Measuring the S Parameter on the Lattice

Transcription

1 Measuring the S Parameter on the Lattice David Schaich (Dissertation: Strong Dynamics and Lattice Gauge Theory) 12 May 2011 Measuring the S Parameter on the Lattice 12 May / 22

2 Setting the scene (U. Toronto) What explains the masses of elementary (non-composite) particles? Measuring the S Parameter on the Lattice 12 May / 22

3 Elementary particle masses Symmetries of nature appear to require massless elementary particles The Higgs mechanism hides these symmetries Englert and Brout, 1964; Higgs, 1964; Guralnik, Hagen and Kibble, 1964 The physics behind the Higgs mechanism remains unknown! Measuring the S Parameter on the Lattice 12 May / 22

4 Technicolor (Steven Weinberg 1976/1979, Leonard Susskind 1979) Proposal: The Higgs mechanism results from the dynamics of a new strong interaction Superficially similar to quantum chromodynamics (QCD), the fundamental theory of the strong nuclear force Despite superficial similarity, the dynamics can be very different... Measuring the S Parameter on the Lattice 12 May / 22

5 Different dynamics (a preview of my main results) S parameterizes information about the Higgs mechanism (more later) QCD-based conventional wisdom: S 1 (red) Experiment (black): S 0.15 ± 0.10 Dynamics of a realistic technicolor theory must be different than QCD... so why is the conventional wisdom based on QCD? Measuring the S Parameter on the Lattice 12 May / 22

6 It s a strong interaction! (Derek Leinweber) Implications Internal structure of composite particles very complicated Standard techniques (perturbation theory) inapplicable Measuring the S Parameter on the Lattice 12 May / 22

7 To perform non-perturbative calculations, define the theory on a space-time lattice (Kenneth Wilson, 1974) (Claudio Rebbi) Measuring the S Parameter on the Lattice 12 May / 22

8 Lattice quantum field theory Numerically evaluate observables O from the defining functional integral O = du O(U) e S(U) du e S(U) Generic workflow of lattice projects U: four-dimensional field configurations S: action giving probability distribution e S 1 Generate and save a stochastic sample of field configurations 2 Carry out measurements on saved configurations (my work) Both steps require inverting large ( 100M 100M) sparse matrices [D(U)] x,y ψ y = η x 90% of cost y Measuring the S Parameter on the Lattice 12 May / 22

9 400M core-hours on clusters and supercomputers Livermore Nat l Lab, NSF Teragrid, USQCD (DoE), BU SCF Measuring the S Parameter on the Lattice 12 May / 22

10 Recap Use lattice quantum field theory Study strongly-interacting physics beyond QCD Apply to the Higgs mechanism Next Parameterize information about the Higgs mechanism S parameter provides most stringent constraint Experiment: S 0.15 ± 0.10 QCD: S = 0.32 ± 0.03 Measuring the S Parameter on the Lattice 12 May / 22

11 The S parameter (Michael Peskin and Tatsu Takeuchi, 1991) S = 4πN D d lim Q 2 0 dq 2 Π V A(Q 2 ) S SM 1 The vacuum polarization function Π V A (Q 2 ) is schematically On the lattice, it is the difference of vector and axial-vector two-point correlation functions 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 2 N D 1 is the number of pairs of contributing techni-fermions 3 S SM is subtracted so that S = 0 in the standard model (the simplest model of the Higgs mechanism) Measuring the S Parameter on the Lattice 12 May / 22

12 The model (last stop before data) Take advantage of experience with lattice QCD QCD is an SU(3) gauge theory with N f = 2 light fermions Study an SU(3) gauge theory with N f = 6 light fermions Compare N f = 2 and N f = 6 calculations to see what changes 1 Match lattice spacing (energy scale) to compare directly 2 Perform calculations with a range of light fermion masses m f Measuring the S Parameter on the Lattice 12 May / 22

13 Polarization function data and fits 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) (Z is a renormalization constant I compute non-perturbatively) Very smooth data fit with functional form a 0 + a 1 Q b 1 Q 2 + b 2 Q 4 Results stable and χ 2 /dof 1 as Q 2 fit range varies extract slope at Q 2 = 0 Measuring the S Parameter on the Lattice 12 May / 22

14 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 P is the pseudoscalar ground state mass at each m f M V 0 is the vector ground state mass in the limit m f 0 (matched between N f = 2 and N f = 6) Measuring the S Parameter on the Lattice 12 May / 22

15 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 For large MP 2 > M2 V 0, techni-fermions are very heavy and have little effect on the dynamics Expect results to agree in this heavy-fermion region Measuring the S Parameter on the Lattice 12 May / 22

16 The last steps from slope to S S = 4πN D d lim Q 2 0 dq 2 Π V A(Q 2 ) S SM 1 N D pairs of techni-fermions contributing to S Most straightforward to have N D = N f /2 (only N D 1 is required by the Higgs mechanism) 2 S SM = 1 4 4M 2 P ds s 1 ( 1 M2 V 0 s ) 3 Θ(s M 2 Diverges logarithmically in the limit MP 2 0 (divergence exactly cancels if N f = 2) Small ( 10%) reduction in this work (MP 2 > 0 on the lattice) V 0 ) Measuring the S Parameter on the Lattice 12 May / 22

17 S parameter, N f = 2 and N f = 6 N D = N f /2 lim M 2 P 0 S = 0.31(2) for N f = 2 Linear fit to light points MP 2 < M2 V 0 guides the eye, accounts for logarithmic divergence as MP 2 0 [ ] ( ) S = A + BMP N 2 f M 2 12π 4 1 log V 0 M 2 P Measuring the S Parameter on the Lattice 12 May / 22

18 Main result: deviation from conventional wisdom Conventional wisdom: S 1 (red) Experiment (black): S 0.15 ± 0.10 This is the most straightforward case with maximal N D = N f /2 = 3 Measuring the S Parameter on the Lattice 12 May / 22

19 Systematic effects Finite volume At most percent-level effect for N f = 2 (except lightest point) Being directly checked for N f = 6 Nonzero lattice spacing a between sites Effects reduced to O(a 2 ) by our lattice action Chiral symmetry breaking Explicitly broken by many lattice actions, in addition to M 2 P > 0 Effects reduced by using (expensive) chiral lattice action M 2 P 0 limit is subtle (recall cancellation from S SM) Autocorrelations and topological effects (!) Exploratory calculations with short (but expensive) simulations Combined statistical and systematic errors of 10 20% result robust (PRL in press) Measuring the S Parameter on the Lattice 12 May / 22

20 Preliminary results for N f = 10 (ongoing work) Measuring the S Parameter on the Lattice 12 May / 22

21 Conclusion Higgs mechanism required by elementary particle masses If the Higgs mechanism involves strong interactions, the lattice can provide non-perturbative information The S parameter provides the most stringent constraint For SU(3) gauge theory with N f = 6 I find S closer to experiment than the conventional wisdom Measuring the S Parameter on the Lattice 12 May / 22

22 Thank you!

23 Acknowledgements At BU Ron Babich, Rich Brower, Michael Cheng, Mike Clark, Saul Cohen, James Osborn, Claudio Rebbi Tom Appelquist, Mike Buchoff, George Fleming, Fu-Jiun Jiang, Joe Kiskis, Meifeng Lin, Ethan Neil, Pavlos Vranas Funding and computing resources

24 The end 1 Introduction 2 Lattice quantum field theory 3 S parameter on the lattice 4 Backup: EWSB 5 Backup: ETC, walking and the conformal window 6 Backup: Lattice topics 7 Backup: More on the S parameter 8 Backup: Lattice Strong Dynamics Collaboration results Backup: Scale setting Backup: NLOχPT Backup: Condensate enhancement Backup: Parity doubling Backup: Conserved currents 9 Backup: Asides and extensions

25 Experimental confirmation of electroweak theory (Gfitter Group)

26 Gauge invariance example: electromagnetism Electric and magnetic fields in terms of potentials Φ and A E = A t Φ B = A E and B are invariant under the gauge transformation Φ Φ Λ t A A + Λ In four-vector notation, A µ = (Φ, A) A µ + µ Λ Photon mass term in lagrangian is 1 2 m2 γa µ A µ = 1 2 m2 γ Forbidden by gauge invariance! (A A Φ 2)

27 Massless fermions from chiral gauge theory Fermion mass term in lagrangian is mψψ = m ( ψ L ψ R + ψ R ψ L ) (Chris Quigg) Forbidden by gauge invariance! ψ L ψ R ( ψ ψ )L (ψ) R

28 Fermion masses in standard model Need to make a gauge-invariant object involving ψ L ψ R ( ψ ψ )L (ψ) R ( ) φ + Standard model solution: stick in a Higgs Φ = With vacuum Φ = λ ψ ( ψ ψ ) ( 0 v/ 2 L ( φ + φ 0 ) (ψ) R φ 0 ), identify m ψ = λ ψ v/ 2. All fermion masses and mixing are arbitrary free parameters!

29 Gauge boson masses in standard model ( Φ = φ 1 + iφ 2 v/ 2 + h + iφ 3 ) L Φ = (D µ Φ) (D µ Φ) + µ 2 Φ Φ λ D µ = ( µ + i 2 g 1B µ ) I + i 2 g 2W a µσ a ( ) 2 Φ Φ v = µ 2 /λ W ± and Z masses pop out of (D µ Φ) (D µ Φ). Relevant terms: v 2 ( g2 Wµ 3 g (0 1) 1 B µ g 2 (Wµ 1 iwµ) 2 8 g 2 (Wµ 1 + iwµ) 2 g 2 Wµ 3 g 1 B µ g2 2 v 2 (0 1) 8 ) 2 ( 0 1 ( 2W + µ 2W µ (g g2 2 )1/2 Z µ /g 2 ) 2 ( 0 1 MW 2 W +µ Wµ M2 Z Z µ Z µ + M W = 1g 2 2v = (M Z /g 2 ) g1 2 + g2 2 M Z cos θ W ) )

30 Gauge boson masses in technicolor Now we have pions with ] L χ = FP [(D 2 Tr µ Σ) (D µ Σ) /4 Σ = exp (2iσ a π a /F P ) q L q R ) D µ = I µ i g 2 2Wµσ a a Wµ a = (W µ, 1 Wµ, 2 Wµ 3 g 1 B µ /g 2 W ± and Z masses pop out of FP 2Tr D µσ 2 /4. Relevant terms: ( µ π a ) 2 F P g 2 ( µ π a )Wµ/2 a + FP 2 g2 2 (Wa µ) 2 /16 = [ F P g 2 Wµ/4 a µ π a] 2 [ = FP 2 g2 2 (Wµ) (Wµ) 2 2] /8 + FP 2 (g2 2 + g2 1 )Z µ/8 2 M 2 W W +µ W µ M2 Z Z µ Z µ F P = v

31 Triviality of fundamental Higgs 1 λ(µ) [1/λ(Λ)] + (3/2π 2 ) log(λ/µ) < 2π 2 3 log(λ/µ) ( ) 4π 2 v 2 Λ M H exp 3M 2 H M H = 115 GeV Λ GeV M H = 700 GeV Λ 1000 GeV

32 (Extended) technicolor in a picture

33 Fermion masses in extended technicolor Integrating out ETC gauge bosons produces four-fermion operators that provide both SM fermion masses and FCNCs Masses: (T T )(qq) M 2 ETC FCNCs: (qq)(qq) M 2 ETC FCNCs required by CKM mixing, limit obtainable SM fermion masses.

34 Walking Technicolor T T METC = T T ( ) METC exp dµ ΛTC Λ TC µ γ(µ) T T ( ) γ METC ΛTC Λ TC γ(µ) 1 for Λ TC µ M ETC enhances fermion masses Implies large, slowly-running ( walking ) coupling, small β function

35 Perturbative Yang Mills β function For SU(N c ) Yang Mills theory with N f fermions in representation r β(g) = µ g µ = β 0g 3 + β 1 g 5 + β 2 g 7 + β 0 = 1 ( 11 (4π) 2 3 N c 4 ) 3 N f C(r) β 1 = 1 [ ( (4π) 4 3 N2 c 3 N c 1 ) ] N N f C(r) c Higher-order β i depend on choice of renormalization scheme C(N) = 1 2 C(Adj) = N c C 2 (N) = d(adj) d(n) C(N) = N2 c 1 2N c

36 Conformal window Strongly-coupled gauge theories can look very different than QCD With many fermions, theory has perturbative IR fixed point; it is in an IR-conformal phase with no spontaneous χsb The conformal window ranges from loss of asymptotic freedom to some (unknown) critical Nf c < Nf AF With N f Nf c, may be approximately conformal (walking!) for some range of scales Visualization of conformal window for SU(N c ) fermions in fundamental rep:

37 Anomalous dimension From rainbow approximation to Schwinger Dyson equation γ(µ) = 1 1 3C 2 (r)α(µ)/π 1 Assume spontaneous chiral symmetry breaking when α(µ) When α(µ) = α χsb, this gives γ(µ) = 1 π 3C 2 (r) α χsb

38 Searching for conformal windows

39 Iwasaki gauge action U x,µ = exp [iaga µ (x + µ/2)] (directed from x + µ to x) [ ] P x,µν = Tr U x,µ U x+bµ,ν U x+bν,µ U x,ν [ ] R x,µν = Tr U x,µ U x+bµ,µ U x+bµ+bν,ν U x+bµ+bν,µ U x+bν,µ U x,ν { S G = 2N c g ) (1 1Nc P x,µν x µ,ν;µ<ν ) (1 } 1Nc R x,µν µ,ν 1 4 F µνf µν + O(a 2 ) as a 0

40 Domain wall fermions Add fifth dimension of length L s Exact chiral symmetry at finite lattice spacing in the limit L s At finite L s, residual mass m res > 0; m = m f + m res

41 Domain wall Dirac operator Dx,y(M W 5 ) = (4 M 5 )δ x,y 1 [ (1 + γ µ )U 2 x,µδ x,y+µ ] + (1 γ µ )U x,µ δ x+µ,y [ ] D s,s (m) = D W [ ] (M 5 ) + 1 δ s,s + P L (1 + m)δs,ls 1δ s,0 δ s+1,s [ ] + P R (1 + m)δs,0 δ s,l s 1 δ s,s +1 D(m) = D W + 1 P L 0 mp R P R D W + 1 P L 0 0 P R D W mp L 0 0 D W + 1 P L = 1 2 (1 γ 5), P R = 1 2 (1 + γ 5); M 5 < 2 is height of domain wall

42 Hybrid Monte Carlo algorithm 1 Generate random momenta with Gaussian distribution e p2 /2 2 Molecular dynamics evolution through fictitious MD time to produce new four-dimensional field configuration 3 Use MD discretization errors in Metropolis accept/reject step Numerically evaluate observables from the defining functional integral O = DU O(U) e S(U) DU e S(U) U: four-dimensional field configurations S: action giving probability distribution e S

43 Finite-volume effects depend on regime (Mike Buchoff)

44 Polarization functions for the S parameter Π VV = 2Π 3e S = 4π lim Q 2 0 Π AA = 4Π 33 2Π 3e ] S SM d [ dq 2 Π VV (Q 2 ) Π AA (Q 2 )

45 Experimentally, S 0 Extract S from global fit to experimental data Z decay partial widths and asymmetries Neutrino scattering cross sections M W, M Z Atomic parity violation (PDG)

46 S from scaling up QCD Analyticity and optical theorem relate polarization functions Π(Q 2 ) to spectral functions R(s) R(s) = σ(e+ e hadrons) σ(e + e µ + µ ) Π(Q 2 ) = Π(0) + Q2 12π 2 S = 4πN D Π V A (0) S SM 0 dsr(s) s + Q 2 S = 1 3π 0 { ( ) 3 ds N D (R V R A ) M2 ( H) } H Θ s M 2 s 4 s Replacing the QCD scale with the electroweak scale, S = 0.32 ± 0.03

47 S parameter on the lattice S = 4πN D Π V A (0) S SM Π VV = 2Π 3Q Π AA = 4Π 33 2Π 3Q On the lattice, correlators involve a single pair of fermions 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 ( ) Q Π µν (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 Conserved currents V and A ensure that lattice artifacts cancel V µa (x)v νa (0) and A µa (x)a νa (0) require O(L s ) inversions Renormalization constant Z computed non-perturbatively Z = 0.85 (2f); 0.73 (6f); 0.71 (10f)

48 Lattice Strong Dynamics Collaboration Argonne James Osborn Berkeley Sergey Syritsyn Boston Ron Babich, Richard Brower, Saul Cohen, Claudio Rebbi, DS Brookhaven Oliver Witzel Fermilab Ethan Neil Harvard Mike Clark Livermore Mike Buchoff, Michael Cheng, Pavlos Vranas, Joe Wasem UC Davis Joseph Kiskis Yale Thomas Appelquist, George Fleming, Meifeng Lin, Gennady Voronov Performing non-perturbative studies of strongly interacting theories likely to produce observable signatures at the Large Hadron Collider

49 LSD Philosophy and Simulation Details Start from what we know (QCD) and use it as a baseline SU(3) gauge theory with N f = 2, 6, 10 Work on large lattices so finite-volume effects are small with m f gives M P L 4 Directly compare the different theories Tune parameters to match chiral symmetry breaking scale Plot results versus M 2 P rather than m = m f + m res Exploratory calculations O(100) independent measurements per point Studying spontaneous chiral symmetry breaking Domain wall fermions with L s = 16 m res (2f); (6f); (10f)

50 Goldstone decay constant

51 Sommer scale, vector and nucleon masses N f = 2 and N f = 6 all match at 10% level

52 Pseudo Nambu Goldstone boson mass

53 NLOχPT for general N f Chiral perturbation theory (χpt) gives an effective field theoretic description of theories with chiral symmetry breaking, in terms of effective low-energy constants. MP 2 { 2m = B 1 + 2mB (4πF ) { 2 F P = F 1 + 2mB (4πF) 2 ψψ = F 2 B { 1 + 2mB (4πF) 2 [ α m + 1 ( )]} 2mB log N f (4πF) [ 2 α F N ( )]} f 2mB 2 log (4πF) [ 2 ( α C N2 f 1 2mB log N f (4πF ) 2 ) ]} α C includes contact term mλ 2 ma 2 NNLO M 2 P coefficients enhanced by N2 f (Johan Bijnens and Jie Lu, 2009)

54 Goldstone decay constant with NLOχPT fit Joint NNLOχPT fit to N f = 2 F P, M 2 P, ψψ

55 Pseudo Goldstone boson mass with NLOχPT fit Slope of MP 2 with m significantly larger for N f = 6 Plot against MP 2, to provide more physical comparison

56 Chiral condensate with NLOχPT fit Joint NNLOχPT fit to N f = 2 F P, M 2 P, ψψ Linear term clearly dominant

57 Chiral condensate enhancement: introduction Search for enhancement through ψψ /F 3 Not RG invariant: keep cutoff fixed in physical units Focus on the ratio R of ψψ /F 3 between N f = 6 and N f = 2 ( 5Mρ R = ( ψψ exp /F 3 ) 6f ( ψψ M ρ = ( /F 3 ) 5Mρ 2f exp M ρ γ(µ) µ γ(µ) µ ) dµ 6f ) dµ 2f MS perturbation theory & perturbative conversion to lattice scheme predicts R = 1.27(7)

58 Enhancement of ψψ /F 3, N f = 2 to N f = 6 Find significant enhancement compared with perturbative R = 1.27(7)

59 NLO χpt fits, N f = 2 and N f = 6 NLOχPT fits work for N f = 2 but not N f = 6 (lighter m f required) GMOR ψψ M = 3 Fπ 3 π q(2m) 3 ψψ = M2 π 2mF π R as m 0 Fit ratios to R [ 1 + m(α XY 10 + α 11 log m) ] where m m 2 m 6

60 Vector and axial spectrum Signs of N f = 6 parity-doubling as M 2 P decreases consistent with reduced S parameter

61 Vector and axial decay constants Parity-doubling involves F V F A in addition to M V M A

62 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) = P L Ψ(x, 0) + P R Ψ(x, L s 1)

63 Ward identities and violations bq µ ˆPx eiq (x+bµ/2) V a µ(x)v a ν (0) = 0 bq µ ˆPx eiq x V a µ(x)v a ν (0) 0 ˆP x eiq (x+bµ/2) ` V a µv a ν Aa µ A a ν bqν 0 ˆP x eiq x ` V a µv a ν Aa µ A a ν bqν 0

64 Single-doublet S parameter

65 Illustration of thermalization M P and M V calculated using 100 configurations Full thermalization analysis considers plaquette, ψψ and correlators themselves Also compare runs starting from random vs. constant field

66 Results from runs with different starting configuration

67 Single-pole approximations to Π V A R V (s) = 12π 2 F 2 V δ(s M2 V ) R A(s) = 12π 2 F 2 A δ(s M2 A ) Π V A (Q 2 ) = F 2 P + Q2 F 2 V M 2 V + Q2 Q2 F 2 A M 2 A + Q2

68 S in χpt, for N f = 2 l 5 is extracted from S = 1 12π [ 1 Π V A (Q 2 ) = FP 2 + Q2 24π 2 J(x) = 1 16π 2 l 5 + log M2 P v 2 FP 2 MH ( l ) + 23 (1 + x)j(x) ] ( 1 + x log [ 1 + x x + 1 ] + 2 (Gasser and Leutwyler) ), x 4MP 2 /Q2 Our N f 6 simulations have M P too large to apply χpt General-N f corrections for l 5 not yet known Must take only two techni-fermions to the massless limit, any others remain massive

69 Comparing Padé and OPE, N f = 2 As Q 2, Π V A (Q 2 ) N c 8π 2 m2 + m ψψ Q 2 + O(α) + O(Q 4 )

70 Corrections to the first Weinberg sum rule, N f = 2 Q 4 term in numerator of (2, 2) Padé is small [ ] a 0 + a 1 Q 2 + a 2 Q b 1 Q 2 + b 2 Q 4 = F0 2 + Q2 F1 2 M1 2 + Q2 F2 2 Q2 M2 2 + Q2

71 Ongoing refinements and future directions Additional runs at m f = to confirm trend Ordered-start runs for N f = 10 (may help understand topology) Direct calculation of finite-volume effects Twisted boundary conditions to probe smaller Q 2 N f = 8 simulations with cheaper (staggered) lattice action Operator product expansion for Π µν V A (Q)