Iden%fying Conformal Gauge Theories (CGT) Roman Zwicky (Southampton) 30 Nov/1 2010, Marseille/Lyon

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1 Iden%fying Conformal Gauge Theories (CGT) Roman Zwicky (Southampton) 30 Nov/1 2010, Marseille/Lyon

2 Overview BSM, Technicolor, Walking-TC study strongly coupled gauge theories (3) General remarks gauge theories - conformal window SUSY & non-susy (4) conformal gauge theories (CGT) -- observables? (1) observables in mass-deformed CGT (8) - hyperscaling laws from RG - mass scaling from Feynmann-Hellmann thm - another look at β-function from trace anomaly - trajectory mass & decay constants - remarks on S-parameter. Del Debbio & RZ PRD 10 & arxiv: Lattice results (3) qq =3 γ where γ mass anomalous dimension Epilogue

3 Beyond the SM centered around the Higgs mechanism of SSB W,Z masses; technical hierarchy problem?* Is the Higgs (object that unitarizes W L W L -scattering) fundamental or composite? fundamental particle small width composite particle large width Supersymmetry opposite statistics partner prototype strong dynamics Technicolour Higgs sector Gauge theory + SU(3)c SU(2)L U(1)Y GTC * Flavour sector, where real hierarchies are present, harder for model building

4 Technicolor Susskind 79 Weinberg 79 Higgs sector strongly coupled gauge theory R GTC L SU(2)L moose notation symmetry breaking: Q L Q R N T C Λ 3 T C masses W,Z bosons through SSB (as in SM!) Λ TC 4πF T F T = v Fermion masses -> Extended TC SM G SM G T C G ET C Dimopoulus Susskind 79 Eichten Lane 80 breaking: G ET C G SM G T C L eff = SM fermion masses FCNC * QCD breaks SU(2)L spontaneously and gives mass to W-boson (orders of mag. too small)

5 Pheno improvement via Walking TC Issues: 1. electroweak precision parameter S ~ 95 Lep 2. dynamical generation of fermion masses and FCNC Extended-TC Improvement Walking-TC: almost reaches IR fixed-point 1. S WTC << S TC 2. γ mass * large Walking results enhancement of No parametric definition (challenge) QQ TC f 3 π(tc) need to know more about strongly coupled near-conformal gauge theories..

6 Gauge theories what theorists can adjust: one coupling theory: N c N F representation g * = 0 either IR (QED-like) or UV (QCD-like asymptotic freedom) fixed-point focus AF-theories (- β 0 < 0) we know how to handle QED-like?? QCD-like IR-conformal?

7 Facts non-susy Conformal Window QCD chiral symmetry is broken (empirical) not in CW! Banks-Zaks 82 (Belavin-Migdal 76) perturbative IR fixed point (conformal) proof of principle If β 0 tuned small α s 2π = β 0 β 1 1 α s 0.02 α β(α s )= β 2 S 0 2π β 1 α3 S (2π) upper line AF (ok) Nf dashed Dyson-Schwinger qq =3 γ 2 lower unitary bound qq 1 via conjectured β-fct Nc

8 SUSY Conformal Window Exact NSVZ 83 β-fct: from β = 0 get γ * 1. Unitarity bound on squark-bound state Δ QQ = 2 -γ * 1 γ * 1 2. Electric magnetic duality N dual = N F - N perturbative electric BZ-fixed point upper boundary (like QCD) perturbative magnetic BZ-fixed point lower boundary!! lower bdry unitarity bound weak-strong coupling duality exist strongly coupled CGT (also from γ *)

9 Two objectives (almost repetition) AF gauge theories 1/2 non-cft + 1/2 CFT (SUSY) (N.B. only known CFT in 4D are GT, coheres with Coleman-Gross Thm) size of conformal window? strong coupling -- value of γ * SUSY N = 1 tells qq =3 γ 2 strongly coupled? size γ * Dyson Schwinger eqn: chiral symmetry breaks qq 2 unitarity bound (Mack 77) qq 1 Is the unitarity bound ever reached? 1. SUSY its because of the squark Δ QQ = 2 -γ 2. DS-eqs. truncation -- ladder approximation... NJL 3. N.B. Δ=1 free field (very strong force...) we want answers lattice simulations

10 Observables in a CFT? Or how to identify a CFT 1. Observables: vanishing β-function & <O(x)O(0)> ~ (x 2 ) -Δ ;Δ=d+γ* 2. Lattice computation finite mquark (& volume anyway) look mass-deformed conformal gauge theories (mcgt) * L = L CGT m qq * hardly related to 2D CFT mass deformation a part of algebra and therefore integrability is maintained

11 Obersvables in mcgt Goal: analytic guidance for lattice (parametric laws) If strongly coupled hadronic spectrum beloved hadronic observables signature of such a theory: each hadronic observable O m η O, η O > 0, η = f(γ ) Let s settle some notation: O = d O + γ O, scaling = physical + anomalous dimension γ m = γ qq, denoted by γ at fixed-point qq =3 γ

12 Hyperscaling laws Consider matrix element: O 12 (g, ˆm, µ) ϕ 2 O ϕ 1 physical states no anomalous dim. 1. O 12 (g, ˆm, µ) =b γ O O 12 (g, ˆm,µ ), RG -transformation* µ = bµ' g = b y g g ˆm = b ym ˆm, y m = 1 + γ, y g < 0 (irrelevant) 2. O 12 (ˆm,µ )=b (d O+d ϕ1 +d ϕ2 ) O 12 (ˆm,µ) change physical units 3. Choose b s.t. ˆm =1 Hyperscaling relations O 12 (ˆm, µ) (ˆm) ( O+d ϕ1 +d ϕ2 )/y m * From Weinberg-like RNG eqs on correlation functions (widely used in critical phenomena)

13 Applications: η O12 =( O + d ϕ1 + d ϕ2 )/y m vacuum condensates: decay constants: φ = H(adronic) N.B. (Δ H = d H = -1 choice) qq m 3 γ 1+γ, G 2 m 4 1+γ more later on... masses from trace anomaly: Adler et al, Collins et al N.Nielsen 77 Fujikawa 81 θα α q 0 on shell = 1 2 βg2 + N f m(1 + γ m ) qq β = 0 & H(p) H(k) =2E p δ (3) (p k) 2M 2 h = N f (1 + γ )m H qq H m 2 (1+γ ) relation reminiscent GMOR-relation

14 ... Summarizing: scaling laws for entire spectrum, decay constants & condensates No SSB of χ-symmetry breaking (no goldstone boson) since condensate triggered by explicit χ-breaking There is no chiral perturbation theory Credits (presentation focused last paper): lowest mass state Miransky 98 quark condensate (just stated) DeGrand 09 all lowest state results DelDebbio RZ 10 May (large time euclidian correlators) all state results DelDebbio RZ 10 Sep A point that can be clarified: M H ~ m 1/(1+γ*) looks a bit like heavy quark physics The definite signature is f P(B-meson) ~ m -1/2 whereas f P(mCGT) ~ m (2-γ*)/(1+γ*)

15 Mass scaling without RG Del Debbio, RZ Sep 10 Hellmann-Feynman-Thm E λ λ = ψ(λ) Ĥ(λ) λ ψ(λ) idea: ψ(λ) ψ(λ) λ =0 applied to our case: m M 2 h m = N f m H qq H combined with GMOR-like.. m M H m = 1 1+γ M H M H m 1 1+γ scaling law without using RG!

16 Generalized Banks-Casher relation Banks & Casher 80 a la Leutwyler & Smilga 92 : Green s function: q(x) q(y) = n qq V = 1 V dx q(x)q(x) = 2m V u n (x)u n (y) m iλ n, where /Du n = λ n u n λ n >0 1 m 2 + λ 2 n V = 2m 0 dλ ρ(λ) m 2 + λ 2 UV-divergences later -- focus IR-physics qq m 0 m η qq ρ(λ) λ 0 λ η qq QCD : η qq =0 ρ(0) = π qq mcgt: another way to measure anomalous dimension Banks, Casher 80 DeGrand 09 DelDebbio RZ 10 May

17 Heuristic look Deconstruct the continuous spectrum of a two point function Infinite sum of adjusted particles can mimick continuous spectrum Stephanov Adding mass term looks like tadpole. find new minimum -- add M n to potential. Delgado, Espinoso, Quiros 07 Solve and reinsert: Λ UV : Δ qq = 3 find quadratic divergence known from Leutwyler-Smilga rep. Λ IR: 1) Λ IR ~ M H ~ m 1/(1+γ) or use (M dyn ) Δqq ~ <qq> generalizing Politzer OPE. and confirm η qq = Δ qq /(1+γ)!

18 A few additional topics

19 Another look at the β-function Consider the again the trace (scale) anomaly: Evaluate it on any hadronic state H> and solve for β: Ratios of A H /G H & B H /G H independent Form β-function close to NSVZ β (for N=1 SUSY gauge theories)

20 Del Debbio, RZ Sep 10 Mass & decay constant trajectory At large-n c neglect width g Hn 0 O H n (decay constant) (q 2 ) x eix q 0 O(x)O(0) 0 = n g H n 2 q 2 +M 2 Hn In limit m 0 (scale invariant correlator) (q 2 )= 0 ds s 1 γ q 2 +s +s.t (q 2 ) 1 γ Solution are given by: MH 2 n α n m 2 1+γ, gh 2 n α n(α n ) 1 γ m 2(2 γ ) 1+γ where α n arbitrary function (corresponds freedom change of variables in ) QCD expect α n ~ n (linear radial Regge-trajectory) (few more words) For those who know: resembles deconstruction Stephanov 07 difference physical interpretation of spacing due to scaling spectrum

21 remarks S-parameter Del Debbio, RZ Sep 10 Analytical guidance S-parameter: S =4πΠ V A (0) pion pole (q µ q ν q 2 g µν )δ ab Π V A (q 2 )=i d 4 xe iq x 0 T (V µ a (x)v ν b (0) (V A)) 0 Π V A (q 2 ) f 2 V m 2 V q2 f 2 A m 2 A q2 m 2 P f 2 P q module (conspiracy) cancellations improve... Π W TC V A (0) O(m 1 ) Π mcgt V A (0) O(m 0 ) Π mcgt V A (q 2 ) m2/y m q 2 for q 2 (Λ U ) 2 Sannino 10 free theory lattice determination coming soon (already some market)

22 Lattice simulations (generic remarks): Ca 7(2) groups UK Swansea/Edbgh), Finnland, Holland, Lin & Onugi USA (LSD,deGRand, Knuti,Fodor, Caterall & Sannino...) IR mass is relevant; coupling irrelevant (principal no tuning necessary) Measure β-fct (stepsize scaling) problem: m 0 so not fixed-pt β-fct not physical measuring zero (cancellations) measure enhancement <QQ>/f π 3 (LSD) parametric control? It would seem longterm mass/decay constant parametric scaling should help

23 Summary of results: See reasonable results scaling in 0 -+,1 -- channels 0 ++ more noisy (as usual) typically γ * ~ 0.4(3?) not too large (upper bound difficult) so-called MinimalWTC looks conformal conformal-tc model building Why & What is simulated: next slide...

24 QCD code older stuff NMWTC = 2S MWTC

25 Epilogue People accept it will take more time to establish CW than foreseen Major goals: 1) size of conformal window 2) how large anomalous dimension 1 Δ qq... will unitary-bound be reached? Is Δ qq 2 border DS? 3) measurement S-parameter Conformal window studies open up theory space model building CTC... More theoretical questions: trajectories etc Maybe by understanding mcgt better we learn sthg about QCD Thanks for your attention!

26 Backup slides...