PADES and LARGE- N c QCD SANTI PERIS (IFAE - UA Barcelona) Luminy, Sep. 2009 LARGE- Nc QCD – p.1/19 PADES and
Introduction ⋆ Q uantum C hromo D ynamics is a Yang-Mills theory with fermions, based on local SU ( N c ) : 1 µν F Aµν 4 g 2 F A L = i ψ a � D ab ψ b − • ψ b : Dirac field, b = 1 , ..., N c • F A µν : Yang-Mills field strength tensor, A = 1 , ..., N 2 c − 1 • g : coupling constant • N c = 3 ⋆ 1 of 7 Millennium Problems worth $ 1M Prize from the Clay Math Inst. ( g 2 N c → constant) ⋆ Although Nature has N c = 3 , the limit N c → ∞ is very interesting. (’t Hooft ’74; Witten ’79) • In this limit, all functions are meromorphic. LARGE- Nc QCD – p.2/19 PADES and
Resonance Saturation < π ( p ′ ) | V µ | π ( p ) > = F ( − q 2 ) ( p ′ + p ) µ q 2 = ( p ′ − p ) 2 , M 2 F ( − q 2 ) V ′ 60 s ) ≃ ( V MD − q 2 + M 2 V 1 + q 2 ∞ C 2 � R ≈ (meromorphic , N c → ∞ ) − q 2 + M 2 R R 1 + ♯ α s ( µ ) log − q 2 α s ( µ ) � � ( q 2 → − ∞ ) − 16 πF 2 ≈ µ 2 + ... + ... π q 2 1 1 ≡ 2 L 9 1 + a q 2 + ... ( q 2 → 0) ≈ ⇒ a = ≃ M 2 (0 . 74 GeV) 2 F 2 π V 0.9 Spacelike ( Q 2 = − q 2 ) 0.8 π P. Zweber Q 2 |F π (Q 2 )| (GeV 2 ) 0.7 Sakurai ’69 Ecker et al. ’89 0.6 VDM Donoghue et al. ’89 0.5 Moussallam ’97 0.4 - - - - - - - - - - - - - 0.3 Knecht, de Rafael ’98 0.2 Perrottet, de Rafael, S.P . ’98 PQCD(asy) 0.1 0 0 2 4 6 8 10 12 14 Q 2 (GeV 2 ) LARGE- Nc QCD – p.3/19 PADES and
∞ C 2 F ( − q 2 ) 1 + q 2 � R = − q 2 + M 2 R R ( ⋆ ) M 2 V ≈ − q 2 + M 2 V • What is this approximation ( ⋆ ) ? (1,2,.... ∞ ) • Does it work for all functions ? • Where in the complex Q 2 plane does ( ⋆ ) converge ? • How are the poles/residues of the approx. ( ⋆ ) related to the physical counterparts? LARGE- Nc QCD – p.4/19 PADES and
High Energy: Weinberg SRs ⋆ < V V − AA > with “Regge” spectrum (large n ) • M 2 A n ∼ M 2 V n ∼ n , for poles n ≫ 1 . • F A n ∼ F V n ∼ const . , for residues n ≫ 1 . N + c N F 2 F 2 � � F 2 − q 2 A n V n q 2 Π( − q 2 ) = � + q 2 � lim − q 2 + M 2 − q 2 + M 2 N →∞ A n V n n n ⋆ Π( − q 2 ) independent of c . ⋆ Analyticity: 2 Im(q ) 2 Re(q ) 2 poles of G(q ) LARGE- Nc QCD – p.5/19 PADES and
High Energy: WSRs (and II) 1 Imposing that q 2 Π( − q 2 ) | q 2 →− ∞ ∼ ( q 2 ) 2 for N finite: N + c N � � ?? − F 2 − � F 2 � F 2 lim A n + = 0 V n N →∞ n n � N N + c � ?? � F 2 A n M 2 � F 2 V n M 2 lim A n − = 0 V n N →∞ n n dependent on c !! (Golterman, S.P . ’03) ⋆ Physical poles and residues do not obey WSRs. LARGE- Nc QCD – p.6/19 PADES and
What is resonance saturation ? • It’s a Pade Approximant to a meromorphic function M 2 ◮ F ( − q 2 ) ≈ V is the PA P 0 1 ( − q 2 ) to F ( − q 2 ) . V − q 2 + M 2 • N A,V resonances in N A N V F 2 F 2 q 2 Π( − q 2 ) = F 2 − q 2 � A + q 2 � V − q 2 + M 2 − q 2 + M 2 A V A V ⇒ P N N ( − q 2 ) with N = N A + N V = ⊕ 1 / ( q 2 ) 2 fall-off = ⇒ P N − 2 ( − q 2 ) N ◮ WSRs are obeyed by PA’s parameters. Parameters (residues + poles) Residues and poles � = of Pade Approx . of physical functions LARGE- Nc QCD – p.7/19 PADES and
Pade Approximants (Physics ⇔ z ≡ − q 2 ) G ( z ) | z → 0 ≈ G 0 + G 1 z + G 2 z 2 + G 3 z 3 + ... . Let Define rational function P M N ( z ) such that N ( z ) ≡ Q M ( z ) R N ( z ) ≈ G 0 + G 1 z + G 2 z 2 + ... + G M + N z M + N + O ( z M + N +1 ) P M If G ( z ) ∼ 1 /z K , choose P M M + K ( z ) . (Pommerenke ’73) Convergence Theorem Let G ( z ) be meromorphic and analytic at the origin. Then, M →∞ P M lim M + K ( z ) = G ( z ) for z ∈ compact set in C , except on isolated points. LARGE- Nc QCD – p.8/19 PADES and
Convergence Map 2 Im(q ) 2 Re(q ) 2 poles of G(q ) LARGE- Nc QCD – p.9/19 PADES and
Convergence Map 2 Im(q ) z 2 Re(q ) 2 poles of G(q ) z* LARGE- Nc QCD – p.9/19 PADES and
Convergence Map 2 Im(q ) z 2 Re(q ) 2 poles of G(q ) z* LARGE- Nc QCD – p.9/19 PADES and
Convergence Map 2 Im(q ) z 2 Re(q ) 2 poles of G(q ) z* _ = pole U zero = ``defect´´ N.B. This is why sometimes residues turn out to be “unexpectedly” small. (Friot, Greynat, de Rafael ’04) LARGE- Nc QCD – p.9/19 PADES and
Convergence Map 2 Im(q ) 2 M P N ~ ~ G(q ) z 2 Re(q ) 2 poles of G(q ) z* Pommerenke ‘73 _ = pole U zero = ``defect´´ LARGE- Nc QCD – p.9/19 PADES and
Toy Model for VV-AA • Not Stieltjes. • Meromorphic. • Spectrum (Regge-like): M 2 V,A ( n ) = m 2 V,A + n Λ 2 QCD Shifman et al. ’98 Golterman, S.P ., ’01 F 2 � � ∞ F 2 F 2 q 2 Π( − q 2 ) = F 2 + q 2 ρ � + q 2 V ( n ) − − q 2 + M 2 − q 2 + M 2 − q 2 + M 2 A ( n ) ρ n =0 where � ’s can be written in terms of ψ ( z ) = Γ ′ ( z ) / Γ( z ) . Can choose realistic numbers so that C 0 − C 2 q 2 + C 4 ( q 2 ) 2 + ... − q 2 Π( − q 2 ) | q 2 → 0 ≈ ( finite radius conv . ) 0 + 0 q 2 + C − 4 ( q 2 ) 2 − C − 6 − q 2 Π( − q 2 ) | q 2 →− ∞ ≈ ( q 2 ) 3 + ... ( no logs , asymptotic) with C ′ s which are calculable ! LARGE- Nc QCD – p.10/19 PADES and
PAs to VV-AA model: Poles and Residues PAs work beautifully. • ∃ Convergence in complex Q 2 plane, away from singularities. • Prediction of physical residues and poles good near the origin but deteriorates very quickly as you move away, eventually becoming complex. Last pole always off. • PAs approximate original function at the expense of altering residues and poles hierarchically (more the farther away from the origin). • Prediction of a global quantity such as � 0 dq 2 q 2 Π( − q 2 ) ∼ ( m π + − m π 0 ) EM −∞ very good. It can even be used as input. LARGE- Nc QCD – p.11/19 PADES and
PAs: predicting the next coeff’s Only with C 0 , C 2 , C 4 : − r 2 r 2 = 3 . 379 × 10 − 3 , P 0 2 = R ) , z R = 0 . 6550 + i 0 . 1732 . ( − q 2 + z R )( − q 2 + z ∗ (natural units: GeV=1) • Poles are complex , i.e. not physical. • One can predict next terms in Taylor expansion at q 2 = 0 and at −∞ . Let’s call X i ≡ C i ( predicted ) . C i ( real ) With P 0 ⇒ X − 4 = 1 . 3 , X 6 = 0 . 97 (not bad !). 2 = • Gone up to P 50 52 (with 103 parameters ): X − 4 , − 6 , − 8 = 1 + O (10 − 52 , − 48 , − 45 ) X 206 = 1 + O (10 − 192 ) , !! Could one always assure this for a Pade ? LARGE- Nc QCD – p.12/19 PADES and
PAs: poles and zeros E.g. Analytic structure of P 50 52 : 2 Im (q ) 15 10 5 (Masjuan, S.P ., ’07) 5 15 10 20 25 2 Re (q ) -5 -1 0 -15 LARGE- Nc QCD – p.13/19 PADES and
Other kind of PAs: Pade-Type Approx. Denominator is fixed with poles at physical masses. Simplest one (3 inputs, C 0 = − F 2 0 , M 2 ρ , M 2 A ) : − F 2 0 M 2 ρ M 2 A T 0 2 ( − q 2 + M 2 ρ )( − q 2 + M 2 A ) • Low- q 2 expansion and integrals over negative q 2 not bad. • Low-order PTAs tend to be worse than PAs, in particular the high- q 2 expansion. Gone up to T 7 9 . • Since poles are predetermined, residues pay full price: ⇒ Residues deteriorate hierarchically (worse the farther from origin). ⇒ Last residue considered, completely off. LARGE- Nc QCD – p.14/19 PADES and
Other kind of PAs: Partial-Pade Approx. • Denominator with some poles fixed and some free. • Interesting example: − r 2 R = 3 . 75 × 10 − 3 , P 0 R r 2 1 , 1 = ρ )( − q 2 + z R ) , with z R = 0 . 8665 . ( − q 2 + M 2 ⇒ M A | Pade = √ z R = 0 . 930 = while M A | exact = 1 . 18 (exactly the same as what is often found in the literature, e.g. Ecker et al. ’89; Friot, Greynat and de Rafael ’04) • In general, PPAs are an intermediate situation between PAs and PTAs. LARGE- Nc QCD – p.15/19 PADES and
Insert: PT prediction in QCD( N c → ∞ ) Assume physical masses from PDG. 0 + 4 L 10 q 2 − 8 C 87 ( q 2 ) 2 + ... q 2 Π ≈ f 2 T n inputs m T 1 f 0 , L 10 ; m ρ , m a , m ρ ′ 3 T 2 ( a ) f 0 , L 10 , δM π ; m ρ , m a , m ρ ′ , m a ′ 4 T 2 ( b ) f 0 , L 10 , F ρ ; m ρ , m a , m ρ ′ , m a ′ 4 T 3 ( a ) f 0 , L 10 , F ρ , δM π ; m ρ , m a , m ρ ′ , m a ′ , m ρ ′′ 5 T 3 ( b ) f 0 , L 10 , F ρ , F a ; m ρ , m a , m ρ ′ , m a ′ , m ρ ′′ 5 T 4 f 0 , L 10 , F ρ , F a , δM π ; m ρ , m a , m ρ ′ , m a ′ , m ρ ′′ , m ρ ′′′ 6 A � Amoros et al. ’00 T 42 a T 42 b T 53 a T 53 b P 20 T 31 T 64 B � Knecht et al. ’01 10 C � Mateu et al. ’07 8 ) A • | Masjuan, S.P ., ’08 C 87 10 3 GeV 2 - − 5 . 7(5) · 10 − 3 GeV − 2 6 (+1 /N c corr ′ s) ← C - B 4 ( (G’lez-Alonso et al. ’08) 2 τ decay ⇒ 4 . 9(2) · 10 − 3 GeV − 2 T 42 a T 42 b T 53 a T 53 b P 20 T 31 T 64 LARGE- Nc QCD – p.16/19 PADES and
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