Strongly Coupled Gauge Theories

1. Strongly Coupled Gauge Theories Richard Kenway

2. times they are a changing… Strongly Coupled Gauge Theories

3. dynamical origins of mass … overview • lattice gauge theories • more than just QCD • non-perturbative dynamics • hierarchy problem 1: why is the proton light? • chiral symmetry • hierarchy problem 2: why are some fermions light? • 2+1 flavour QCD • QCD is correct: which quarks are light? • low-energy constraints on BSM physics • precision tests of the SM • technicolor/non-QCD-like strongly interacting theories • hierarchy problem 3: why are some quarks heavy? Strongly Coupled Gauge Theories

4. lattice gauge theories a U(x) q(x) U(x) • Euclidean space-time lattice regularisation • lattice spacing a, spatial size L • g2 and mf are fundamental parameters of the SM Strongly Coupled Gauge Theories

5. lattice gauge theories L a • Monte Carlo approximation to path integral • N gauge configurations • lattice spacing must be extrapolated to zero, keeping the box large enough • by approaching a critical point ξ Strongly Coupled Gauge Theories

6. lattice gauge theories lattice gauge theory • hadronic scheme (QCD) • at each value of g2, fix quark masses mf by matching Nf hadron mass ratios to experiment • one dimensionful quantity fixes the lattice spacing in physical units fermion masses + gauge coupling physical observables Strongly Coupled Gauge Theories

7. continuum limit • SU(N) gauge theory with Nf fermions in the fundamental representation • β0 > 0: UV stable fixed point at g = 0 (‘asymptotic freedom’ in QCD) • dimensionless ratios become independent of g2 if a is small enough (scaling) N = 3 Sommer hep-ph/0607088 Strongly Coupled Gauge Theories

8. hierarchy problem 1 • why is the proton mass ≈ 1 GeV so far below the Planck scale?  why is gravity weak? • asymptotic freedom permits hadron masses to be exponentially smaller than the Planck scale • lattice QCD computes the dimensionless ratios Strongly Coupled Gauge Theories

9. universality • symmetries of the lattice theory define the universality class • which defines the continuum theory non-perturbatively • Lorentz invariance is an “accidental” symmetry as a → 0 • there are no relevant operators to break it • gauge invariance is preserved at the sites of the lattice • by Wilson’s construction • chiral and flavour symmetries can be realised correctly • 5D formulations realise full chiral symmetry at a ≠ 0 on 4D domain walls • confinement of quarks • for SU(3) there is no phase transition into an unconfined phase as mf, g are tuned to the critical line (a → 0) Strongly Coupled Gauge Theories

10. hierarchy problem 2 • why are some fermions light? • how is chiral symmetry realised non-perturbatively? • 4D lattice theories have the fermion doubling problem • if the lattice fermion action is bilinear, local, translation invariant, Hermitian, and has continuous chiral symmetry, then the continuum limit contains fermions in opposite chirality pairs • Nature is not like this! • 5D lattice theories do not have chiral symmetry • γ5 appears in the action • Wilson’s explicit dimension 5 term removes the doublers Strongly Coupled Gauge Theories

11. hierarchy problem 2 y,z,t L5 • left and right handed fermions may be localised as zero modes on a 4D domain wall in 5D RBC, UKQCD arXiv:0705.2340 exponential suppression of ρ(0) in QCD • chiral symmetry arises naturally (without fine tuning in 5D) • light quarks may be a hint of extra dimensions Strongly Coupled Gauge Theories

12. 2+1 flavour QCD • Rational Hybrid Monte Carlo (RHMC) • exact for any number of flavours • highly optimised → much faster than previous algorithms Strongly Coupled Gauge Theories

13. easily computed quantities decays asymptotically with energy of lightest state created by O determines matrix elements such as at large time separations, 2 >> 1 >> 0, can isolate matrix elements such as • no general method for multi-hadron final states, eg K  Strongly Coupled Gauge Theories

14. explore different versions of QCD • Edinburgh plot for 2+1 flavours RBC, UKQCD arXiv:0804.0473 Strongly Coupled Gauge Theories

15. QCD is the correct theory of hadrons • 2004 • quenched QCD (Nf = 0) disagrees with experiment • Nf = 2+1 QCD agrees within 1σ HPQCD, MILC, FNAL hep-lat/0304004 Strongly Coupled Gauge Theories

16. effective theories mQ>> QCD lattice QCD a, mq, mQ, L QCD scale QCD L >> QCD-1 mq <<QCD • simulations at physical parameter values are too expensive • use effective theories to extrapolate results from parameter regimes where systematic errors can be controlled to the physical regime • determine region of validity of effective theories HQET/NRQCD Lüscher finite volume effective theory Symanzik effective field theory a << QCD-1 chiral perturbation theory Strongly Coupled Gauge Theories

17. which quarks are light? • chiral perturbation theory • expansion in quark masses and momenta SU(2) x SU(2) SU(3) x SU(3) • SU(3) NLO fits do not work up to ms RBC, UKQCD arXiv:0804.0473 Strongly Coupled Gauge Theories

18. simulations at the physical point • … almost • mPS≥ 156 MeV (input mπ, mK, mΩ) • confirms that NLO corrections to SU(3) chiral perturbation theory are large at ms • but L = 2.9 fm may be too small PACS-CS arXiv:0807.1661 Strongly Coupled Gauge Theories

19. quark masses 2.527(47) 72.72(78) 28.8(4) Strongly Coupled Gauge Theories

20. lattice QCD is a predictive tool • quantities computed before experimental measurement • give confidence in using lattice QCD to search for new physics Strongly Coupled Gauge Theories

21. precision flavour physics Compare RGI values: Theory: BK = 0.72 ± 0.04 Experiment: BK = 0.78 ± 0.09 ^ ^ • CP violation RBC, UKQCD arXiv:0804.0473 • lattice QCD is reducing uncertainties in SM parameters Strongly Coupled Gauge Theories

22. precision flavour physics  e+ Vxy W+ qx qy • semileptonic B decays • lattice is needed to compute deviations from HQET at zero recoil  FNAL, MILC arXiv:0808.2519 Strongly Coupled Gauge Theories

23. precision flavour physics • hadronic matrix elements of all ΔS = 2 four-fermi operators can constrain models of new physics • to differentiate between models (eventually) + lattice QCD Strongly Coupled Gauge Theories

24. a signal of new physics? • D(s)→ ℓνℓdecays • Vcs and Vcd are well constrained by the unitarity triangle → experimental determination of decay constants • fDsdiffers by 3.8σ • 45-parameter chiral/continuum extrapolation HPQCD, UKQCD arXiv:0706.1726 Strongly Coupled Gauge Theories

25. proton decay • SUSY/GUTs mean protons decay • hadronic matrix elements of baryon-number violating operators constrained by SM symmetries dim 4 forbidden by R parity: dim 5 colour triplet Higgsino exchange Strongly Coupled Gauge Theories

26. proton decay models quenched 2 flavours 2+1 flavours • minimal SUSY SU(5) is ruled out dim 6 gauge boson exchange (SUSY and non-SUSY) Murayama, Pierce hep-ph/0108104 RBC, UKQCD arXiv:0806.1031 Strongly Coupled Gauge Theories

27. hierarchy problem 3 • new strong dynamics may drive electroweak symmetry breaking • ΛTC = scale of technicolor which generates EWSB ~ 1 TeV • ΛETC = scale of extended TC which generates fermion masses • resulting low-energy effective interactions include some quarks may be heavy Strongly Coupled Gauge Theories

28. non-QCD-like strong dynamics • fermions in fundamental representation of gauge group • SU(3): conformal window starts between Nf = 8 and 12 • walking occurs close to the conformal window (IR stable fixed point) Nf = 8 Nf = 12 Appelquist et al. arXiv:0712.0609 Strongly Coupled Gauge Theories

29. non-QCD-like strong dynamics • fermions in higher representations • SU(3): Nf = 2 sextet fermions discrete β function 2-loop • IR stable fixed point at g2 ~ 2.0 • not visible in perturbation theory Shamir et al. arXiv:0803.1707 Strongly Coupled Gauge Theories

30. conclusion • progress = push the scale of unknown physics to higher energies • understanding = generate the scale of known physics dynamically Strongly Coupled Gauge Theories