Renormalization Group Optimized Perturbation: some applications at zero and finite temperature Jean-Lo¨ ıc Kneur (Lab. Charles Coulomb, Montpellier) e Neveu ( T = 0 ) and Marcus Pinto ( T � = 0 ) with Andr´ Rencontres Physique des Particules Jan. 2015, IHP, Paris
1. Introduction, Motivations 2. (Variationally) Optimized Perturbation (OPT) 3. Renormalization Group improvement of OPT (RGOPT) 4. F π / Λ QCD and α S MS 5. chiral quark condensate � ¯ qq � (preliminary!) 6. λφ 4 ( T � = 0) : Pressure at two-loops (preliminary!) Summary, Outlook
1. Introduction/Motivations General goal: get approximations (of reasonable accuracy?) to ’intrinsically nonperturbative’ chiral sym. breaking order parameters from unconventional resummation of perturbative expansions Very general: relevant both at T = 0 or T � = 0 (also finite density) → address well-known problem of unstable thermal perturbation theory: (here illustrate for λ Φ 4 , next goal: real QCD for Quark Gluon Plasma: thermodynamic quantities, comparison with Lattice results).
Chiral Symmetry Breaking ( χ SB ) Order parameters Usually considered hopeless from standard perturbation: qq � 1 / 3 , F π ,.. ∼ O (Λ QCD ) ≃ 100–300 MeV 1. � ¯ → α S (a priori) large → invalidates pert. expansion 2. � ¯ qq � , F π ,.. perturbative series ∼ ( m q ) d � n,p α n s ln p ( m q ) vanish for m q → 0 at any pert. order (trivial chiral limit) 3. More sophisticated arguments e.g. (infrared) renormalons (factorially divergent pert. coeff. at large orders) n (ln p 2 µ 2 ) n ∼ n ! dp 2 � ⇒≃ � + ... = All seems to tell that χ SB parameters are intrinsically NP • Optimized pert. (OPT): appear to circumvent at least 1., 2., and may give more clues to pert./NP bridge
T � = 0 : perturbative Pressure (QCD or λφ 4 ) Know long-standing Pb: poorly convergent and very scale-dependent (ordinary) perturbative expansion QCD (pure glue) pressure at successive pert. orders bands=scale-dependence µ = πT − 4 πT
2. (Variationally) Optimized Perturbation (OPT) L QCD ( g, m q ) → L QCD ( δ g, m (1 − δ )) ( α S ≡ g/ (4 π )) 0 < δ < 1 interpolates between L free and massless L int ; (quark) mass m q → m : arbitrary trial parameter • Take any standard (renormalized) QCD pert. series, expand in δ after: m q → m (1 − δ ) ; α S → δ α S then take δ → 1 (to recover original massless theory): BUT a m -dependence remains at any finite δ k -order: fixed typically by optimization (OPT): ∂ ∂m ( physical quantity ) = 0 for m = m opt ( α S ) � = 0 Manifestation of dimensional transmutation ! Expect flatter m -dependence at increasing δ orders... But does this ’cheap trick’ always work? and why?
Simpler model’s support + properties • Convergence proof of this procedure for D = 1 λφ 4 oscillator (cancels large pert. order factorial divergences!) Guida et al ’95 particular case of ’order-dependent mapping’ Seznec+Zinn-Justin ’79 (exponentially fast convergence for ground state energy E 0 = const.λ 1 / 3 ; good to % level at second δ -order) • In renormalizable QFT, first order consistent with Hartree-Fock (or large N ) approximation • Also produces factorial damping at large pert. orders (’delay’ infrared renormalon behaviour to higher orders)( JLK, Reynaud ’2002 ) • Flexible, Renormalization-compatible, gauge-invariant: applications also at finite temperature (phase transitions beyond mean field approx. in 2D, 3D models, QCD...) (many variants, many works)
Expected behaviour (Ideally...) Physical quantity Exact result (non−perturbative) 2d order etc... 3rd order OPT 1st order 0 m O( Λ ) But not quite what happens.. (except in simple oscillator) Most elaborated calculations (e.g T � = 0 ) (very) difficult beyond first order: → what about convergence? Main pb at higher order: OPT: ∂ m ( ... ) = 0 has multi-solutions (some complex!), how to choose right one??
3. RG improved OPT (RGOPT) Our main new ingredient (JLK, A. Neveu 2010) : Consider a physical quantity (perturbatively RG invariant), e.g. pole mass M: ∂ ∂ m M ( k ) ( m, g, δ = 1) | m ≡ ˜ in addition to OPT Eq: m ≡ 0 Require ( δ -modified!) series at order δ k to satisfy a standard perturbative Renormalization Group (RG) equation: � � M ( k ) ( m, g, δ = 1) RG = 0 with standard RG operator: RG ≡ µ d d µ = µ ∂ ∂µ + β ( g ) ∂ ∂g − γ m ( g ) m ∂ ∂m [ β ( g ) ≡ − 2 b 0 g 2 − 2 b 1 g 3 + · · · , γ m ( g ) ≡ γ 0 g + γ 1 g 2 + · · · ]
→ Combined with OPT, RG Eq. takes a reduced form: � � µ ∂ ∂µ + β ( g ) ∂ M ( k ) ( m, g, δ = 1) = 0 ∂g Note: OPT+RG completely fix m ≡ ˜ m and g ≡ ˜ g (two constraints for two parameters). � � µ ∂ ∂µ + β ( g ) ∂ • Now Λ MS ( g ) satisfies by def. Λ MS ≡ 0 ∂g consistently at a given pert. order for β ( g ) . Thus equivalent to: � M k ( m, g, δ = 1) � M k ( m, g, δ = 1) ∂ � ∂ � = 0 ; = 0 ∂ m Λ MS ( g ) ∂ g Λ MS ( g ) –
OPT + RG main features • OPT: (too) much freedom in the interpolating Lagrangian?: m → m (1 − δ ) a in most previous works: linear case a = 1 for ’simplicity’... [exceptions: Bose-Einstein Condensate T c shift, calculated from O (2) λφ 4 , requires a � = 1 : gives real solutions +related to critical exponents (Kleinert,Kastening; JLK,Neveu,Pinto ’04) • OPT, RG Eqs. are polynomial in ( L ≡ ln m µ , g = 4 πα S ) : serious drawback: polynomial Eqs of order k → (too) many solutions, and often complex, at increasing δ -orders • Our compelling way out: require solutions to match standard perturbation (i.e. Asympt. Freedom for QCD): 1 α S → 0 , | L | → ∞ : α S ∼ − 2 b 0 L + · · · → at arbitrary RG order, AF-compatible RG + OPT branches only appear for a specific universal a value: γ 0 QCD ( n f = 3) = 4 γ 0 2 b 0 ; m → m (1 − δ ) (e.g. 9 ) 2 b 0 + Removes spurious solutions incompatible with AF! –
Pre -QCD guidance: Gross-Neveu model • D = 2 O (2 N ) GN model shares many properties with QCD (asymptotic freedom, (discrete) chiral sym., mass gap,..) ΨΨ) 2 ( massless ) 2 N ( � N L GN = ¯ Ψ i � ∂ Ψ + g 0 1 ¯ Standard mass-gap (massless, large N approx.): consider V eff ( σ ) , σ ∼ ¯ ΨΨ ; σ ≡ M = µe − 2 π g ≡ Λ MS • Mass gap known exactly for any N : 1 M exact ( N ) (4 e ) 2 N − 2 = Λ MS 1 Γ[1 − 2 N − 2 ] (From D = 2 integrability: Bethe Ansatz) Forgacs et al ’91 –
Massive GN model Now consider massive case (still large N ): M ( m, g ) ≡ m (1 + g ln M µ ) − 1 : Resummed mass ( g/ (2 π ) → g ) µ + ln 2 m = m (1 − g ln m µ + g 2 (ln m µ ) + · · · ) (pert. re-expanded) • Only fully summed M ( m, g ) gives right result, upon: -identify Λ ≡ µe − 1 /g ; → M ( m, g ) = m m ˆ Λ ; Λ ≡ g ln M ln M Λ ) = F e F Λ ∼ F for ˆ m ( F ≡ ln M -take reciprocal : ˆ m → 0 ; m ˆ → M ( ˆ m → 0) ∼ m 2 ) = Λ MS m/ Λ+ O ( ˆ ˆ never seen in standard perturbation: M pert ( m → 0) → 0 ! • But (RG)OPT gives M = Λ MS at first (and any) δ -order! (at any order, OPT sol.: ln m µ = − 1 g , RG sol.: g = 1 ) • At δ 2 -order (2-loop), RGOPT ∼ 1 − 2% from M exact ( any N ) –
4. QCD Application: Pion decay constant F π Consider SU ( n f ) L × SU ( n f ) R → SU ( n f ) L + R for n f massless quarks. ( n f = 2 , n f = 3 ) Define/calculate pion decay constant F π from i � 0 | TA i µ ( p ) A j ν (0) | 0 � ≡ δ ij g µν F 2 π + O ( p µ p ν ) qγ µ γ 5 τ i where quark axial current: A i µ ≡ ¯ 2 q F π � = 0 : Chiral symmetry breaking order parameter Advantage: Perturbative expression known to 3,4 loops (3-loop Chetyrkin et al ’95; 4-loop Maier et al ’08 ’09, +Maier, Marquard private comm.) x x x x x x x x x x –
(Standard) perturbative available information π ( pert ) MS = N c m 2 4 π (8 L 2 + 4 − L + α S F 2 3 L + 1 � 6 ) 2 π 2 4 π ) 2 [ f 30 ( n f ) L 3 + f 31 ( n f ) L + f 32 ( n f ) L + f 33 ( n f )] + O ( α 3 +( α S � S ) L ≡ ln m µ , n f = 2(3) Note: finite part (after mass + coupling renormalization) not π , as defined, mixes with m 2 1 separately RG-inv: (i.e. F 2 operator) → (extra) renormalization by subtraction of the form: S ( m, α S ) = m 2 ( s 0 /α S + s 1 + s 2 α S + ... ) where s i fixed 3 requiring RG-inv order by order: s 0 = 16 π 3 ( b 0 − γ 0 ) , s 1 = ... Same feature for � ¯ qq � , related to vacuum energy, needs an extra (additive) renormalization in MS -scheme to be RG consistent. –
Warm -up calculation: pure RG approximation neglect non-RG (non-logarithmic) terms: � � 4 π (8 L 2 + 4 π ( RG-1 , O ( g )) = 3 m 2 − L + α S 8 π ( b 0 − γ 0 ) α S − 5 1 F 2 3 L ) − ( 12 ) 2 π 2 → F 2 π ( m → m (1 − δ ) γ 0 / (2 b 0 ) , α S → δα S , O ( δ )) | δ → 1 = � � 4 π (8 L 2 + 4 3 m 2 29 L + α S − 102 π 841 α S + 169 348 − 5 3 L ) 2 π 2 OPT+RG: ∂ m ( F 2 π / Λ 2 MS ) , ∂ α S ( F 2 π / Λ 2 MS ) ≡ 0 : have a unique AF-compatible real solution: ˜ µ = − γ 0 L ≡ ln ˜ m α S = π 2 b 0 ; ˜ 2 8 π 2 ) 1 / 2 ˜ α S ) = ( 5 → F π ( ˜ m, ˜ m ≃ 0 . 25Λ MS • Includes higher orders +non-RG terms: ˜ m opt remains O (Λ MS ) (rather than m ∼ 0 ): RG-consistent ’mass gap’, And OPT stabilizes α opt ≃ . 5 : more perturbative values S –
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