From Lattice Strong Dynamics to Electroweak Phenomenology

Transcription

1 From Lattice Strong Dynamics to Electroweak Phenomenology David Schaich, 4 November 2013 PRL 106: (2011) [ ] PRD 85: (2012) [ ] and work in progress with the Lattice Strong Dynamics Collaboration See also: USQCD white paper Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

2 arxiv: signal strengths Higgs boson Standard-model-like No BSM discovered Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

3 Suppose new strong dynamics composite Higgs This scenario is disfavored, but hard to rule out (non-perturbative!) USQCD white paper arxiv: reviews some possibilities Would expect many new composite states... around 4πv 3 TeV Complementary approach: low-energy effective theory New strong dynamics fix Low-Energy Constants of effective theory If heavy composite states can t be directly observed at the LHC, first signs of new physics may appear through these LECs Many strongly-coupled systems produce similar effective theories with range of LEC values Non-perturbative lattice calculations can explore possibilities Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

4 Scale setting & electroweak effective theory (Appelquist & Wu, PRD 48:3235, 1993) Integrating out resonances around 4πv scale gives chiral lagrangian Dynamical d.o.f. are Goldstones π a to be eaten by W and Z, appear through matrix field U exp [ 2iT a π a /F ] L LO = F 2 [ ] 4 Tr D µ U D µ U + F 2 B [ 2 Tr m (U + U )] Decay constant F sets electroweak scale, W & Z masses F = v = 246 GeV in simplest case (one electroweak doublet) Chiral condensate ψψ F 2 B involved in fermion mass generation Caveat 125 GeV is not around 4πv = work in progress to extend L χ Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

5 Higher-Order Low-Energy Constants With T Uτ 3 U and V µ (D µ U) U, next-to-leading order includes oblique corrections S α 1, T β 1, U α 8 triple gauge vertices and dominant contributions to WW scattering: L 1 = α 1 2 g 1g 2 B µν Tr (TW µν ) L 2 = iα 2 2 g 1B µν Tr (T [V µ, V ν ]) L 3 = iα 3 g 2 Tr (W µν [V µ, V ν ]) L 4 = α 4 {Tr (V µ V ν )} 2 L 5 = α 5 {Tr (V µ V µ )} 2 L 6 = α 6 Tr (V µ V ν ) Tr (TV µ ) Tr (TV ν ) L 7 = α 7 Tr (V µ V µ ) Tr (TV µ ) Tr (TV ν ) L 8 = α 8 4 g2 2 {Tr (TW µν)} 2 L 9 = iα 9 2 g 2Tr (TW µν ) Tr (T [V µ, V ν ]) L 10 = α 10 2 {Tr (TV µ) Tr (TV ν )} 2 L 11 = α 11 g 2 ɛ µνρλ Tr (TV µ ) Tr (V ν W ρλ ) L 1 = β 1 4 g2 2 F 2 {Tr (TV µ )} 2 Outline for this talk: S parameter and WW scattering, work with Lattice Strong Dynamics Collaboration Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

6 Lattice Strong Dynamics Collaboration Argonne Meifeng Lin, Heechang Na, James Osborn Berkeley Sergey Syritsyn Boston Richard Brower, Michael Cheng, Claudio Rebbi, Evan Weinberg, Oliver Witzel Colorado Ethan Neil Livermore Chris Schroeder, Pavlos Vranas, Joe Wasem NVIDIA Ron Babich, Mike Clark Syracuse DS UC Davis Joseph Kiskis U Wash. Mike Buchoff, Saul Cohen Yale Thomas Appelquist, George Fleming, Gennady Voronov Exploring the range of possible phenomena in strongly-coupled gauge theories Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

7 IBM Blue Kraken Cray Ridge Results to be shown are from state-of-the-art lattice calculations O(100M core-hours) invested overall Many thanks to DOE (through USQCD & Livermore), NSF (through XSEDE) USQCD Ds Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

8 Lattice Strong Dynamics strategy SU(3) gauge theory, N F = fund. (only N F = 2 and 6 today) Systematically explore trends as N F increases from QCD Match m 0 IR scales to allow more direct comparisons domain wall fermions good chiral & flavor symmetries (but expensive!) Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

9 The S parameter (Peskin & Takeuchi, 1991) Oblique correction to vacuum polarization of neutral EW gauge bosons new S defined to vanish for the standard model with M H = 125 GeV Experiment: S = 0.03 ± 0.10 Z decay widths and asymmetries Neutrino scattering cross sections M W, M Z Atomic parity violation Scaling up QCD (and imposing M H = 125 GeV) predicts S 0.43 Lattice calculations can predict S for strong dynamics beyond QCD Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

10 S on the lattice S = 4πN D d lim Q 2 0 dq 2 Π V A(Q 2 ) S SM (M H ) new L χ α 1 2 g 1g 2 B µν Tr [TW µν ] Π V A (Q 2 ) 1 Π V A (Q 2 ) is transverse component of vacuum polarization tensor Π µν V A (Q) = Z [ ] e iq (x+bµ/2) Tr V µa (x)v νb (0) A µa (x)a νb (0) x (correlators mix conserved and local domain wall currents for efficiency) 2 N D 1 is the number of doublets with chiral electroweak couplings 3 S SM (M H ) subtracted so that S = 0 in the standard model Removes three eaten Goldstones, depends on Higgs mass Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

11 Polarization function data and fits, N F = 2 and N F = 6 S = 4πN D d lim Q 2 0 dq 2 Π V A(Q 2 ) S SM (M H ) Uncorrelated fits to Padé-(1, 2) rational function (cf. Aubin et al.) Π V A (Q 2 ) = a 0 + a 1 Q b 1 Q 2 + b 2 Q 4 = m=0 a mq 2m 2 n=0 b nq 2n (motivated by meson pole dominance and Weinberg sum rules / OPE) Can already see contrast between N F = 2 and N F = 6... Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

12 Fit results for Π V A (0), N F = 2 and N F = 6 Vertical axis: 4πΠ V A (0) where Π d (0) = lim Q 2 0 dq 2 Π(Q2 ) S = 4πN D Π V A (0) S SM Horizontal axis: MP 2/M2 V 0 gives a more physical comparison than m M V 0 lim m 0 M V is matched between N F = 2 and N F = 6 Expect agreement in the quenched limit M 2 P Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

13 From slopes to S for M H = 125 GeV S = 4πN D d lim Q 2 0 dq 2 Π V A(Q 2 ) S SM (M H ) 1 N D doublets with chiral electroweak couplings contribute to S Scaled-up QCD often considers maximum N D = N F /2 but only N D 1 is required for electroweak symmetry breaking 2 S SM = 1 4 4M 2 P ds s [ ( ) 3 ] 1 1 M2 V 0 Θ(s MV 2 s 0) 1 ( M 2 ) 12π log V 0 M 2 H Integral diverges logarithmically as M 2 P 0 to cancel contribution of three eaten modes First term assumes M H M V 0 TeV; second term corrects for M H = 125 GeV TeV Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

14 S parameter, N F = 2 and N F = 6 Direct N F = 2 result lim S = 0.42(2) MP 2 0 matches scaled-up QCD Linear fit to light points (M P M V 0 ) guides the eye, accounts for any chiral logs remaining after S SM (M H ) S = A + B M2 P MV ( ) ( ) NF M 2 12π 2 1 log V 0 0 MP 2 for N D = 1 Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

15 Phenomenologically-relevant chiral limit Direct N F = 2 result lim M 2 P 0 S = 0.42(2) matches scaled-up QCD Significant N F = 6 reduction Lattice calculation involves NF 2 1 degenerate pseudoscalars Only three massless Goldstones eaten by W and Z, NF 2 4 must be given non-zero masses Imagine freezing those NF 2 4 masses at the blue curve s minimum, and taking only three to zero mass Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

16 Connection to effective theory approach The effective theory predicts the functional form of Π V A (Q 2 ), which we could have fit directly to determine α 1 = S/16π 2 Instead we used a rational function ansatz and had to subtract S SM Advantages of effective theory analysis: 1 Eliminates model dependence of Q 2 rational function ansatz 2 Incorporates S SM (M H ) directly into fit (resolving potential ambiguity from radiative corrections to M H ) Drawback: the analysis becomes more involved Let s consider the simpler case of WW scattering to see how it goes Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

17 WW scattering from the lattice: The Big Picture WW scattering guaranteed to contain information about EWSB Very direct probe (though not easiest) at LHC On the lattice, restricted to low-energy scattering (M. Buchoff) Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

18 WW scattering from the lattice: EFT matching Hadronic chiral lagrangian has m > 0 and g = 0 Electroweak chiral lagrangian has m = 0 and g > 0 Both reduce to same form in the limit m 0 and g 0 Hadronic EFT EW EFT m d 0 p 2 M 2 ds,m 2 ss g, g 0 p 2 M 2 ds,m 2 ss f 2 4 tr( µu µ U)+α 5 tr( µ U µ U) 2 + α4 tr( µ U ν U) 2 Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

19 Pseudoscalar scattering on the lattice: Target Simplest, cleanest, and best scattering process We focus on S-wave scattering of identical charged pseudoscalars (the I = 2 channel for N F = 2) Other isospin channels (e.g., I = 0) can involve fermion-line-disconnected diagrams Extremely expensive to evaluate on lattice Other spin channels (e.g., D-wave) have smaller signals, require higher precision We want to extract LECs l 1 and l 2 related to α 4 and α 5 in L χ These hide in the scattering length a PP Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

20 Pseudoscalar scattering on the lattice: Procedure Maiani & Testa, 1990 No asymptotically non-interacting in and out states in euclidean spacetime Lehmann Symanzik Zimmermann scattering formalism inapplicable In a finite volume, measure energy of two-pseudoscalar state E PP, projecting each correlator onto zero momentum for S-wave scattering Access scattering phase shift δ from energy shift E PP (Lüscher, 1986) E PP = E PP 2M P = 2 k 2 + MP 2 2M P k cot δ = 1 Λ J 1 πl j 2 k 2 L 2 /(4π 2 ) 4πΛ j j 0 ( 1 Low-energy scattering length is a PP = k cot δ + O ) k 2 Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26 M 2 P

21 Joint chiral fit to M 2 P, F P, ψψ and MP a PP M P 2 / 2 m N f =2 LO+NLO N f =2 N f =6 F P N f =2 LO+NLO N f =2 N f =6 M P/ m k cot δ m N f =2 LO+NLO N f =2 N f = m m a PP 1/ k cot δ ψψ in backup slide Only N F = 2 fit feasible Fit restricted to solid points, 0.01 m f 0.02 χ 2 /dof = 83/6 Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

22 N F = 2 NLO contribution to WW scattering Chiral fit predicts sum of hadronic LECs l 1 + l 2 EFT matching discussed above relates this to the sum α 4 + α 5 Matching involves one-loop standard model calculation [ ( ) ] ( ) α 4 + α 5 = 3.34 ± MH 2 128π 2 log v 2 + O(1) SM (dominant systematic error from chiral fit range) Context for our N F = 2 result Unitarity bounds [hep-ph/ ]: α 4 + α α Expected LHC bounds [hep-ph/ ]: (99% CL; 100/fb; 14 TeV) 7.7 < α < < α < 10 Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

23 Complications for N F > 2 As for the S parameter, only charge one chiral doublet d Here we take the other N F 2 to be electroweak singlets s, leading to N 2 F 4 pseudoscalars with masses M ds and M ss Hadronic chiral perturbation theory (χpt) now involves 9 LECs with more complicated relations to α 4 and α 5 Higher-order terms in χpt increase with N F Leads to smaller radius of convergence Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

24 Strategy: Reorganize chiral expansion Replace low energy constants B and F by measured M P and F P Expansion parameter is M 2 P /F 2 P, leading order is M Pa PP = M2 P 16πF 2 P M P/ k cot δ LO -1 N f =2 N f = ( M P / F P ) 2 An old story in QCD (Weinberg, 1966) Somewhat controversial: leading-order relation persists well beyond expected radius of convergence As a result, we can directly compare N F = 2 and N F = 6 LECs: N F = 6 scattering length only slightly smaller, but chiral logs differ... Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

25 Possible enhancement of WW scattering for N F = 6 Combined LEC b PP must increase from N F = 2 to N F = 6, to get similar a PP despite different chiral logs b PP = 256π2 [L 0 + 2L 1 + 2L 2 + L 3 2L 4 L 5 + 2L 6 + L 8 ] contains α 4 and α 5, but we aren t able to isolate them 0-2 N f =2 N f =6 b r PP (µ = a-1 ) b PP m = 4.67 ± (2f); b PP = 7.81 ± (6f) Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

26 Wrap-up & ongoing work: extend L χ with light Higgs Effective theory approach helps lattice calculations of LECs make contact with electroweak phenomenology (S, WW scattering) Matching EFTs provides rigor but requires complicated analyses Now we have another complication: a Higgs too light to integrate out Work in progress to include in extended effective theory Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

27 Backup: Light Higgs from lattice calculations LatKMI Collaboration SU(3) with N F = 8 fundamental M H v = 4(4) Lattice Higgs Collaboration SU(3) with N F = 2 sextet (two-index symmetric) M H v = 2(1) arxiv: claims radiative corrections could shift M H 600 GeV 125 GeV (that is, M H v 2 0.5) Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

28 Backup: Matching IR scales in the chiral limit Vector mass, nucleon mass, and inverse Sommer scale all match at 10% level between N F = 2 and N F = 6 M V 0 = 0.216(2) [2f]; 0.199(3) [6f]; (0.170(3) [8f] & 0.148(22) [10f]) Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

29 Backup: More on domain wall fermions Domain wall fermions add fifth dimension of length L s, a significant computational expense Exact chiral symmetry at finite lattice spacing in the limit L s, with nice continuum-like currents and flavor symmetries At finite L s = 16, residual mass m res m f ; m = m f + m res 10 5 m res = 2.6 [2f]; 82 [6f]; 268 [8f]; 170 [10f] Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

30 Backup: Electroweak vacuum polarization functions Π VV = 2Π 3e S = 4πN D lim Q 2 0 Π AA = 4Π 33 2Π 3e ] S SM (M H ) d [ dq 2 Π VV (Q 2 ) Π AA (Q 2 ) Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

31 Backup: Scaling up QCD gives S 0.4 N F 2 fermions in fundamental rep of SU(N) for N 3, with 1 N D N F /2 doublets given chiral electroweak charges S 0.3 N F 2 ( ) ( N 3 + N D 1 12π log MV 2 MP π log TeV 2 MH 2 1 Resonance contribution uses QCD phenomenology to model R(s) d 4π lim Q 2 0 dq 2 Π V A(Q 2 ) = 1 3π 0 ) ds s [R V (s) R A (s)] (essentially vector meson dominance with large-n scaling) 2 Chiral-log contribution based on leading-order chiral pert. theory GeV Higgs contributes 0.1 (leading-order estimate) May be subtlety regarding M H (cf. arxiv: ) for strong sector in isolation (no EW or radiative corrections) Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

32 Backup: More on current correlators S = 4πN D d lim Q 2 0 dq 2 Π V A(Q 2 ) S SM (M H ) new Π µν V A (Q) = Z x ( Π µν (Q) = [ ] e iq (x+bµ/2) Tr V µa (x)v νb (0) A µa (x)a νb (0) ) Q δ µν µ Q ν Q Π(Q 2 µ Q ν ) Π L (Q 2 ) Q = 2 sin (Q/2) Q 2 Q 2 Renormalization constant Z evaluated non-perturbatively Chiral symmetry of domain wall fermions = Z = Z A = Z V Z = 0.85 [2f]; 0.73 [6f]; 0.70 [8f]; 0.71 [10f] Conserved currents V and A ensure that lattice artifacts cancel... Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

33 Backup: Conserved and local domain wall currents Conserved currents: V a µ(x) = L s 1 s=0 j a µ(x, s) A a µ(x) = L s 1 s=0 ( sign s L ) s 1 j a 2 µ(x, s) j a µ(x, s) = Ψ(x + µ, s)p +µ τ a U x,µψ(x, s) Ψ(x, s)p µ τ a U x,µ Ψ(x + µ, s) where P ±µ 1 ± γ µ 2 Local currents: V a µ(x) = q(x)γ µ τ a q(x) A a µ(x) = q(x)γ µ γ 5 τ a q(x) q(x) = 1 γ 5 2 Ψ(x, 0) γ 5 Ψ(x, L s 1) 2 Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

34 Backup: Non-conservation of local currents Π µν V A (Q) = Z x e iq (x+bµ/2) Tr [ V µa (x)v νb (0) ] A µa (x)a νb (0) Local currents are simply qγ µ q, defined on the domain walls No Ward identity: Qµ [ x eiq x V a µ(x)v a ν (0) ] 0 Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

35 Backup: Ward identity for conserved currents Π µν V A (Q) = Z x e iq (x+bµ/2) Tr [ V µa (x)v νb (0) ] A µa (x)a νb (0) Conserved currents are point-split, summed over fifth dimension Obey Ward identity, PCAC: Qµ [ x eiq (x+bµ/2) V a µ(x)v a ν (0) ] = 0 Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

36 Backup: Lattice artifacts cancel in mixed correlators Plot shows divergence of local current [ in each correlator, e.g., e iq (x+bµ/2) Vµ(x)V a ν a (0) ] Q ν 0 x Cancellation seems due to conserved currents forming exact multiplet, also possible with overlap even staggered (Y. Lattice 2013) Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

37 Backup: Padé fit Q 2 -range dependence Uncorrelated fits to Padé-(1, 2) rational function, Π V A (Q 2 ) = a 0 + a 1 Q b 1 Q 2 + b 2 Q 4 = m=0 a mq 2m 2 n=0 b nq 2n 4Π ' Χ 2 dof Max Max Results reported above use Q 2 Max = 0.4 Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

38 Backup: Twisted boundary conditions for Π V A (Q 2 ) Twisted boundary conditions (TwBCs) Introduce external abelian field (add phase at lattice boundaries) Allows access to arbitrary Q 2, not just lattice modes 2πn/L Correlations = TwBCs do not improve Padé fit results for slope May help fits to chiral perturbation theory, where we need both small M P and small Q 2 Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

39 Backup: Spurious S 0 from finite volume effects Compare N F = 6 results on and lattice volumes: L = 16 results crash to zero as m 0, attributable to volume-induced parity doubling In tandem, M P freezes around M P L 5.5 L = 32 results show no such distortion of the spectrum, but we want more quantitative control over finite-volume effects Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

40 Backup: Finite-volume diagnostic plot Expect finite-volume effects to push points up and to the right M N êf P N f =2 m! 0 N f =6 N f = 10 N f = M P êf P Finite-volume effects may be significant in lightest N F = 6 point (N F = 12 data from Fodor et al., PLB 703:348 (2011) [ ]) Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

41 Backup: 8f results for S parameter Assuming M V 0 > 0 Expect agreement in the quenched limit M 2 P 10f analysis requires careful treatment of frozen topological charge Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

42 Backup: N F = 8 domain wall on staggered USBSM is investigating N F = 8 on volumes up to (initial results recently released in arxiv: ) I am carrying out mixed-action measurements for S May provide more information about finite-volume effects Lightest point shows clear finite-volume effects N F = 6 reduction began only for M 2 P M2 V 0 Also working on calculation with staggered fermions Currently measuring smaller masses on , generating Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

43 Backup: Mixed action procedure (LHPC, arxiv: ) HYP smear to reduce m res and get renormalization factors Z 1 Tune domain wall height M 5 and length L s of fifth direction so that residual chiral symmetry breaking m res m Tune bare valence mass m f so that M P matches unitary value M 5 = 1.8 and L s = 16 Z V Z A 1.08 m res m f /13 Need m f > m s to match M P : 1.7 m/m s 2.05 where m m f + m res Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

44 Backup: Valence domain wall m res and Z A Want m res m Want Z A 1 Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

45 Backup: NLO chiral expansions For general N F, M P a PP = 2mB 16πF 2 M 2 P = 2mB { 1 + 2mB F P = F ψψ = F 2 2mB 2m A = 2 N F + 2NF 2 + N3 F { [ 1 + 2mB (4πF) 2 b PP 2 N F 1 NF 2 + A ( ) ]} 2mB NF 2 log µ 2 [ (4πF ) 2 b M + 1 ( )]} 2mB log N F µ [ 2 (4πF ) 2 b F N ( )]} F 2mB 2 log µ { [ mB (4πF) 2 b C N2 F 1 ( ) ]} 2mB log N F µ 2 { 1 + 2mB LECs b are all linear combinations of low-energy constants L i LECs dependence on scale µ cancels the corresponding logs b C includes contact term mλ 2 m/a 2 NNLO MP 2 coefficients enhanced by N2 F (arxiv: ) Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

46 Backup: Chiral condensate with chiral fit Joint NNLOχPT fit to N F = 2 F P, M 2 P, ψψ Linear term clearly dominant Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

47 Backup: Reorganized chiral expansion for M P a PP Solve chiral expansions for measured M P and F P replace low-energy constants 2mB and F by M 2 P /F 2 P : M2 P M P a PP = 16πFP 2 { 1 + M2 P 16π 2 F 2 P [ b PP 2N f 1 N 2 f N f + N 2 f N 2 f log ( M 2 P µ 2 ) ]} Now b PP = 256π2 [L 0 + 2L 1 + 2L 2 + L 3 2L 4 L 5 + 2L 6 + L 8 ] No explicit factors of N f in b PP, all N f dependence due to dynamics affecting LECs L i Unable to untangle L i to recover l 1, l 2 α 4, α 5 Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26

48 Backup: Chiral perturbation theory for Π V A (Q 2 ) Π V A (Q 2 ) in hadronic χpt: Π V A (M 2 dd, Q2 ) = F 2 P Q2 [ { [ 8L r 10 (µ) + 1 M 2 24π 2 log dd µ 2 ] H ( 4M 2 dd Q 2 [ ( )] x 1 H(x) = (1 + x) + x log x )}] Match with S = 16π 2 α 1 in electroweak chiral lagrangian: ( { [ ] }) S(µ, M ds ) = [ 192π 1 2 L r 10 12π (µ) + 1 M 2 384π 2 log ds µ [ ] µ 2 + log 1 ]. M H 6 Phenomenology from Lattice Strong Dynamics Theory Seminar, 4 November / 26