Yuhsin Tsai Cornell/Fermilab Warped Penguins 1/38

Warped Penguins Yuhsin Tsai Fermilab / Cornell Fermilab Theory Seminar, 18 Aug 2011 BG: Wall paper from www.layoutsparks.com 1 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 1/38 38

This talk includes the two papers Warped Penguins arxiv:1103.0240 Csaba Csaki, Yuval Grossman, Flip Tanedo, YT The Birds and the Bs in RS Work in progress Monika Blanke, Bibhushan Shakya, Flip Tanedo, YT 2 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 2/38 38

Why are warped penguins interesting? 3 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 3/38 38

Why are warped penguins interesting? Flavor & Finiteness 3 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 3/38 38

For the finiteness One generic feature of extra dimension models: They are non-renormalizable! M = a 0 + a 1 log(λ/m) + a 2 (Λ/M) + a 3 (Λ/M) 2 +... 4 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 4/38 38

For the finiteness One generic feature of extra dimension models: They are non-renormalizable! M = a 0 + a 1 log(λ/m) + a 2 (Λ/M) + a 3 (Λ/M) 2 +... The cutoff dependance is important. 4 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 4/38 38

Penguins live in extra dimension Penguin diagram is a loop induced process. µ γ ē 5 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 5/38 38

Penguins live in extra dimension Penguin diagram is a loop induced process. µ γ ē In SM Renormalizable, the leading diagram must be finite. 5 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 5/38 38

Penguins live in extra dimension Penguin diagram is a loop induced process. µ γ ē In extra dimension Non-renormalizable, the leading diagram must be...? 5 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 5/38 38

The finiteness issue Would XD penguins explode? µ γ ē Previous Analysis RS Penguinswith brane-higgs loop is UV sensitive! Cannot get a physical result. 6 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 6/38 38

The finiteness issue Would XD penguins explode? µ γ ē Previous Analysis The one-loop RS penguins are FINITE. Can get a physical result. Interesting Yukawa bounds. 6 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 6/38 38

Randall-Sundrum in one slide Randall, Sundrum (99); ds 2 = ( ) 2 R z (dx 2 dz 2 ) 7 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 7/38 38

Randall-Sundrum in one slide Gauge Boson ds 2 = ( ) 2 R z (dx 2 dz 2 ) Randall, Sundrum (99); Davoudiasl, Hewett, Rizzo (99); 7 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 7/38 38

Randall-Sundrum in one slide 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) 7 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 7/38 38

The pros and cons Flavor changing & the mass hierarchy 4D Yukawa Coupling: Y L i HE j f Li (R ) f Ej (R ) R 4D Gauge Coupling: g ii L i ZL i dz R ( ) 4 R f Li(z)f Z (z)f Li(z) z 7 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 7/38 38

Anarchic Flavor in RS For an interesting model, we want... Yij = Y ij is an ancharic matrices with O(1) numbers. wave function decides the hierarchy. M kk is not too heavy. relevant to LHC. 8 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 8/38 38

Anarchic Flavor in RS For an interesting model, we want... Yij = Y ij is an ancharic matrices with O(1) numbers. wave function decides the hierarchy. M kk is not too heavy. relevant to LHC. Allowed by flavor constraints? γ µ e µ z e 8 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 8/38 38

Lepton Flavor Violation : Loop 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 9 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 9/38 38

Lepton Flavor Violation : Tree Two dominant parmeters µ e e Z, Z Fermion µ e Gauge Boson 48 22 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 10 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 10/38 38

Complementary tree- and loop-level bounds Possible tension between tree- and loop-level processes In the lepton sector: Tree-level bound: ( ) 2 ( ) 3 TeV 2 < 0.5, M KK Y Penguin bound: ay 2 + b ( ) 2 3 TeV 0.015 M KK Can test anarchic flavor ansatz. 11 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 11/38 38

Penguins into Details Answering the questions of Finiteness & Flavor Penguins in a warped XD 5D calculation Diagram parade The flavor bounds Lepton sector Quark sector The finiteness The finiteness Matching 5D to 4D Conclusion 12 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 12/38 38

Penguins in a Warped Extra Dimension 13 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 13/38 38

Operator analysis: 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 kl 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 14 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 14/38 38

4D KK vs. 5D Y kk µ e Y Y N 1 N 2 kk=1 kk=1 d 4 k 15 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 15/38 38

4D KK vs. 5D Y kk µ e Y Y N 1 N 2 kk=1 kk=1 d 4 k R R dz d 4 k 15 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 15/38 38

The 5D propagator A mixed propagator with momentum & position space One example: brane-to-brane SU(2) L fermion propagator: (p, z = R, z = R, c) = i/p [ π R 5 2R 4 S c + S c S c + ]. 16 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 16/38 38

The 5D propagator A mixed propagator with momentum & position space One example: brane-to-brane SU(2) L fermion propagator: (p, z = R, z = R, c) = i/p [ π R 5 2R 4 S c + S c S c + ]. The propagator carries Bessel functions: S c ± = J c± 1 2 S c + = J c+ 1 2 (pr ) Y c 1 2 (pr) Y c+ 1 2 (pr ) J c 1 2 (pr ) J c+ 1 2 (pr ) Y c± 1 (pr ) 2 (pr) (pr ) Y c+ 1 2 16 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 16/38 38

The 5D propagator The meaning of the bessel functions The Bessel function part contains e p(x x ) as the usual propagator in position space. It also contains zero- and KK-mode wave functions: (p, R, R, c) i The is important. /p p 2 f (0) c f (0) c + n=1 i /p p 2 + (n/r ) 2 f (n) f (n). 17 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 17/38 38

How to calculate the 5D loop? Feynman s parametrization with Bessel functions?? You must be kidding...however, we can Taylor expand the propagator into powers ( of the ) external momentum (p µ, q µ 1 ). (k 2 +2k q) = 1 k 1 2k q 2 k +.... 2 Isolate the p µ + p µ e terms. Get the coefficient a. ɛ µ M µ a ɛ µ ū ( p µ + p µ e (m µ + m e )γ µ ) u Obtain the a numerically. ( ) a R 2 e 2 v fl YY Yf E ū/ɛu 16π 2 18 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 18/38 38

Diagrams for µ e γ: a and b coefficients Yes, we actually calculated all of these... 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 ± 19 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 19/38 38

The Flavor Bounds 20 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 20/38 38

Leading order µ e γ The following diagrams with external mass insertions dominate 1 W L e E µ L e ν N ν Z Z Y N Y N Y E Y E Y E Y E Y E Three coefficients (a W, 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 1 We thank Martin Beneke for pointing this out. 21 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 21/38 38

Leading order b s γ Different from the lepton case H Y u Y u Y d Y d Y d Y d Y d Three coefficients (a H, a Z,g, 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 22 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 22/38 38

Bounds for µ e γ: b = 0 Y 5 4 3 2 1 (Left of this ruled out) ( ) 2 ( ) 3 TeV 2 M KK Y < 0.5, 1.6 Generic Custodial a W a Z a W a Z ay 2 ( 3TeV M KK ) 2 <.015 M kk 0 0 5 10 15 20 23 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 23/38 38

Bounds for µ e γ: b 0 Y 5 (Left of this ruled out) 4 3 2 µ eγ, b = b 1σ ay 2 + b ( ) 2 3TeV M KK <.015 µ eγ, b = + b 1σ 1 M kk 0 0 5 10 15 20 24 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 24/38 38

Bounds for b s γ: b = 0 14 12 ( 3 TeV M KK ) 2 ( 2 Y ) < 0.025 (PRELIMINARY) ay 2 ( 3TeV M KK ) 2 < 0.1 10 a 0.5 Y 8 6 4 bound from Ε k The strongest bound comes from the ɛ K in the K K mixing. 2 5 10 15 20 25 0 kk 25 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 25/38 38

Some tuning is necessary M. Blankea, A. Burasa, B. Dulinga, S. Goria, and A. Weiler (08) The tuning of ɛ K in the RS model with Custodial Protection. with M kk = 2.5 TeV, 0.3 < y < 3. Question How many points can pass the tree-level & b sγ bound? 26 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 26/38 38

A PRELIMINARY result The size of B X s γ in the RS model with custodial protection. Using 1228 points passing the F = 2 bound with 0.3 < y < 3. Allowed Br(b sγ) < (0.4 ± 0.7) 10 4. NOT surprised Most of the points pass the bound due to small Yukawas. 27 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 27/38 38

Divergent? 28 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 28/38 38

The finiteness in the KK picture Naïve NDA in mass basis KK SM Loop integral ( )d 4 k +5 +4 Gauge invariance (p + p ) 1 1 Fermion propagators 1 2 Higgs propagator 2 2 Total degree of divergence 0 2 KK H Hard to see the finiteness in KK. Infinitely large Yukawa matrix with the KK tower. 29 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 29/38 38

The finiteness in the KK picture Naïve NDA in mass basis KK SM Loop integral ( )d 4 k +5 +4 Gauge invariance (p + p ) 1 1 Even number of k 0 1 Fermion propagators 2 2 Higgs propagator 2 2 0 2 Yukawa 1 + Mi 1 1 Total degree of divergence 1 2 KK H Hard to see the finiteness in KK. Infinitely large Yukawa matrix with the KK tower. 30 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 30/38 38

The finiteness in 5D Some tricks... Large momentum pulls the vertices together in 5D. Loop with mass insertion shrink to the brane. Loop in the bulk 5D theory without compactification. fermion: k 0, scalar: k 1, vector: k 1. Bulk vertices (dz): k 1. 31 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 31/38 38

Finiteness: bulk 5D fields Neutral Charged Loop integral (d 4 k) +4 +4 Gauge invariance (p + p ) 1 1 Bulk boson propagator 1 2 Bulk vertices (dz) 3 3 Overall z-momentum +1 +1 Derivative coupling 0 +1 Mass insertion/eom 1 1 Total degree of divergence 1 1 Note: this all carries over to the KK picture 32 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 32/38 38

Finiteness: brane-localized Higgs Neutral Charged W H ± Loop integral (d 4 k) +4 +4 +4 Gauge invariance (p + p ) 1 1 1 Brane boson propagators 2 4 2 Bulk boson propagator 0 0 1 Bulk vertices (dz) 1 0 1 Derivative coupling 0 +1 0 Brane chiral cancellation 1 0 0 Brane MW 2 cancellation 0 2 0 Total degree of divergence 1 2 1 33 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 33/38 38

Divergent? 34 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 34/38 38

No, they re finite! 34 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 34/38 38

Matching the 5D to 4D KK One mistake people made before When calculating in the KK picture, if we do N M = d 4 k n=1 0 ˆM (n) (k) The leading order result is M 1 (M KK ) 2 v 2 M 2 kk which is different from the 5D result M R 2! 35 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 35/38 38

Matching the 5D to 4D KK One mistake people made before When calculating in the KK picture, if we do N M = d 4 k n=1 0 ˆM (n) (k) The leading order result is M 1 (M KK ) 2 v 2 M 2 kk which is different from the 5D result M R 2! 5D KK!? 35 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 35/38 38

The KK result should be the same as 5D The problem comes from the Wrong UV limit! Momentum cutoff KK cutoff. e.g., the 5D propagator in a flat extra dimension can be expand into: ( p / + iγ 5 ) cos p(2π z) z = ( /p + iγ 5 ) [ ] 1 z sin pπr 2p 2 + cos(znl) p 2 (n/l) 2 n=0 36 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 36/38 38

The KK result should be the same as 5D The problem comes from the Wrong UV limit! Momentum cutoff KK cutoff. One way to do the integral, N N MKK lim d 4 k N n=1 0 ˆM (n) (k) This gives the same result as in 5D. 37 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 37/38 38

The KK result should be the same as 5D The problem comes from the Wrong UV limit! Momentum cutoff KK cutoff. One way to do the integral, N N MKK lim d 4 k N n=1 0 This gives the same result as in 5D. ˆM (n) (k) 5D = KK 37 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 37/38 38

Conclusion Warped penguins are important. Cutoff structure Flavor bounds Full 5D calculations are useful. Easier to see the cutoff structure. Avoid mistakes from the KK sum. One loop penguins are finite. Can be easily seen in the 5D way. The two loop result decides the cutoff scale. Interesting flavor bounds on Y, M kk. Tension from the loop- and tree-level bounds. Will have more results for the quark sector. 38 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 38/38 38

Backup Slides 38 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 38/38 38

The allowed cutoff scales When will the 1-loop diagrams really be the leading ones? M = M NDA ( a + y 2 16π 2 b +... ), need a > 1 16π 2 b. For µ e γ, a is suppressed by (vr ) 2. In the WORST CASE: Bulk-higgs: b = log(λr ) log(λr ) 7...ok Brane-higgs: b = ΛR ΛR 7...dangerous For b s γ, a is 0.5. In the WORST CASE: Bulk-higgs: b = log(λr ) log(λr ) < 80...ok Brane-higgs: b = ΛR ΛR < 80...ok The 2-loop calculation is necessary for µ e γ in the brane-higgs case. 38 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 38/38 38

µ e γ: 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 3 10 3 A 3 10 3 A 3 10 3 A 3 10 3 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 10 2 38 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 38/38 38

µ e γ: b coefficient Z Z W W, H ± AB 10 2 A 10 1 AB 10 2 AB 10 2 A. Mass insertion 10 1 per insertion (cross) B. No sum over internal flavors 10 1 38 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 38/38 38

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 38 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 38/38 38

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. 38 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 38/38 38

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: ˆψ J ˆχ J ˆχ 2 ˆψ 1 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. 38 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 38/38 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 23 0 0 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 1 0 1 y 1 0 1 0 0 0 38 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 38/38 38

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 : 1 0 1 1 ɛ 1 1 1 + ɛ 1 + ɛ ŷ 1 0 1 1 1 + ɛ 0 0 0 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! 38 / Yuhsin Tsai Cornell/Fermilab Warped Penguins 38/38 38