Fun with the S parameter on the lattice

Transcription

1 Fun with the S parameter on the lattice David Schaich (Boston Colorado Syracuse) from work with the LSD Collaboration and USQCD BSM community arxiv: & arxiv: Origin of Mass 2013 Lattice BSM Workshop, Odense, 7 August David Schaich (Colorado) Lattice S Parameter Origin of Mass 2013, 7 August 1 / 10

2 Overview You saw this plot yesterday Today I informally summarize how I made it (From the 2013 USQCD white paper Lattice Gauge Theories at the Energy Frontier) David Schaich (Colorado) Lattice S Parameter Origin of Mass 2013, 7 August 2 / 10

3 S on the lattice: definitions S = 4πN D d lim Q 2 0 dq 2 Π V A(Q 2 ) S SM (M H ) 1 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) (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 Results below (from arxiv: ) take M H 1000 GeV This morning: May not need to shift all the way to 125 GeV David Schaich (Colorado) Lattice S Parameter Origin of Mass 2013, 7 August 3 / 10

4 Cancellation of lattice artifacts 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 (Conserved DWF currents satisfy vector Ward identity & PCAC) x Cancellation seems due to conserved currents forming exact multiplet, also possible with overlap even staggered (Y. Lattice 2013) David Schaich (Colorado) Lattice S Parameter Origin of Mass 2013, 7 August 4 / 10

5 Slope at Q 2 = 0 from fit to rational function Very smooth data fit to Padé-(1, 2) functional form (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 (motivations: single-pole dominance, Weinberg sum rules) Results stable and χ 2 /dof 1 as Q 2 fit range varies Sannino, arxiv: : take m 0 first in conformal window David Schaich (Colorado) Lattice S Parameter Origin of Mass 2013, 7 August 5 / 10

6 S parameter, N F = 2 and N F = 6 lim S = 0.31(4) MP 2 0 for N F = 2 Linear fit to light points (M P M V 0 ) guides the eye, accounts for any chiral logs remaining after S SM S = A + B M2 P MV ( ) ( ) NF M 2 12π 2 1 log V 0 0 MP 2 for N D = 1 David Schaich (Colorado) Lattice S Parameter Origin of Mass 2013, 7 August 6 / 10

7 Phenomenologically-relevant chiral limit lim S = 0.31(4) MP 2 0 for N F = 2 Lattice calculation involves NF 2 1 degenerate pseudoscalars Only three massless NGBs eaten in Higgs mechanism, NF 2 4 must be massive PNGBs Imagine freezing N 2 F 4 PNGB masses at the blue curve s minimum, and taking only three to zero mass David Schaich (Colorado) Lattice S Parameter Origin of Mass 2013, 7 August 7 / 10

8 Beware spurious S 0 from finite volume effects Compare N F = 6 results on vs L = 16 results crash to zero as m 0 As L = 16 results crash to zero, M P freezes around M P L 5.5 L = 32 results show no such behavior, but we want more quantitative control over finite-volume effects David Schaich (Colorado) Lattice S Parameter Origin of Mass 2013, 7 August 8 / 10

9 Initial results for N F = 8 domain wall on staggered Last week I spoke about the first USBSM project: (slides) N F = 8 staggered on larger volumes ( , ) I am carrying out mixed-action measurements for S May provide more information about finite-volume effects for N F = 6 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 David Schaich (Colorado) Lattice S Parameter Origin of Mass 2013, 7 August 9 / 10

10 Recapitulation You saw this plot yesterday (from 2013 USQCD white paper) Today I informally summarized how I made it Many backup slides follow in case there are questions

11 Backup: Polarization functions for the S parameter Π VV = 2Π 3e S = 4πN D lim Q 2 0 Π AA = 4Π 33 2Π 3e ] S SM d [ dq 2 Π VV (Q 2 ) Π AA (Q 2 )

12 Backup: Experimentally, S = 0.03(10) (Gfitter, arxiv: ) For the old reference Higgs mass scale of 1 TeV, S 0.15(10) (2010 PDG) Z decay partial widths and asymmetries Neutrino scattering cross sections M W, M Z Atomic parity violation

13 Backup: Scaling up QCD gives S 0.3 N F 2 fermions in fundamental rep of SU(N) for N 3, all N D = N F /2 doublets given chiral electroweak charges S 0.3 N F 2 ( ) ( ) N NF 2 12π 4 1 MV 2 log MP 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 ds 3π 0 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 Both contributions are non-negative; first is 0.3, second 0 Can reduce many-n F logs by giving chiral charges to only one doublet

14 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 = 0.215(3) [2f]; 0.209(3) [6f]

15 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

16 Backup: Conserved and local domain wall currents Conserved currents: V µa (x) = L s 1 s=0 Local currents: j µa (x, s) A µa (x) = L s 1 s=0 ( sign s L ) s 1 j µa (x, s) 2 j µa (x, s) = Ψ(x + µ, s) 1 + γµ τ a U 2 x,µψ(x, s) Ψ(x, s) 1 γµ τ a U x,µ Ψ(x + µ, s) 2 V µ (x) = q(x)γ µ τ a q(x) A µ (x) = q(x)γ µ γ 5 τ a q(x) q(x) = 1 γ5 2 Ψ(x, 0) γ5 Ψ(x, L s 1) 2

17 Backup: More on current correlators S = 4πN D d lim Q 2 0 dq 2 Π V A(Q 2 ) S SM 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 = Z A = Z V for chiral fermions Computed non-perturbatively: Z = 0.85 (2f); 0.73 (6f) Conserved currents V and A ensure that lattice artifacts cancel

18 Backup: More on local domain wall 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 simple qγ µ q, defined on domain walls No Ward identity: Qµ [ x eiq x V a µ(x)v a ν (0) ] 0

19 Backup: More on conserved domain wall 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

20 Backup: Padé fit Q 2 -range dependence Fitting 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 here used Q Max = 0.4

21 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 in conformal window, where we want to extrapolate m 0 at small fixed Q 2

22 Backup: Chiral perturbation theory for Π V A (Q 2 ) (Used by JLQCD & RBC UKQCD in QCD projects) Effective field theory predicting dependence on MP 2 Need both MP 2 and Q2 smaller than ours N F > 2 produces complications and Q2 For N F = 2, (Gasser and Leutwyler, 1984) S = 1 l 5 + log M2 P v 2 FP 2 12π MH [ ( 1 Π V A (Q 2 ) = FP 2 + Q2 l 5 1 ) + 23 ] 3 (1 + x)j(x) J(x) = 1 16π 2 24π 2 ( [ ] x 1 + x log x + 1 ), x 4MP 2 /Q2

23 Backup: 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: M 2 P /M2 V 0 gives a more physical comparison than m f 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

24 Backup: From slopes to S S = 4πN D d lim Q 2 0 dq 2 Π V A(Q 2 ) S SM 1 N D doublets with chiral electroweak couplings contribute to S Scaled-up QCD usually 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 1 ( 1 M2 V 0 s ) 3 Θ(s M 2 V 0 ) No direct dependence on N F or N D Diverges logarithmically in the limit M 2 P 0 to cancel contribution of three eaten modes Small ( 10%) reduction in this work (M 2 P > 0)

25 Backup: More on finite-volume effects Range of accessible masses determined by lattice volume If masses get too small, finite-volume effects significant m! 0 N f = 10 M N êf P N f = N f = M P êf P In this diagnostic, finite-volume effects push points up and to the right by increasing the masses but decreasing F P

26 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

27 Backup: Valence domain wall m res and Z A Want m res m Want Z A 1

28 Backup: pseudoscalar decay constant For N F = 8 staggered, 1 F P L 2.1 Fitted points: 1.25 F P L Intercept /48 For N F = 2 domain wall, 0.8 F P L 1.2, intercept /32 For N F = 6 domain wall, 0.7 F P L 1.6, intercept /32 All simple linear extrapolations NLOχPT may have significant effect

29 Backup: Connection to parity-doubling 4πΠ V A (0) = 1 3π 0 ds s [R V (s) R A (s)] 4π [ F 2 V M 2 V F 2 A M 2 A for R(s) 12πF 2 δ(s M 2 ) ] Signs of N F = 6 parity-doubling consistent with reduced S Direct checks of finite-volume effects underway

30 Backup: More parity-doubling plots

31 Backup: N F = 10 thermalization and autocorrelations We generate a Markov chain of gauge field configurations = Nearby links in the chain are correlated From initial state, system thermalizes to equilibrium distribution Independent measurements require autocorrelations to die off Independent ensembles starting from either random or ordered states = Find thermalization time from convergence to equilibrium

32 Backup: N F = 10 thermalization and autocorrelations Random and ordered initial states Goal: convergence to equilibrium = thermalization time Result: two independent samples