero mode source field (elementary) KK modes CFT bound states (composites) but wait... Mixing through operator ϕ 0 (x)o(x) = Mass eigenstates are elementary/composite mixtures Cornell 1/30/08 8

10 The Holographic Recipe Step 1: Step 2: Evaluate bulk action for arbitrary boundary conditon Φ(x, y 0 ) = ϕ 0 (x) to obtain Γ(ϕ 0 ) Take functional derivatives to compute correlation functions of CFT operators ( OO (p) = δ2 δϕ 2 exp 0 = ip ) ϕ 0 O CFT ( J ip ) ( ) ipe πkr b 1 k Yb 1 k ( J ip ) ( ) ipe πkr b 2 k Yb 1 k = δ2 δϕ 2 exp [ Γ(ϕ 0 ) ] 0 ( Y ip ) ( ipe πkr b 1 k Jb 1 k ( Y ip ) ( ipe πkr b 2 k Jb 1 k ) ) Step 3: Interpret OO (p) Cornell 1/30/08 9

11 Operator Dimension = a = 2 + b b 2 {}}{ {}}{ ϕ 0 (x) O(x) b < 1 or b > 3 = irrelevant mixing 2 1 Relevant mixing 1 < b < 3 = relevant mixing b Cornell 1/30/08 10

12 Two branches in dual theory = 2 + b 2 b < 2 : - source field ϕ 0 (x) massless - zero mode primarily elementary - Nearly all RS phenomenological examples are described by b < 2 (fermions too!) b > 2 : - source field ϕ 0 (x) massive M 0 k - zero mode primarily composite - Higgs; perhaps t R in some models Cornell 1/30/08 11

13 Partial compositeness of SM fields UV IR e Aµ KK H t UV localized mostly elementary IR localized mostly composite Can we quantify source/cft (elementary/composite) mixing? Cornell 1/30/08 12

14 KK decomposition: Kaluza-Klein mass eigenbasis Φ(x, y) = φ n (x)f n (y), n=0 BC : (++) ( 5 bk)f n (y) 0,πR = 0 Localized massless mode: f 0 (y) e (b 1)ky, < b < The fields φ n (x) are the mass eigenstates - Spectrum: ( mn ) ( mn e πkr ) J b 1 Y b 1 k k Y b 1 ( mn k ) ( mn e πkr ) J b 1 k = 0 Cornell 1/30/08 13

15 Holographic basis Basic idea: Expand the bulk field directly in terms of a source field ϕ s (x) and composite CFT states ϕ n CF T (x): Φ(x, y) = ϕ s (x)g s (y) + ϕ n CF T (x)g n (y) n=1 Leads to kinetic and mass mixing in 4D effective theory Mass eigenstates will be a mixture of ϕ s (x) and ϕ n CF T (x) Cornell 1/30/08 14

16 Source profile g s (y) g s (y) can be determined from mass of source 0 for b < 2 Ms 2 = 4(b 2)(b 3)k 2 for b > 2 = g s (y) e ky e (4 )ky = e (b 1)ky for b < 2 e (3 b)ky for b > 2 Cornell 1/30/08 15

17 Source profiles mimic operator dimensions: = b UV IR b < 1 b = 1 b = 2 b = 3 b > 3 - Indicates when mixing is relevant, marginal, or irrelevant Cornell 1/30/08 16

18 CFT composite profiles g n (y) CFT spectrum obtained from poles in 2-point function: J b 2 ( Mn k ) ( Mn e πkr ) Y b 1 k Y b 2 ( Mn k ) ( Mn e πkr ) J b 1 k = 0 Note different from KK spectrum! Identical to the spectrum obtained with the following BC for g n (y): BC : ( +) g n (y) = 0 0 ( 5 bk)g n (y) = 0 πr Cornell 1/30/08 17

19 Effective 4D Lagrangian in the holographic basis L = 1 2 ϕt Z ϕ 1 2 ϕt M 2 ϕ, where ϕ T = (ϕ s, ϕ 1 CF T, ϕ2 CF T, ) Z = 1 z 1 z 2 z 3 z z z , M2 = M 2 s µ 2 1 µ 2 2 µ 2 3 µ 2 1 M µ M µ M Notice kinetic mixing = nonorthogonal basis z n and µ 2 n computed from wavefunction overlap integrals Diagonalization leads to KK basis Cornell 1/30/08 18

20 γ ρ mixing in SM Vector Meson Dominance L = 1 4 (F µν) (ρ µν) z γ ρ F µν ρ µν 1 2 m2 ρρ µ ρ µ Physical photon can be viewed as partly composite Cornell 1/30/08 19

21 Graviton h µν f 0 (y) e ky b = 0; = 4 = irrelevant mixing h0 h 1. = 1 e πkr h s h 1(CF T ). 4D graviton h 0 µν(x) elementary source ; compositeness negligible KK modes are purely composite Cornell 1/30/08 20

22 b = 1; = 3 = marginal mixing Gauge field A µ f 0 (y) = 1 πr A 0 µ A 1 µ A 2 µ. = A s µ A 1(CF T ) µ A 1(CF T ) µ. massless eigenstate A 0 µ(x) is primarily elementary KK modes are purely composite Cornell 1/30/08 21

23 Bulk Fermions Bulk mass m ψ = ck KK mass eigenbasis: ψ ± (x, y) = ψ±(x)f n ±(y), n n=0 Chiral zero mode ψ 0 +(x); wavefunction f 0 +(y) e (1 2 c)ky Cornell 1/30/08 22

24 Fermion holography Contino, Pomarol 04 Operator dimension: = c If < 5/2 = relevant mixing Holographic basis ψ + (x, y) = ψ s (x)g s (y) + λ n +(x)g+(y), n n=1 ψ (x, y) = χ(x)g χ (y) + λ n (x)g (y), n n=1 Cornell 1/30/08 23

25 Two branches in dual theory = c c > 1/2 : - source field ψ s (x) chiral; χ(x) absent from theory - zero mode primarily elementary - Nearly all bulk fermions described by c > 1/2 c < 1/2 : - field χ(x) marries with source field ψ s (x) to become massive M 0 k - zero mode primarily composite - perhaps t R in some models Cornell 1/30/08 24

26 Right-handed top t R f 0 (y) = e (1 2 c)ky m ψ = ck Take e.g. c = 0.7; = 1.7 = relevant mixing t (0) R t (1) R t (2) R t (3) Ṛ. = t t t t s R CF T (1) R CF T (2) R CF T (3) R. massless eigenstate t 0 R (x) roughly equal mixture of source/cft KK modes contain elementary component Cornell 1/30/08 25

27 Example: RS GIM mechanism Important point on inverse transformation: - Source field contains zero mode ψ s (x) = ψ 0 +(x) + m=1 ω sm + ψ m + (x) - Composite modes do not; entirely composed of KK modes λ n +(x) = m=1 ω nm + ψ m + (x) Cornell 1/30/08 26

28 Gauge interactions g A ψψ Contain SM fermions gs ss g*n ss gs ns gs nn g*m g*m sn ln Cornell 1/30/08 27

29 1.2 * * 1 1s ss * 1 s ss, s s 1s 0 1 Light fermions have exponentially suppressed couplings to composites Cornell 1/30/08 28

30 s ss *1 ss 1 3-source vertex dominates Cornell 1/30/08 29

31 RS GIM mechanism For light fermions, c > 1/2, 3-source vertex dominates: s g ss s s 0 g g πr f 1 (0) 1 s 0 0 KK gauge boson couplings are approx. universal for light fermions = FCNCs suppressed Cornell 1/30/08 30

32 Flavor violation Important to track nonuniversal contribution to coupling g*n ss g n ss = πr Sum over all composite modes = 0 e ky g s (y)g n (y)g s (y) g 1 nonuniversal = n=1 g n ss ω n1 = g 1 g 1 universal Cornell 1/30/08 31

33 Flavor violation - cont d Near c 1/2, first composite mode saturates nonuniversal piece: g 1 nonuniversal g 1 ssω 11 g 2πkR ( ) 2c 1 e (1 2c)πkR. 2 2c Works well - order few % - for c < 0.6 (e.g. tau, muon) Deviates for c > 0.6 (e.g. smaller anyway electron), but nonuniversal contributions Thanks to K. Agashe for discussions Cornell 1/30/08 32

34 Conclusions Holographic basis: bulk field expanded in source and CFT resonances Quantitatively describe elementary/composite mixing in warped duals Explain warped physics in terms of strong gauge dynamics Things to do: Other applications: Higgsless models, warped SUSY, Gauge-Higgs models (QCD?) Loop diagrams - important for EWPT, gauge coupling unification etc. Brane localized kinetic terms - could modify composite content More general geometries? Cornell 1/30/08 33