Roman Zwicky (Southampton) , City College London (workshop integrable models)

Transcription

1 Iden%fying Conformal Gauge Theories (CGT) Roman Zwicky (Southampton) , City College London (workshop integrable models)

2 Prologue: Object of interest: (4d) gauge theories fundamental forces in nature described gauge theories strong, weak and electrom. force SU(3)c SU(2)L U(1)Y (Gravity special gauge theory) New gauge forces at the LHC? e.g. Technicolour gauge theories are interesting per se (e.g. Applications mathematical physics) Renewed interest in the phase diagram of gauge theories conformal window Phases: Parameters: Tools: a) IR: chiral symmetry broken (QCD) known b) IR: conformal less known Nc,Nf, gauge group fermion representation a) lattice monte carlo b) Functional RGE Dyson-Schwinger eqs (truncation?) This talk: analytic tools to identify IR-conformal theories on lattice.

3 Overview LHC -- gauge theories (Technicolour) General remarks gauge theories - conformal window non-susy & SUSY conformal gauge theories (CGT) -- observables? observables in mass-deformed CGT (8) - hyperscaling laws from RG - mass scaling from Feynmann-Hellmann thm - another look at β-function from trace anomaly - remarks on S-parameter. Del Debbio & RZ PRD 10 & arxiv: Lattice results qq =3 γ where γ mass anomalous dimension Epilogue

4 LHC: why there ought to be sthg s WL-WL scattering +... perturbative unitarity S-wave A 0 = g2 2 16πm 2 W s Unitarity bound: Re[A 0 ] 1 2 s crit (1.2 TeV) δ H A 0 = g2 2 16πm 2 W s Non-unitarity is merely a consequence of loss of gauge invariance without Higgs. So...

5 LHC: Origin mass for W,Z-boson and chiral fermions Is the object that unitarizes W L W L scattering fundamental or composite? fundamental particle small width composite particle large width Supersymmetry opposite statistics partner prototypes without hierarchy problem strong dynamics Technicolour Higgs sector Gauge theory + SU(3)c SU(2)L U(1)Y GTC

6 Technicolor: Higgs sector strongly coupled gauge theory Susskind 79 Weinberg 79 R GTC L SU(2)L moose notation techni-fermions QL * QR same charges as the Higgs how strong? symmetry breaks at Λ EW = Λ TC Q L Q R N c Λ 3 TC SSB: SU(2)L U(1)Y U(1)EM masses W,Z bosons (as in SM!) N.B. QCD breaks SU(2)L spontaneously and gives mass to W-boson (3 orders of mag. too small) 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

7 Pheno improvement via Walking TC* Issues: 1. electroweak precision parameter S too large Cern 95) 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.. * Also a scheme called Conformal Technicolor on the market (Luty 04)

8 Hot of the press: CDF-paper 4th of April 11 Detector signal: 2 dijets (q q*), lepton & missing energy q.. is a Higgs-hunter signal q q H q W W l ν.. signal 300 times too high to be a Higgs statistical significance 3.2σ Attention QCD backgrounds.. also not seen in case qq replaced by: 1) ll leptophobic 2) bb (b-tagging) beautyphobic 6th of April 11

9 To take away: Gauge theories can replace Higgs sector Phenomenologically a) near conformal (walking) b) conformal are attractive (evading constraints & fermion masses)

10 Phases of gauge theories what theorists can adjust: Nc,Nf, gauge group (e.g. SU(N)) fermion representation one coupling theory: g * = 0 either IR or UV fixed-point (unless scale invariant β(g * ) = 0 candidate N=4 SUSY 4d) QED-like?? QCD-like IR-conformal? focus AF-theories (- β 0 < 0) we know how to handle (well-defined lattice) asymptotic safety non-trivial UV fixed-point also well-defined in principle How to guess Lagrangian? (lately indications gravity could be asymptotically safe from functional RGE)

11 Two remarks before we go on: Scale invariance (T μμ = μ L μ )?? Conformal invariance (T μμ = 0) e.g. Polchinski 88 Rychkov, Rattazzi, Vinchi We use scale invariance but it is customary to use conformal - Though: bounds on scaling dimension of operators follow from CFT-OPE & bootstrap Ryttov, Rattazzi, Vinchi Unitarity bound CFT from SO(4,2) representation theory: scaling dimension Δ O for a scalar operator O ΔO 1 (=1 then free field) Mack 77

12 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 lost (solid) Nf dashed line Dyson-Schwinger predicts bound qq =3 γ 2 lower line unitary bound qq 1 via conjectured β-fct (conjecture=not solid) Nc

13 N=1 -- SUSY conformal window Nf - upper line AF - just below (perturbative) BZ fixed point - lower boundary (new tool) electromagnetic dual (perturbative) BZ fixed point Nc Assume scale invariance persists in between Use exact NSVZ 83 β-fct to get γ * from β = 0 β NSV Z = 1 3T G T R (1+γ) 16π 2 1 T G /(8π 2 ) Remarks: - Lower boundary saturates unitarity bound squark-squark scalar Δ QQ = 2 -γ * 1 γ * 1 - Weak-strong coupling duality exist strongly coupled CGT (also from γ *)

14 Summarizing... two primary goals are: Size of the conformal window SUSY looks AF gauge theories 1/2 non-cft + 1/2 CFT (The expected picture is non-susy theories is similar) (SUSY) Strong coupling large value of γ * Recall unitarity bound Δqq = 3-γ * 1 ( γ * 2) so far γ * max 1... from theory 1. SUSY its because of the squark unitarity bound Δ QQ = 2 -γ * 1 2. DS-eqs. truncation -- ladder approximation etc? Is the unitarity bound ever reached? lattice simulations

15 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

16 Observables in mcgt Goal: analytic guidance for lattice (parametric laws) finite m q quarks decouple pure YM confines (string tension confirmed lattice) 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

17 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)

18 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 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

19 Let s pause: comparison with QCD-like spectrum QCD-like (chiral symmetry broken) mcgt spectrum Goldstone boson No SSB of χ-symmetry breaking (no goldstone boson) since condensate triggered by explicit χ-breaking L = mq L q R +h.c. There is no chiral perturbation theory 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*)

20 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!

21 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

22 A few additional topics

23 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) β NSV Z = 1 3T G T R (1+γ) 16π 2 1 T G /(8π 2 )

24 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 modulo (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)

25 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) --- no parametric control It would seem longterm mass/decay constant parametric scaling should help

26 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...

27 QCD code older stuff NMWTC = 2S MWTC

28 Summary 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

29 Epilogue for the workshop Why so many CFT in 2D whereas in 4D: N=4 SUSY what else? IR fixed point lot of gauge theories... ( the prize of asymptotic freedom Coleman-Gross 73) Is a more systematic deformation theory (beyond RGE) known for non 2D CFT? Are there cases of asymptotic safety (non-trivial UV fixed-point)known in 2D? (Thirring model at large-n in 3D is asymptotically safe) Thanks for your attention!

30 Backup slides...

31 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)!

32 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

33 Addendum (bounds scaling dimension) assume add L = mqq (N.B. not a scalar under global flavour symmetry!) using bootstrap ( associative OPE on 4pt function) possible to obtain upper-bound on scaling dimension Δ of lowest operator in OPE non-singlet Rattazzi, Rychkov Tonni & Vichi 08 singlet Δ 4 allows for Δ qq to be: Rattazzi, Rychkov & Vichi 10 very close to unitarity bound! 1.35 still rather close to unitarity bound good news for Luty s conformal TC