FLIGHT OF THE WARPED PENGUINS. In collaboration with Csaba Csáki, Yuval Grossman, and Yuhsin Tsai. DAMTP/Cavendish HEP Seminar, 26 May 2011

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1 FLIGHT OF THE WARPED PENGUINS Phys. Rev. D83, arxiv: Cornell University In collaboration with Csaba Csáki, Yuval Grossman, and Yuhsin Tsai DAMTP/Cavendish HEP Seminar, 26 May / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 1/28 28

2 Warped Penguins Randall-Sundrum & flavor anarchy Defying anarchy in µ eγ UV finite 5D loops Remarks on current work 2 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 2/28 28

3 Lepton Flavor Violation Br (µ eγ) SM = 0 Current bound: Br (µ eγ) < MEGA, LAMPF Later this year from MEG: Br (µ eγ) < / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 3/28 28

4 Part III Ben Allanach s Essay Prompt Essay 74, The Phenomenology of Extra Dimensions. Extra dimensional models provide an interesting playground for model building and investigating collider signatures. Candidates are invited to provide an overview of one of the following extra-dimensional models: ADD (Arkani-Hamed, Dimopoulos and Dvali), UED (Universal Extra Dimensions) or Randall-Sundrum I. The candidate should include a calculation of a matrix element squared for a collider signature of the model. 4 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 4/28 28

5 Reminder: Randall-Sundrum Randall, Sundrum (99); ds 2 = ( ) 2 R z (dx 2 dz 2 ) 5 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 5/28 28

6 Reminder: Randall-Sundrum Fermion Gauge Boson ds 2 = ( ) 2 R z (dx 2 dz 2 ) Randall, Sundrum (99); Davoudiasl, Hewett, Rizzo (99); Grossman, Neubert (00); Gherghetta, Pomarol (00); Bulk Higgs: Agashe, Contino, Pomarol (04); Davoudiasl, Lille, Rizzo (05), Agashe, Okui, Sundrum (08) 5 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 5/28 28

7 Reminder: Yukawa matrices Fermion Gauge Boson Y (4D) ij = f i Y ij f j f i = 1 2c i 1 (R/R ) 1 2c i Flavor: Huber, Shafi (03); Burdman (03); Kalil, Mohapatra (04); Agashe, Perez, Soni (04); Chen (05); Agashe, Blechman, Petriello (06); Davidson, Isidori, Uhlig (07); Csáki, Falkowski, Weiler (08); Chen, Yu (08); Agashe, Okui, Sundrum (08); Chen, Mahanthappa, Yu (09),... 6 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 6/28 28

8 Anarchic Flavor in RS Definition: anarchic matrix All entries O(1) with arbitrary phase. The product of anarchic matrices is also anarchic. Assumption: true in all preferred bases. Y (4D) ij = f i Y ij f j f i = 1 2c i 1 (R/R ) 1 2c i The 5D parameters Y ij are anarchic matrices, Y ij = Y ij Mass hierarchy m i = f i Y ii f i v set by exponentially small overlap of the zero-mode fermions with the Higgs vev; controlled by the fermion bulk masses, c i / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 7/28 28

9 Lepton Flavor Violation Controlled by two dominant parameters Flavor is dominantly controlled by: Y and M KK Y µ e Y Y M loop ( 1 M KK ( 1 M KK ) 2 f L Y 3 f E ) 2 Y 2 m 8 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 8/28 28

10 Lepton Flavor Violation Two dominant parmeters µ e e Z, Z Fermion µ e Gauge Boson Ti Z, Z e ( )2 ( ) 1 1 M tree M KK Y If we increase Y, must maintain SM mass spectrum push fermion profiles to UV Less overlap with the FCNC part of the Z 9 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 9/28 28

11 Complementary tree- and loop-level bounds Possible tension between tree- and loop-level processes Tree-level bound: ( ) 2 ( ) 3 TeV 2 < 0.5, 1.6 (Custodial) M KK Y Penguin bound: 2 ay + b ( ) 2 3 TeV M KK What the heck is this? Can test anarchic flavor ansatz. 10 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 10/28 28

12 Operator analysis of µ eγ Match to 4D EFT: R 2 e 16π 2 v ( 2 f Li akl Y ik Y kl Y ) lj + b ij Y L(0) ij f Ej i σ µν E (0) j F µν (0) Y ij is a spurion of U(3) 3 lepton flavor Indices on a ij and b ij encode bulk mass dependence Flavor structure a ij Y ik Y kl Y lj gives a generic contribution Depends only on Y and M KK New: b ij Y ij is aligned up to structure of b ij f i Y ij f j m ij, so this term is almost diagonal in the mass basis This depends on the particular flavor structure of the anarchic Y 11 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 11/28 28

13 Alignment vs FCNC Definition: anarchic matrix, All entries O(1) with arbitrary phase. The product of anarchic matrices is also anarchic. Assumption: this is true in all preferred bases. R 2 e v ( 2 f 16π 2 Li akl Y ik Y kl Y ) lj + b ij Y ij f Ej L (0) i σ µν E (0) Compare to zero mode mass matrix: m ij = f Li Y ij f Ej v j F µν (0) b terms have the flavor structure of 4D mass terms, up to bulk masses Alignment: b ij term almost diagonalized in the mass basis Flavor structure important, defying anarchy Alignment in RS: Agashe, Perez, Soni 04; Agashe, Azatov, Zhu / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 12/28 28

14 A bunch of diagrams: a and b coefficients R 2 e v ( 2 f 16π 2 Li akl Y ik Y kl Y ) lj + b ij Y ij f Ej L (0) i σ µν E (0) j F µν (0) H 0, G 0 H 0, G 0 H ± H ± Z Z W H ±, W Z Z W W, H ± 13 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 13/28 28

15 The structure of RS penguins: a coefficient H 0, G 0 H 0, G 0 H ± H ± B 10 4 AC 10 4 B 10 4 AD 10 3 Z Z W H ±, W A 2 B 10 5 A A 2 B 10 5 A 2 B 10 5 A. Mass insertion 10 1 per insertion (cross) B. Equation of motion 10 4 (external arrows point same way) C. Higgs/Goldstone cancellation 10 3 (H 0, G 0 diagram only) D. Proportional to charged scalar mass / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 14/28 28

16 The structure of RS penguins: b coefficient Z Z W W, H ± BE 10 4 A 10 1 BE 10 4 AE 10 2 A. Mass insertion 10 1 per insertion (cross) B. Equation of motion 10 4 (external arrows point same way) E. No sum over internal flavors / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 15/28 28

17 Leading order diagrams H ± H ± L µ E i Z E e L j E e E µ L e L e ν N Y E Y N Y N Y E Y E Y E Z Y E Three coefficients (a H, a Z, b) with arbitrary relative signs Defined ay 3 = k,l a kly ik Y kl Y lj and by = k,l (U L) ik b kl Y kl (U R ) lj So, just calculate these: (many details in paper) 5D position/momentum space: external zero modes Mass insertion approximation, but sum over all KK modes Gauge invariance: only identify (p + p ) µ coefficient 16 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 16/28 28

18 Representative Bounds: b = 0 Y * (Left of this ruled out) ( ) 2 ( ) 3 TeV 2 M KK Y < 0.5, 1.6 Generic Custodial a H a Z µ eγ, average ay 2 + b ( 3TeV M KK ) 2 < For M KK = 3 TeV, b = 0 a =.001 and generic 3.7 Y 4 a =.016 and custodial 1 Y M KK (TeV) 17 / 28 KK bound Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 17/28

19 Representative Bounds: b 0 Y * (Left of this ruled out) ( ) 2 ( ) 3 TeV 2 M KK Y < 0.5, 1.6 Favorable b µ eγ, b = b 1σ (Above this ruled out) µ eγ, average ay 2 + b ( 3TeV M KK ) 2 <.015 µ eγ, b = + b 1σ M KK (TeV) 18 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 18/28 28

20 Finiteness: naïve dimensional analysis 4D Naïve: d 4 k F γ µ F B log(λ) Really log divergent? No, finite. Here s why: Gauge invariance: q µ M µ = 0. Lorentz invariance: d 4 k /k = 0. k2n Indeed, M 4D Λ 2. Suspect that M 5D Λ 1. 5D bulk, i.e. d 4 k d 5 k 19 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 19/28 28

21 Finiteness: bulk 5D fields Neutral Charged Loop integral (d 4 k) Gauge invariance (p + p ) 1 1 Bulk boson propagator 1 2 Bulk vertices (dz) 3 3 Overall z-momentum Derivative coupling 0 +1 Mass insertion/eom 1 1 Total degree of divergence 1 1 Note: this all carries over to the KK picture 20 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 20/28 28

22 Finiteness: brane-localized Higgs Neutral Charged W H ± Loop integral (d 4 k) Gauge invariance (p + p ) Brane boson propagators Bulk boson propagator Bulk vertices (dz) Derivative coupling Brane chiral cancellation Brane MW 2 cancellation Total degree of divergence / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 21/28 28

23 The structure of RS penguins: a coefficient H 0, G 0 H 0, G 0 H ± H ± B 10 4 AC 10 4 B 10 4 AD 10 3 Z Z W H ±, W A 2 B 10 5 A A 2 B 10 5 A 2 B 10 5 A. Mass insertion 10 1 per insertion (cross) B. Equation of motion 10 4 (external arrows point same way) C. Higgs/Goldstone cancellation 10 3 (H 0, G 0 diagram only) D. Proportional to charged scalar mass / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 21/28 28

24 Finiteness: brane-localized Higgs The M 2 W cancellation comes from the form of the photon coupling to the brane-localized H ± : (2k p p ) µ [(k p ) 2 MW 2 ][(k p)2 MW 2 ] = (p + p ) µ [ k 2 ] 1 (k 2 MW 2 )2 k 2 MW 2 = M2 W (p + p ) µ (k 2 M W 2 O(1/k6 ) )3 We have used the fact that the (p + p ) µ coefficient gives the complete gauge-invariant contribution. 22 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 22/28 28

25 Finiteness: brane-localized Higgs Neutral Charged W H ± Loop integral (d 4 k) Gauge invariance (p + p ) Brane boson propagators Bulk boson propagator Bulk vertices (dz) Photon Feynman rule Brane chiral cancellation Brane MW 2 cancellation Total degree of divergence / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 23/28 28

26 Finiteness: brane-localized Higgs The chiral cancellation comes from the UV structure of the sum of the two diagrams: Fermion propagator goes like /k + kγ 5, numerator structures are M a /kγ µ /k/k kγ µ k/k = k 2 (/kγ µ γ µ /k) M b /k/kγ µ /k /kkγ µ k = k 2 (γ µ /k /kγ µ ) KK picture: appears as v/m KK in the mass-basis Yukawa See Agashe et al / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 24/28 28

27 Finiteness: brane-localized Higgs Neutral Charged W H ± Loop integral (d 4 k) Gauge invariance (p + p ) Brane boson propagators Bulk boson propagator Bulk vertices (dz) Photon Feynman rule Brane chiral cancellation Brane MW 2 cancellation Total degree of divergence / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 24/28 28

28 Perturbativity and the 2-loop result Yin-yang and double rainbow topologies. Insert a photon and odd number of mass insertions. Dotted line represents gauge or Higgs boson. Purely bulk fields: Loop integrals (d 4 k) +8 Gauge invariance (p + p ) 1 Bulk boson propagators 2 Bulk vertices (dz) 5 Total degree of divergence 0 Must do full calculation Like 1-loop, hard to determine brane Higgs power counting. It may not be unreasonable to expect 1-loop cancellations to carry over to 2-loop. Log Λ large perturbative regime 25 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 25/28 28

29 The disappearing KK term 5D Lorentz invariance: must take the M n = nm KK and Λ = λm KK cutoffs together. Otherwise might lose leading term! M H 0 = gv 16π 2 f µf e ū e (p + p ) µ u µ 1 [ ( v ) ] 2 M 2 c 0 + O M c 0 = λ 2 N N n=1 m=1 ( n ) 2 ĉ 0 (n, n) for n λ λ ( n ) 0 ĉ 0 (n, n) for n λ λ ( ) 4 λ ĉ 0 (n, n) for n λ. n λ 2 (n 2 + m 2 ) + 2n 2 m 2 4 (n 2 + λ 2 ) 2 (m 2 + λ 2 ) 2 1 N N ĉ λ 2 0 (n, m), n=1 m=1 Dominant contribution from n λ. Taking λ for fixed n will lose this term! This is not a non-decoupling effect, just EFT. 26 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 26/28 28

30 Flight of the Warped Penguins Future directions with local collaborators 1. Bulk Higgs models (integrals are much nastier) 2. b sγ (operator mixing with b sg) H 0, G 0, H ± No Goldstone cancellation! 27 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 27/28 28

31 Conclusion Calculation of µ eγ in a warped extra dimension: Near tension between loop- and tree-level bounds on Y, M KK b ij Y ij is highly non-anarchic Finite at one-loop, suspect perturbative Certain features more transparent in 5D Thanks! 28 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 28/28 28

32 Gauge invariance This is a dipole operator and the Ward identity forces the gauge invariant amplitude to take the form M = ɛ µ M µ ɛ µ ū p [(p + p ) µ (m µ + m e )γ µ ] u p Thus it is sufficient to calculate the coefficient of the (p + p ) µ term in M µ to determine the overall gauge invariant amplitude. Diagrams which are not 1PI, such as external photon emissions, are gauge redundant to the 1PI diagrams. Lavoura / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 28/28 28

33 The standard µ eγ EFT Traditional parameterization for the µ eγ amplitude ic L,R 2m µ ū L,R σ µν u R,L F µν, For the case of RS, C L,R = ( ) ay 3 + by R 2 e 16π 2 v 2 2m µ f L2,1 f E1,2 Br(µ eγ) = Trick: C 2 L + C 2 R 2C L C R 12π2 (G F m 2 µ) 2 ( C L 2 + C R 2 ) < Br(µ eγ) 6 ay 2 + b 2 α 4π ( ) R 2 2 m e 28 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 28/28 28 G F m µ

34 Mixed 5D position/momentum space Mixed position/momentum space: (p µ, z) Due to the explicit z-dependence of the geometry and the localization of the Higgs, it is natural to work in mixed space. d dk i k 2 e ik (x x ) i d k z k 2 kz 2 e ikz (z z ) Usual momentum space in Minkowski directions Propagator dimension: [ 5D ] = [ 4D ] + 1 Each vertex: perform dz overlap integral 1/k External states carry zero-mode z-profile 28 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 28/28 28

35 5D Feynman rules See our paper for lots of appendices on performing 5D calculations. = ig 5 ( R z ) 4 γ µ = ie 5 (p + p ) µ = i 2 e 5g 5 v η µν = i ( R R ) 3 Y5 = k (z, z ) = iη µν G k (z, z ) = ɛ µ (q)f (0) A ( ) 2 ( ) c = fc z z R R R u(p) g 2 5 = g 2 SMR ln R /R e 5 f (0) A = e SM Y 5 = RY 28 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 28/28 28

36 Analytic expressions M(1MIH ± ) = M(3MIZ) = M(1MIZ) = i 16π 2 (R ) 2 f cl Y E Y N Y ev Nf ce 2I 1MIH ± 2 i ( 16π 2 (R ) 2 f cl Y E Y E Y ev E f ce 2 i ( 16π 2 (R ) 2 ev f cl Y E f ce g 2 ln R 2 R g 2 ln R R ) I 1MIZ. ) ( R v 2 ) 2 I 3MIZ Written in terms of dimensionless integrals. See paper for explicit formulae. 28 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 28/28 28

37 Finiteness in the KK picture Power counting for the brane-localized Higgs Charged Higgs: same M 2 W cancellation argument as 5D Neutral Higgs: much more subtle! A basis of chiral KK fermions: χ = ( χ (0) L i, χ (1) R i, χ (1) ) L i ψ = ( ψ (0) R i, ψ (1) R i, ψ (1) ) L i Worry about the following type of diagram: ˆχ 2 ˆψ J ˆχ J ˆψ1 ŷ 2J MJJ ŷ J1 The (KK) mass term in the propagator can be Λ. Have to show that the mixing with large KK numbers is small. 28 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 28/28 28

38 Finiteness in the KK picture Power counting for the brane-localized Higgs A basis of chiral KK fermions: ψ = ( ψ (0) R i, ψ (1) R i, ψ (1) ) L i χ = ( χ (0) L i, χ (1) R i, χ (1) ) L i Mass and Yukawa matrices (gauge basis, ψmχ + h.c.): m 11 0 m 13 M = m 21 M KK,1 m M KK,2 The zeroes are fixed by gauge invariance. ŷ 1J ŷ J2 = 0 Indices run from 1,..., 9 labeling flavor and KK number y / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 28/28 28

39 Finiteness in the KK picture Power counting for the brane-localized Higgs ( ) ψ = ψ (0) R i, ψ (1) R i, ψ (1) L i ( ) M = m11 0 m 13 m 21 M KK,1 m 23 χ = χ (0) L i, χ (1) R i, χ (1) L i 0 0 M KK,2 Rotating to the mass basis, ɛ v/m KK : Now we have y 1J y J2 ɛ, good! ɛ 1 ŷ ɛ 28 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 28/28 28

40 Finiteness in the KK picture Power counting for the brane-localized Higgs ( ) ψ = ψ (0) R i, ψ (1) R i, ψ (1) L i ( ) M = m11 0 m 13 m 21 M KK,1 m 23 χ = χ (0) L i, χ (1) R i, χ (1) L i 0 0 M KK,2 Rotating to the mass basis, ɛ v/m KK : ɛ ɛ 1 + ɛ ŷ ɛ ɛ 1 ɛ Must include large rotation of m 21 and m 13 blocks representing mixing of chiral zero modes into light Dirac SM fermions. This mixes wrong-chirality states and does not affect the mixing with same-chirality KK modes. Indeed, O(1) factors cancel: y 1J y J2 ɛ, good! 28 / Flip Tanedo pt267@cornell.edu Flight of the Warped Penguins 28/28 28