All order pole mass and potential

Transcription

1 All order pole mass and potential Taekoon Lee Seoul National University, Korea QWG3, Beijing, China, Oct , 2004 Defining pole mass and potential through Borel summation How to perform the Borel summation Systematic summation of renormalons 1

2 Divergence of perturbative expansions The heavy quark pole mass is well defined perturbatively. p/ Σ(p/) p/=mpole = 0 = m pole = m MS (µ) [ 1 + p n (µ)α s (µ) n+1 ] n=0 Gauge and RG invariant order by order. Convergence is bad! Asymptotically, (n + ν)! [ p n = N m (2β 0 ) n 1 + c 1 ν! n + c ] 2 n , ν = β 1 2β 2 0 2

3 m pole = m MS ( ); α s = 0.22, N f = 4 The factorial divergence causes an intrinsic uncertainty of order m MS exp ( ) 1 2β 0 α s (m MS ) Λ QCD The static potential also suffers from IR renormalon V (r) = 1 r n=0 V n α s (1/r) n+1 3

4 Asymptotically, (n + ν)! [ V n = N V (2β 0 ) n 1 + c 1 ν! n + c ] 2 n , ν = β 1 2β 2 0 Convergence is not good. 4

5 Definition of pole mass through Borel summation Pole mass can be defined rigorously with Borel summation. No unique pole mass definition: Anything that is indistinguishable, perturbatively, from the perturbative pole mass can be defined as a pole mass. In Borel summation this ambiguity appears through the ambiguity in defining the integration contour. The ambiguity, however, does not appear in physical amplitudes: renormalon cancellation. Simplest definition: m BR = m MS [1 + Re ( 1 β 0 +iɛ 0+iɛ e b/β 0α s M(b)db )] M(b) = n=0 p n n! ( b β 0 ) n 5

6 Problem: How to perform the Borel integral? The Borel transform from low order perturbations is only good near the origin. Several tens order of perturbations required for accurate computation. This Calculability problem (not the ambiguity) led to abandonment of pole mass, and adoption of renormalon free (short distance) masses such as M S mass, PS mass, 1S mass, RS mass,... The concept of IR insensitivity should be applied first of all to the fundamental parameters of the QCD Lagrangian, the coupling constant and the quark masses. In this respect we have concluded that the pole mass definition should be abandoned even for heavy quarks, because it is more sensitive to long distances than many processes involving heavy quarks. Beneke, Phys. Rep. 317 (1999) 1 The calculability problem can be resolved 6

7 Why care about BR mass? Fundamental conceptual difference: Not a short distance mass Demonstrative example of Borel summation beyond bubble approximation Scale separation 7

8 How to compute Borel integral? The Borel transform has UV and IR renormalon singularities. -1 1/2 3/2 The large order behavior is dominated by the first IR singularity. To compute the pole mass to an accuracy better than O(Λ QCD ) an accurate description of the Borel transform is required in the region that contains the origin and the first IR renormalon. 8

9 x 1/2 b Equivalent to summing the divergent series to all orders. (n + ν)! [ p n = N m (2β 0 ) n 1 + c 1 ν! n + c ] 2 n , ν = β 1 2β 2 0 9

10 Bilocal expansion The RG invariance of the renormalon ambiguity determines the nature of singularity (Beneke(1995)) M(b) = N m (1 2b) 1+ν [ 1 + c1 (1 2b) + c 2 (1 2b) 2 + ] + (analytic part) c 1 = β2 1 β 0 β 2 4νβ 4 0 c 2 = β β 3 0β 1 β 2 2β 0 β 2 1β 2 + β 2 0(β 2 2 2β 3 1) 2β 3 β ν(ν 1)β

11 ``Analytic part'' denotes terms analytic on the disk b 1/2 < 1, and the series expansion converges on the same disk. Combine the expansions about the origin and the singularity via an interpolation: Bilocal expansion(lee, PRD 67 (2003)014020, JHEP 0310:044 (2003)) M(b) = lim N,M M N,M (b), where M N,M (b) = N n=0 h n n! ( b β 0 ) n + N m (1 2b) 1+ν [ 1 + M i=1 c i (1 2b) i ]. Demanding the bilocal expansion have the correct perturbative expansion about the origin we get h 0 = p 0 N m (1 + c 1 + c 2 ), 11

12 h 1 = p 1 2N m β 0 [1 c 2 + ν(1 + c 1 + c 2 )], h 2 = p 2 4N m β 2 0[2 + ν(3 + c 1 c 2 ) + ν 2 (1 + c 1 + c 2 )], h 3 = p 3 8N m β 3 0(1 + ν)[6 + ν(5 + 2c 1 c 2 ) + ν 2 (1 + c 1 + c 2 )], etc. Since the expansion about the origin is convergent on the disk b < 1/2 and so is the expansion about the IR singularity on the disk b 1/2 < 1, bilocal expansion is expected to give a good approximation. Bilocal expansion provides systematic improvement of the Borel transform: higher order calculations of pole mass and β function gives better accuracy to the normalization constant and bilocal expansion. Crucial to the scheme is the computation of the the normalization constant of the large order behavior. With an accurate N m an enormous speed-up in Borel transform can be achieved. 12

13 Normalization constant of large order behavior Infinite number of renormalon diagrams contributes to N m. Being a pure number, no small parameter to expand with. Nevertheless, can be computed perturbatively (Lee, PRD 56 (1997) 1091, PLB 462 (1999) 1). Introduce R(b) (1 2b) 1+ν M(b). Then N m = R(1/2) = n=0 r n ( 1 2) n. 13

14 The right-hand-side can be evaluated in power series, since R(b) is analytic on the disk b < 1/2 and bounded at b = 1/2. Applied to Adler function in QCD N f = 1 N f = 2 N f = 3 N f = 4 N f = 5 LO NLO NNLO Converges very well for the pole mass! (Pineda(2001)) N m = { = , (nf = 5) = , (n f = 4). Estimates of higher or coefficients suggest N m few percents. accurate to a 14

15 Borel summed mass With known NNLO expansions, the interpolating Borel transforms M 0,2 (b), M1,2 (b), and M 2,2 (b) are known. Numerical computations are done using a conformal mapping w = 1 + b 1 3b/2 1 + b + 1 3b/2 Infinity x 1 1/5 15

16 Result: m BR m MS = { = ± = ± at α [5] s (m MS ) = and α [4] s (m MS ) = 0.22, respectively. Errors estimated assuming 5% uncertainty in N m and turning c 2 off. Compare them to the ordinary series m pole = { mms ( ), (n f = 5) m MS ( ), (n f = 4). Precise relation between the pole mass and MSbar mass at a given α s. Static potential (Lee(2003)) 16

17 17

18 Scale separation Short distance mass: Renormalon in hard quantity (pole mass) is subtracted by that in soft quantity (potential). This inevitably mixes hard and soft scales. Example: Heavy quarkonium energy H = 2m pole + V (r, rµ, α s (µ)) = n k n (µ/m, rµ, m, r)α s (µ) n+1 Expansion of H in α s (µ) is free of the renormalon, but suffers from an ambiguity of choosing µ over a wide range 1/r µ m = powers of ln(rm) ln(v) in the coefficients. In Borel summation, pole mass and potential, being RG invariant, can be summed independently at their optimal scales, µ = m for pole mass and µ = 1/r for the potential. = No mixing of scales. H = 2m BR + V BR (r) 18

19 May be useful in top threshold production 19

20 Conclusions Normalization constant of large order behavior can be calculated perturbatively Bilocal expansion provides a framework for systematic summation of renormalons Scale separation is automatic in Borel summation 20