1 Lattice Explorations of TeV Physics Argonne April 13, 2009

2 Disclaimer : I am not a lattice gauge theorist 1. Collaboration with George Fleming and Ethan Neil 1) arXiv: , PRL 100, , 2) arXiv:  PRD 2. LSD collaboration

3 Possible Lattice Directions ● Higgs sector of the standard model (eg. Triviality bounds) ● Supersymmetry ● New Strong Dynamics Electroweak Breaking New, SM-singlet sector

4 Strong Dynamics Electroweak breaking ● Technicolor, ETC, Higgless theories, ADS/CFT approaches ● Near-conformal infrared behavior: walking technicolor New, SM-singlet Sector ● Conformal infrared behavior

5 QCD-Like Theories SU(N) Gauge Theories with N f Massless Fermions (fundamental representation) Asymptotically-free (can take lattice spacing to zero) N f < N af Vary N f and study how the infrared behavior changes (chiral symmetry breaking and confinement versus infrared conformal behavior.)

6 Conformal Window N af > N f > N fc (Fermion screening) Below this range, conformal/chiral-symmetry breaking and confinement. Standard-Model Non-singlets with N f < N fc : Dynamical Electroweak Symmetry Breaking. ( N f → N fc : Walking Technicolor)? QCD is NOT Conformal.

7 Infrared conformal behavior (IR fixed point) can emerge already in the two-loop β function, depending on the number of massless fermions N f Gross and Wilczek, antiquity Caswell, 1974 Banks and Zaks, 1982 Many Others Reliable if N f is just below the value at which asymptotic freedom sets in.

8 Cartoons α* increases as N f decreases. When N f reaches N fc, α* becomes large enough to trigger chiral symmetry breaking and confinement. Not necessarily accessible in perturbation theory ↑

9 Questions 1.Value of N fc ? 2.Order of the phase transition? 2.Correlation functions and anomalous dimensions inside the Conformal window 3.Physical states below and near the transition? Implications for electroweak precision studies (S parameter etc)? Implications for the LHC?

10 N fc in SU(N) QCD Degree-of-Freedom Inequality (Cohen, Schmaltz, TA 1999). Fundamental rep: N fc  4 N[1 – 1/18N ² + …] Gap-Equation Studies, Instantons (1996): N fc  4 N Lattice Simulation (Iwasaki et al, Phys Rev D69, ): 6 < N fc < 7 For N = 3

11 Some Quasi-Perturbative Studies of the Conformal Window in QCD-like Theories 1. Gap – Equation studies in the mid 1990s 2. V. Miransky and K. Yamawaki hep-th/ (1996) 3. E. Gardi, G. Grunberg, M. Karliner hep-ph/ (1998) 4. E. Gardi and G. Grunberg JHEP/004A/1298 (2004) 5. Kurachi and Shrock, hep-ph/ H. Terao and A. Tsuchiya arXiv: [hep-ph] (2007)

12 Lattice-Simulation Study of the Extent of the Conformal window in an SU(3) Gauge Theory with Dirac Fermions in the Fundamental Representation Fleming, Neil, TA

13 Some Previous Lattice Work with Many Light Fermions 1.Brown et al (Columbia group) Phys. Rev. D12, 5655 (1992). 2R. Mahwinney, hep/lat (1) Nucl.Phys.Proc.Suppl.83:57-66,2000. e-Print: hep- lat/ C. Sui, Flavor dependence of quantum chromodynamics. PhD thesis, Columbia University, New York, NY, UMI Iwasaki et al, Phys. Rev, D69, (2004)

14 Focus: Gauge Invariant and Non- Perturbative Definition of the Running Coupling Deriving from the Schroedinger Functional of the Gauge Theory ALPHA Collaboration: Luscher, Sommer, Weisz, Wolff, Bode, Heitger, Simma, …

15 Using Staggered Fermions as in U. Heller, Nucl. Phys. B504, 435 (1997) Miyazaki & Kikukawa Focus on N f = multiples of 4: 16: Perturbative IRFP 12: IRFP “expected”, Simulate 8 : IRFP uncertain, Simulate 4 : Confinement, ChSB

16 The Schroedinger Functional Transition amplitude from a prescribed state at t=0 to one at t=T (Dirichlet BC). (m = 0) Euclidean path integral with Dirichlet BC in time and periodic in space (L) to describe a constant chromo-electric background field. η↓ η↓

17 Picture η

18 Schroedinger Functional (SF) Running Coupling on Lattice Define: Response of system to small changes in the background field.

19 SF Running Coupling Then, to remove the O(a) bulk lattice artifact Depends on only one large scale L!

20 Perturbation Theory

21 At three loops IRFP at No perturbative IRFP

22 Lattice Simulations MILC Code (Heller) Staggered Fermions Range of Lattice Couplings g 0 ² (= 6/β ) and Lattice Sizes L/a → 20 O(a) Lattice Artifacts due to Dirichlet Boundary Conditions

23 Statistical and Systematic Error 1. Numerical-simulation error 2. Interpolating-function error 3. Continuum-extrapolation error Statistics Dominates “Don’t Mis-underestimate.”

24 Lattice Data N f = 8

25 Perturbation Theory Polynomial Fit N f = 8, 12

26 Data with Fits

27 Renormalization Group (Step Scaling)

28 N f =8 Extrapolation Curve

29 N f = 8 Continuum Running

30 N f = 8 Features 1.No evidence for IRFP or even inflection point up through. 2.Exceeds rough estimate of strength required to break chiral symmetry, and therefore produce confinement. Must be confirmed by direct lattice simulations. 3.Rate of growth exceeds 3 loop perturbation theory. 4.Behavior similar to quenched theory [ALPHA N.P. Proc. Suppl. 106, 859 (2002)] and N f =2 theory [ALPHA, N.P. B713, 378 (2005)], but slower growth as expected.

31 Lattice Data N f = 12

32 Data with Fits

33 N f =12 Extrapolation Curve

34 N f = 12 Continuum Running

35 Approach to Fixed Point

36 Conclusions 1.Lattice evidence that for an SU(3) gauge theory with N f Dirac fermions in the fundamental representation 8 < N fc < N f =12: Relatively weak IRFP 3. N f =8: Confinement and chiral symmetry breaking. Employing the Schroedinger functional running coupling defined at the box boundary L

37 Things to Do 1.Refine the simulations at N f = 8 and 12 and examine other values such as N f =10. Study the phase transition as a function of N f. 2. Consider other gauge groups and representation assignments for the fermions Sannino et al; DeGrand, Shamir (08), Svetitsky; Hietanen et al (09), …….. 3. Examine physical quantities such as the static potential (Wilson loop) and correlation functions. LSD

38 5. Examine chiral symmetry breaking directly: / f 3 for N f = 2,4,6,8,……. LSD Parity doubling? 6. Apply to BSM Physics. Is S naturally small as N f →N fc due to approximate parity doubling? LSD

39 LSD Collaboration Lattice Strong Dynamics J. C. Osborn Argonne National Laboratory R. Babich, R. C. Brower, M. A. Clark, C. Rebbi, D. Schaich Boston University M. Cheng, T. Luu, R. Soltz, P. M. Vranas Lawrence Livermore National Laboratory T. Appelquist, G. T. Fleming, E. T. Neil Yale University

40 Thomas DeGrandExploring the phase diagram of sextet QCD Albert DeuzemanThe physics of eight flavours Michael EndresNumerical simulation of N=1 supersymmetric Yang-Mills theory Philipp GerholdHiggs mass bounds from a chirally invariant lattice Higgs-Yukawa model with overlap fermions Joel GiedtDomain Wall Fermion Lattice Super Yang Mills Ari HietanenSpectrum of SU(2) gauge theory with two fermions in the adjoint representation Kieran HollandProbing technicolor theories with staggered fermions Xiao-Yong JinLattice QCD with Eight Degenerate Quark Flavors Biagio LuciniOrientifold Planar Equivalence: The Chiral Condensate Ethan NeilThe Conformal Window in SU(3) Yang-Mills Daniel NogradiNearly conformal electorweak sector with chiral fermions Elisabetta PallanteSearching for the conformal window Agostino PatellaFermions in higher representations. Some results about SU(2) with adjoint fermions. Benjamin SvetitskyNonperturbative infrared fixed point in sextet QCD Rapidly Growing Activity Lattice-2008 Applications beyond QCD