Renormalon Structure. of Correlation Functions in QCD. Matthias Jamin ICREA & IFAE Universitat Autònoma de Barcelona. Renormalon Structure

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1 of Correlation Functions in QCD ICREA & IFAE Universitat Autònoma de Barcelona 1

2 The vector correlator is central to the τ hadronic width. It s general perturbative expansion reads: Π (1+0) V (s) = N c 1π where a µ α s (µ)/π. n=0 Defining the Adler function as one arrives at a n n+1 µ k=0 D (1+0) V (s) s d D (1+0) V (s) = N c 1π n=0 c n,k L k, ds Π(1+0) V a n n+1 µ k=1 (s), L ln s µ k c n,k L k 1.

3 Resumming the Log s with the scale choice µ = s Q : D (1+0) V (Q ) = N c 1π c n,1 aq n. n=0 This shows that only the coefficients c n,1 are independent. c 0,1 = c 1,1 = 1, c,1 = 1.640, c 3,1 = 6.371, c 4,1 = (Baikov, Chetyrkin, Kühn 008) All other c n,k with k > 1 are related to lower c m,1 (m < n) and β-function coefficients through the RG equation. Numerically at Q = M τ : (α s (M τ ) = ) 4π D (1+0) V (Q ) =

4 Define general τ moments (without factor V ud S EW ): R w V /A (s 0) 6πi s =s 0 [ ds w (s) Π (1+0) s V /A (s) + s 0 (s 0 + s) Π(0) V /A ]. (s) For R τ,v /A, the kinematic weight reads: ( w τ (s) = 1 s Mτ ) ( 1 + s M τ And the general decomposition of R w τ,v /A (s 0): R w V /A (s 0) = N c [ δw tree + δ w (0) (s 0 ) + D ). ] δ (D) w,v /A (s 0) + δw,v DV /A (s 0). 4

5 Introducing the dimensionless variable x s/s 0 : δ (0) dx = πi x (1 x)3 (1 + x)d (1+0) V (Mτ x). x =1 Inserting the general expansion of D (1+0) V (M τ x): δ (0) = a n n 1 µ k c n,k πi n=1 k=1 x =1 dx x (1 x)3 (1+x)ln k 1 Setting the renormalisation scale µ = M τ, FOPT follows: with δ (0) FO = τ n=1a(m ) n n k c n,k J k 1, k=1 ( M τ x µ ). J 0 = 1, J 1 = 19 1, J = π, J 3 = π. 5

6 Setting the renormalisation scale µ = M τ x, CIPT follows: with Jn a (Mτ ) 1 πi x =1 δ (0) CI = c n,1 Jn a (Mτ ), n=1 dx x (1 x)3 (1 + x)a n ( M τ x). Numerically at α s (M τ ) = : a 1 a a 3 a 4 a 5 δ (0) FO = (+0.006) = 0.196(0.0) δ (0) CI = (+0.003) = 0.181(0.185) Geometric growth of δ (0) FO : c 5,1 83. (Also c 4,1 5!) 6

7 Large-β 0 limit Introduce new function to discuss Borel transform: 4π D (1+0) V (s) 1 + D(s) = 1 + n=1 c n,1 a(q ) n. Then the Borel transform is defined by: B[ D](u) 1 ( ) c n+1,1 u n π. n! β n=0 1 D(a) is given by the integral representation: D(a) = π du e β u 1a B[ D](u). β 1 0 7

8 In large-β 0, closed solution for B[ D](u): (Broadhurst 1993) B[ D](u) = 3 3π e Cu ( u) k= ( 1) k k [k (1 u) ]. Scheme-dependent constant: C MS = 5/3. Poles for positive integer u (IR renormalons) and negative integer u (UV renormalons). IR renormalons: fixed-sign contribution to c n,1. UV renormalons: alternating-sign contribution to c n,1. High orders dominated by poles close to u = 0. (u = and u = 1.) 8

9 δ (0) Borel sum FO perturbation theory CI perturbation theory Smallest term Perturbative order n δ (0) FO = n=1[ cn,1 + g n ] a(m τ ) n. c n+1,1 = g n+1 = ( ) n [ β1 n! ( 4 9 e 5/3 ( 1) n n + 7 ) ] + e10/3 n +..., ( ) n β1 n! [ 49 ( e 5/3 ( 1) n n + 16 ) ] e10/3 5 n

10 General term in the Operator Product Expansion: Ĉ Od (a Q ) Ôd γ (1) O d [Ĉ(0) ] β Q d = [a Q ] 1 O + Ĉ(1) d O a d Q + Ĉ() O a d Q +... Ô d Q d. Express Q-dependence in terms of a Q : { Ĉ Od (a Q ) Q d ĈO d (a Q )e d β 1 a Q [aq ] d β β 1 exp d ĈO d (a Q )e β d 1 a Q [aq ] d β β 1 a Q 0 } [ ] 1 β (a) 1 + β β 1 a da β1 a [1 + b 1 a Q + b a Q +... ], with b 1 = d β 3 1 ) (β β 1β 3, b = b 1 d ( ) β1 4 β 3 β 1β β 3 + β1 β 4. 10

11 Take Ansatz for Borel transform of IR renormalon pole: B[ D IR p ](u) dp IR [ (p u) 1+ γ 1 + b 1 (p u) + b ] (p u) The imaginary ambiguity takes the form: [ DIR ] Im p (a Q ) e p β 1 a Q [aq ] [1+ γ b β 1 1 γ a Q + b β ] 1 4 γ( γ 1)a Q We can identify: p = d, γ = p β β 1 γ(1) O d, β 1 b 1 = (b 1 + c 1 ), b = 4(b + b 1 c 1 + c ) β 1 γ β1 γ( γ 1). with c 1 Ĉ(1) O d /Ĉ(0) O d, c Ĉ() O d /Ĉ(0) O d. 11

12 (Beneke, MJ 008) To incorporate known renormalon structure, use Ansatz: B[ D](u) = B[ D UV 1 IR IR PO ](u) + B[ D ](u) + B[ D 3 ](u) + d0 + d 1 PO u. Fitting c 1,1 to c 5,1, the parameters are found to be: d1 UV = , d IR IR = 3.16, d3 = 13.5, d PO 0 = 0.781, d PO 1 = Dropping input for c 5,1 and d PO 1 yields: c 5,1 80. Also stable result adding IR pole at u = 4 and dropping d PO 1. 1

13 0.3 ^D(αs ) Borel sum Perturbative series Smallest term Perturbative order n α s (M τ ) = , c 5,1 =

14 Tau width δ (0) Borel sum FO perturbation theory CI perturbation theory Smallest term Perturbative order n α s (M τ ) = , c 5,1 =

15 Model Dependence The behaviour of the Borel model crucially depends on the residue of the gluon-condensate renormalon pole. Assuming some sensitivity to the u= pole at intermediate orders (3-5), a fit to the known c n,1 yields d IR 3.. For small d IR, models can be constructed for which Contour-improved PT is the preferred resummation. Investigate alternate model with d IR 0 and an additional IR renormalon pole at u=4. 15

16 w(x) = (1 x) (1 + x) = w τ FOPT CIPT Borel sum FOPT CIPT Borel sum (Beneke, Boito, MJ 013) 16

17 w(x) = (1 x) 3 (1 + x) = w (1,0) FOPT CIPT Borel sum FOPT CIPT Borel sum (Beneke, Boito, MJ 013) 17

18 0.010 w(x) = (1 x) 3 x (1 + x) = w (1,) FOPT CIPT Borel sum FOPT CIPT Borel sum (Beneke, Boito, MJ 013) 18

19 To make progress regarding CIPT versus FOPT, the value of d IR should be corroborated. Two possible routes: i) As the renormalon ambiguity is universal, employ PT series of other correlators to obtain additional information. ii) Determine d IR from the lattice. Not possible directly for the Adler function, but for the plaquette. (Bali et al. 014) In view of bad perturbative behaviour, the classical ALEPH moments should be avoided. 19

20 To make progress regarding CIPT versus FOPT, the value of d IR should be corroborated. Two possible routes: i) As the renormalon ambiguity is universal, employ PT series of other correlators to obtain additional information. ii) Determine d IR from the lattice. Not possible directly for the Adler function, but for the plaquette. (Bali et al. 014) In view of bad perturbative behaviour, the classical ALEPH moments should be avoided. Thank You! 0