Overview Intro: Motivation & introduction conformal window - PowerPoint PPT Presentation

Hyperscaling rela.on for the conformal window Roman Zwicky (Southampton) 15.2.12, strong-BSM workshop Bad Honnef

Overview • Intro: Motivation & introduction conformal window studies [4 slides] • Part I: mass-deformed conformal gauge theories (observables) [8 slides] - hyperscaling laws of hadronic observables e.g. f[0 ++ ] ~ m η ( γ *) • Part II: the quark condensate -- various appraoches [4 slides] . lattice material (Del Debbio’s talks) Del Debbio & RZ walking technicolour (Shrock, Sannino, others) PRD’10 & PLB’11

types of gauge theories Adjustable: gauge group SU(N c ) -- N f (massless) fermions -- fermion irrep Focus on asymptotically free theories (not many representations) 2) well-defined on lattice 2) chance for unification in TC n F 16 11? N c =3 IR-conformal walking-type QCD-type TC-models: Dietrich-Sannino’04 Susskind-Weinberg ‘79 Holdom ‘84 Luty-Okui’04 confinement & chiral symmetry breaking SU L (n F ) x SU R (n F ) → SU V (n F ) the latter is dynamical eletroweak symmetry breaking M W = g f π (TC)

Conformal window (the picture) “SU(N)” non-SUSY N=1 SUSY N f N f fund fund 15 8 2A 6 10 2A 4 5 upper line AF QCD 2 2S 2S adj adj N c N c 2 3 4 5 6 7 8 2 3 4 5 6 7 8 . β 0 tuned small α ∗ β 0 2 π = − β 1 ≪ 1 s just below pert. BZ/BM fixed pt: g n i p l u o c k a e w g n i l p u o c g n lower line BZ/BM fixed pt o r lower line Dyson-Schwinger eqs t s “electromagnetic dual” predict chiral symmetry breaking (lattice results later ...) assume in between conformal use β NSVZ ( γ *)= 0 to get γ * γ *| strong =1 (unitarity bound QQ state) γ * | strong ≈ 1 DS eqs ladder

anomalous dimension canonical dimension d: (classical) e.g. fermion field q: composite: d q = D − 1 d ¯ qq = D − 1 2 anomalous dimension γ : (quantum corrections) scheme-dependent physical γ = γ ∗ + γ 0 ( g − g ∗ ) + O (( g − g ∗ ) 2 ) ... - significance: change of renormalization scale μ → μ ’ : scaling leading correction � µ � γ ∗ O ( µ ) = O ( µ ′ ) (1 + O (ln µ, /µ ′ )) µ ′ - QCD: UV-fixed point (asymptotic freedom) - γ * ( g * = 0)= 0 (trivial) - correction RGE (logarithmic) - our interest: IR fixed point non-trivial - γ * ≠ 0 (large?)

scaling dimension scaling dimension: Δ scaling = d canonical + γ anomalous 4 Δ � L = C i ( µ ) O i ( µ ) i relevant marginal irrrelevant • Value of Δ O is a dynamical problem (= d O at trivial fixed-point, g * =0) - unitarity bounds Δ Scalar ≥ 1 etc Mack’77 - Δ OO’ ≠ Δ O Δ O’ generally (except SUSY and large-N c ) • gauge theory: expect to be most relevant operator ¯ qq Δ qq = 3 + γ qq = 3 - γ m ( γ m = γ * at fixed point) γ * is a very important parameter for model building Ward-identity

Part I: Observables for Monte Carlo (lattice) for conformal gauge theories -- parametric control --

or how to identify Part I: Observables in a CFT? a CFT (on lattice) pure-CFT: Vanishing β -function & correlators (form 2 & 3pt correlators known) e.g. <O(x)O(0)> ~ (x 2 ) - Δ deformed-CFT: Lattice: quarks massive / finite volume ⇒ consider 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

• if mass-deformation relevant Δ qq = 3 - γ * < 4 theory flows away from fixed-point (likely) physical picture: ( Miransky ’98) finite m q ; quarks decouple ➭ pure YM confines (string tension confirmed lattice) ➭ hadronic spectrum signature: hadronic observables (masses, decay constants) hypothesis : hadronic observables → 0 as m q → 0 (conformality restored) O ∼ m η O (1 + .. ) , η O ( γ ∗ ) > 0 If fct η O known: a) way to measure γ * b) consistency test through many observable

Mass scaling from trace anomaly & Feynman-Hellman thm 2 β G 2 + N f m (1 + γ m )¯ α | on − shell = 1 trace/scale anomaly: θ α qq Adler et al, Collins et al � H ( p ) | H ( k ) � = 2 E p δ (3) ( p − k ) ⇒ β = 0 & N.Nielsen ’77 Fujikawa ’81 reminiscent 2 M 2 h = N f (1 + γ ∗ ) m � H | ¯ qq | H � GMOR-relation ∂λ = � ψ ( λ ) | ∂ ˆ H ( λ ) ∂ E λ Feynman-Hellman thm: | ψ ( λ ) � ∂λ ∂ � ψ ( λ ) | ψ ( λ ) � idea: = 0 ∂λ m ∂ M 2 ∂ m = N f m � H | ¯ qq | H � H ★ applied to our case ( λ ➣ m) : m ∂ M H 1 1+ γ ∗ M H ★ combined with GMOR-like: ∂ m = scaling law for all masses 1 M H ∼ m 1+ γ ∗

Brief comparison with QCD QCD-spectrum mCGT-spectrum • m ALL = m 1/(1+ γ * ) O( Λ ETC γ * /(1+ γ * ) ) • m ρ = O( Λ QCD ) + m q m B = m b + O( Λ QCD ) m π = O( (m q Λ QCD ) 1/2 ) • Breaking global flavour symmetry : SU L (n F ) x SU R (n F ) → SU V (n F ) (chiral symmetry) QCD: mCGT: CGT: spontaneous yes * no no explicit yes yes no (mass term) confinement yes yes no * F π ≠ 0 m → 0 order parameter ⇒ no chiral perturbation theory in mCGT (pion not singled out -- Weingarten-inequality still applies)

Hyperscaling laws from RG physical states no anomalous dim. O 12 ( g, ˆ m ≡ m µ , µ ) ≡ � ϕ 2 |O| ϕ 1 � • local matrix element: RG -trafo O 12 1 . O 12 ( g, ˆ m, µ ) = b − γ O O 12 ( g ′ , ˆ m ′ , µ ′ ) , µ = bµ' g ′ = b 0+ γ g g m ′ = b 1+ γ ∗ ˆ ˆ y m = 1 + γ ∗ , γ g < 0 (irrelevant) m , change physical units m ′ , µ ′ ) = b − ( d O + d ϕ 1 + d ϕ 2 ) O 12 ( ˆ 2 . O 12 ( ˆ m ′ , µ ) m ′ = 1 ⇒ trade b for m 3 . Choose b s.t. ˆ “master equation” Hyperscaling relations m ) ( ∆ O + d ϕ 1 + d ϕ 2 ) / (1+ γ ∗ ) O 12 ( ˆ m, µ ) ∼ ( ˆ ⇒ * From Weinberg-like RNG eqs on correlation functions (widely used in critical phenomena)

Applications: • master formula (local matrix element): � ϕ 1 |O| ϕ 2 � ∼ m ( ∆ O + d ϕ 1 + d ϕ 2 ) / (1+ γ ∗ ) alternative derivation ... 2 2 M 2 h = N f (1 + γ ∗ ) m � H | ¯ qq | H � ∼ m 1. hadronic masses: 1+ γ ∗ more later 3 − γ ∗ 4 � G 2 � ∼ m 1+ γ ∗ , � ¯ qq � ∼ m 2. vacuum condensates: on... 1+ γ ∗ 3. decay constants: | φ 〉 = |H(adronic) 〉 N.B. ( Δ H = d H = -1 choice)

Remarks S-parameter: S = 4 π Π V − A (0) − [pion − pole] � ( q µ q ν − q 2 g µ ν ) 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 � f 2 f 2 f 2 V A P A ( q 2 ) ≃ P − q 2 + ... . Π V − hadronic representation: V − q 2 − A − q 2 − m 2 m 2 m 2 • difficulties: a) non-local b) difference (density not positive definite) pion-pole Π W − TC (0) ∼ O ( m − 1 ) V − A (conspiracy) cancellations modulo Π mCGT (0) ∼ O ( m 0 ) V − A improve on non- perturbatve computations ( q 2 ) ∼ m 2 /y m (lattice, FRG, DSE...) for − q 2 ≫ ( Λ ETC ) 2 Π mCGT V − A q 2 Sannino’10 free theory

Summary - Transition • low energy (hadronic) observables carry memory of scaling phase 1 1+ γ ∗ m H ∼ m 2 − γ ∗ f H (0 − ) ∼ m 3 − γ ∗ 1+ γ ∗ � ¯ qq � ∼ m 1+ γ ∗ • “all” quantities scale with one parameter -- witness relations between the zoo critical exponents α , β , γ , ν .. = hyperscaling • clarify: heavy quark phase and mCGT are parametrically from 1 m B (0 − ) ∼ m b m H (0 − ) ∼ m similar: � = 1+ γ ∗ 2 − γ ∗ f B (0 − ) ∼ m − 1 / 2 f H (0 − ) ∼ m distinct: � = 1+ γ ∗

Part 2: story quark condensate <qq> the most relevant operator • important for conformal TC/ partially gauged TC models

How does CFT react to a perturbation Unparticle area L e ff ∼ C O| H | 2 VEV → C O v 2 I. couple CFT Higgs-sector: 1 B ≃ Cv 2 Λ ∆ O Λ 4 Λ B ∼ ( Cv 2 ) 4 − ∆ O criteria breaking (NDA): ⇒ B Fox, Rajaraman, Shirman ’07 II. Heuristics: deconstruct the continuous spectrum of a 2-function. Stephanov’07 2.0 Infinite sum of adjusted particles can mimick continuous spectrum 1.5 1.0 0.5 n = δ 2 ( M 2 � n ) ∆ − 2 0.5 1.0 1.5 2.0 2.5 3.0 f 2 � O ( x ) ∼ f n ϕ n ( x ) ; � ϕ n |O| 0 � ∼ f n , M 2 n = n δ 2 n ➭ tadpole & mass term as potential ⇒ find new minimum . Delgado, Espinoso, Quiros’07 n M 2 n ϕ 2 V e ff = − m � n f n ϕ n − 1 / 2 � n

minimise - solve - reinsert: mf n + M 2 � ϕ n � = − mf n /M 2 δ ϕ n V e ff = 0 ⇒ ⇒ n ϕ n = 0 n � Λ 2 f 2 δ → 0 IR s ∆ O − 3 ds � O � ∼ � n f n � ϕ n � − m � → − m UV n Λ 2 n M 2 n result depends on IR and UV physics ➪ need model(s) III. Within conformal gauge theory Sannino RZ ’08 generic 0 ++ -operator: O → ¯ qq within gauge theories N.B. 4D only ”known” CFT gauge theories. why? - Λ UV → Λ ETC where Δ qq → 3 ⇒ ( Λ ETC ) 2 -effect - Λ IR : (m constituent ) Δ qq ~ <qq> generalising Politzer OPE Solve self-consistency condensate eqn OPE extension 1 1 4 g 2 / A / = q − B q + of pole mass q 4 m = B A - ’10: Λ IR : = m H m 1/(1+ γ * ) O( Λ ETC γ * /(1+ γ * ) ) agrees depending on value γ * Del Debbio, RZ Sep’10