The Pγ transition form factor in QCD

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1 The Pγ transition form factor in QCD P. Kroll Fachbereich Physik, Univ. Wuppertal and Univ. Regensburg Jefferson Lab, May 2011 Outline: The πγ trans. form factor in coll. factorization The new BaBar data Ways out The mod. pert. approach Generalization to η,η,η c Summary PK 1

2 e e e Measuring the πγ form factor e + e + e + space-like time-like γ γπ vertex: Γ µν = ie 2 F πγ (Q 2 ) ǫ µναβ q α q β data from: TPC/2γ(90), CELLO(91) CLEO(95,98) Q 2 < 8GeV 2 BaBar(09) 4< Q 2 < 38GeV 2 also data on ηγ, η γ, η c γ L3(97), CLEO(95,98), BaBar(10) BaBar(06) ηγ and η γ at s = 112GeV 2 two-photon decay width of the mesons: normalization of FF at Q 2 = 0 PK 2

3 µ Theory: collinear factorization È È F πγ (Q 2 ) = T NLO H = µ 1 Φ π (x,µ F ) = 6x(1 x) 2fπ 3 0 { 1 α s (µ R ) 1+C xq 2 F 2π [ dxφ π (x,µ F )T H (x,q 2,µ R ) at large Q 2 1+ n=2,4,... [ 1 2 ln2 x xlnx 2(1 x) 9 2 +( 3 2 +lnx) ln Q2 µ 2 R ( ) γn /β 0 αs ] (µ F ) a n (µ 0 ) Cn 3/2 (2x 1) α s (µ 0 ) f π pion decay constant; µ F,µ R,µ 0 factorization, renormalization, initial scale a n embody soft physics convenient choice: µ F = µ R = Q MS scheme γ n anomalous dimensions (pos. fractional numbers, growing with n) LO: Brodsky-Lepage (80) NLO: del Aguila-Chase (81); Braaten (83) Kadantseva et al(86) PK 3 ] }

4 LO: Q 2 F πγ = [ 2fπ 3 1/x 1/x = 3 1+ ] a n (µ F ) due to evolution relative weights of the a n vary with lnq 2 for lnq 2 Φ π 6x(1 x) = Φ AS Q 2 F πγ 2f π PK 4

5 Q 2 = 1 2 (Q2 +Q 2 ) ; ω = Q2 Q 2 Q 2 +Q 2 F πγ (Q 2,ω) = Two virtual photons 2fπ 3Q dx Φ π (x,µ F ) 1 (2x 1) 2 ω 2 for ω 0 : Q 2 2fπ [ F πγ = 1 α ] s 3 π ω 0 limit Cornwall(66) a n contribute to order ω n Diehl-K-Vogt(01) α s corrections del Aguila-Chase (81) αs 2 corrections Melic-Müller-Passek (03) [ 1+ α s(µ R ) π +O(ω 2 ) ] K(ω, x) parameter-free QCD prediction (not end-point sens., power corr. small) theor. status comparable with R = σ(e + e hadrons)/σ(e + e µ + µ ), Bjorken sum rule, Ellis-Jaffe sum rule,.. but no data PK 5

6 Situation before the advent of the BaBar data Q 2 F πγ CELLO CLEO97 mhsa Q 2 [GeV 2 ] remaining 10% can be explained easily but differently: Q 2 = 0 close to NLO result evaluated from asymptotic distribution amplitude non-asymptotic DA, low renormalization scale, twist-4 effects, quark transverse momenta,... K-Raulfs (95), Ong (96), Musatov-Radyushkin (97), Brodsky-Pang-Robertson (98), Yakovlev-Schmedding (00), Diehl et al (01), Bakulev et al (03),... PK 6

7 The new situation Q 2 F(Q 2 ) [GeV] a 2, a 4 a 2 AS fit Q 2 [GeV 2 ] BABAR CLEO BaBar (09) strong increase with Q 2 green: 2f π ( Q 2 /10GeV ) 0.25 (to guide the eyes) AS: NLO corr. < 0 a 2 (1GeV) = 0.39 a 2 (1GeV) = 0.39, a 4 = 0.24 we have to worry: a substantial increase of FF is difficult to accomodate in fixed order pqcd corr. due to a n (> 0) only shift NLO pred. upwards, don t change shape (except unplausible solutions like a 2 4, a in conflict with lattice QCD: a 2 (1GeV) = 0.252±0.143 Braun et al (06)) PK 7

8 Ways out flat DA Φ 1: Polyakov(09) Q 2 F πγ 1 0 dx[ x+m 2 /Q 2] 1 = ln[q 2 /M 2 +1] Radyushkin(09) with Gaussian w.f. k 2/x(1 x) Q 2 F πγ 1 0 dx {1 exp [ xq2 ] } ln[q 2 /2σ] x 2(1 x)σ broad DA also found from AdS/QCD x(1 x) Brodsky-de Teramond(06) but not by Mikhailov et al (10) QCD sum rules Dorokhov(10) non-pert. effects (chiral quarks, instantons) ln[q 2 /M 2 q] dispersion (LCSR) approach: Khodjamirian(09), Agaev et al (10) corrections due to long-distance, hadron-like component of photon quark-transverse momenta and resummation of Sudakov-like effects: Li-Mishima(09), K(10) PK 8

9 exp[ s( l ; ~ b l ; Q)] 0 1 The modified perturbative approach LO pqcd + quark transv. momenta + Sudakov suppr. Sterman et al (89,92) = coll. fact. a. for Q 2 (k fact. based on work by Collins-Soper) Sudakov factor: higher order pqcd in NLL, resummed to all orders S ln ln(xq/ 2Λ QCD ) ln(1/bλ QCD ) +NLL+RG(µ F,µ R ) = e S exponentiation in b space (q q separation) with e S = 0 for b > 1/Λ QCD 1 l 0 ˆΨ π (x,b,µ F ) = 2π f π Φ π (x,µ F )exp[ x(1 x)b2 ] 6 4σπ 2 fact. scale µ F = 1/b (µ R = max( xq,1/b)) b plays role of IR cut-off: interface between soft gluons (in wave fct) ξ = x,1 x 0 1 F πγ = 1 0 ~ b l QCD dx 1/ΛQCD 0 and (semi-)hard gluons in Sudakov f. and T H db 2 ˆΨπ [ 2 3π K 0 ( xqb) ] e S PK 9

10 A remarkable property with the Gegenbauer expansion Q 2 F πγ = 2f π C 0 (Q 2,µ 0,σ π ) [1+ ] a n (µ 0 )C n /C 0 n=2,4, C n /C 0 n Q 2 [GeV 2 ] Q 2 : C 0 1 and C n 0 due to evolution low Q 2 : strong suppr. of higher terms increasing Q 2 : higher n terms become gradually more important intrinsic k omitted only lowest few Gegenbauer terms influence results on F πγ Φ AS suffices for low Q 2 (see fit to CLEO data) PK 10

11 Nature of corrections in m.p.a. can be understood by replacing e S by Θ(1/Λ QCD b) and wave fct δ(k 2 ): Λ 1 QCD bdbk 0 ( xqb) = 1 [ 1 xq 2 xq ( xq K 1 Λ QCD 0 xq ΛQCD : K 1 term exponentially suppressed xq ΛQCD : K 1 term of O(1) multiplication with distr. ampl. and integration over x Λ QCD ) ] (coll. fact: suppression of F πγ 1+a 2 +a Λ2 QCD Q 2 (1+6a 2 +15a )+O large b only by pert. prop.) ( Λ 4 ) QCD Q 4 Sudakov factor provides series of power suppressed terms which come from region of soft quark momenta (x,1 x 0) and grow with Gegenbauer index n intrinsic transverse momentum: power suppressed terms from all x which do not grow with n Agaev et al (10): similar observation in the framework of LCSR large non-pert. corrections suppress higher order Gegenbauer terms PK 11

12 Suppression of soft regions Fπγ(µc)/Fπγ(0) µ c [GeV] accumulation profile at Q 2 = 10GeV 2 µ c cut-off parameter - any contribution to form factor is set to zero if µ R < µ c PK 12

13 Fit to BaBar data present data allow to fix only one Gegenbauer coefficient fit to CLEO and Babar data (initial scale µ 0 = 1GeV): a 2 = 0.25 (fixed from lattice Braun(06)) a 4 = 0.01±0.06 σ π = 0.40±0.06GeV 1 (trans. size parameter) dashed line: Φ AS K.-Raulfs(95); dotted line: coll. NLO result 0.3 πγ Q 2 F(Q 2 ) [GeV] BaBar CLEO Q 2 [GeV 2 ] PK 13

14 Extension to ηγ and η γ P = η,η : F Pγ = FPγ 8 +F1 Pγ FPγ i as F πγ except of diff. wave fct. and charge factors octet-singlet basis favored because of evolution behavior: flavor-octet part as for pion flavor-singlet part: due to mixing with the two-gluon Fock component if intrinsic glue is small a g n(µ 0 ) 0: evolution with γ n (+) γ n to NLO: also direct contribution from gg Fock state (K-Passek(03)) quark-flavor mixing scheme (Feldmann-K-Stech (98)) F ηγ = cosθ 8 F 8 sinθ 1 F 1 2 Q 2 F 8 3 f 8 F η γ = sinθ 8 F 8 +cosθ 1 F 1 Q 2 F 1 4 f 1 3 f 8 = 1.26f π f 1 = 1.17f π θ 8 = 21.2 θ 1 = 9.2 PK 14

15 Results for ηγ and η γ Q 2 F(Q 2 ) [GeV] ηγ BaBar CLEO Q 2 [GeV 2 ] Q 2 F(Q 2 ) [GeV] η γ BaBar CLEO L Q 2 [GeV 2 ] data BaBar (10) dashed: Φ AS Feldmann-K.(97) dotted: asymptotic behavior solid: σ 8 = 0.84±0.14GeV 1 ; a 8 2(µ 0 ) = 0.06±0.06 σ 1 = 0.74±0.05GeV 1 ; a 1 2(µ 0 ) = 0.07±0.04 Q 2 F(Q 2 ) [GeV] F 1 F Q 2 [GeV 2 ] BaBar CLEO PK 15

16 Extension to η c γ F(Q 2 )/F(0) BaBar Q 2 [GeV 2 ] η c γ data BaBar(10) 2nd large scale 2 6e 2 c T H = xq 2 +(1+4x(1 x))m 2 c +k 2 (x 1/2) Φ ηc = Nx(1 x)exp [ σ 2 2ηc M 2ηc x(1 x) Wirbel-Stech-Bauer(85) (N = 8.849, σ ηc = 0.44GeV 1 ) Sudakov unimportant ] solid (dashed, dotted) line: m c = 1.35(1.49,1.21)GeV in contrast to πγ form factor: behavior predicted Feldmann-K(97) NLO corrections at low Q 2 : Shifman-Vysotskii(81), Braaten(83) F(Q 2 ) / F AS η π η η c Q 2 [GeV 2 ] PK 16

17 The time-like region collinear factorization to LO accuracy: time-like = space-like (at s = Q 2 ) within m.p.a. ( as proposed by Gousset-Pire(94) for pion elm. FF) 1/(xQ 2 +k 2 ) 1/( xs+k 2 ıǫ) or K 0 ( xqb) ıπ 2 H(1) 0 ( xsb) analytic continuation of Sudakov f. not well understood (Magnea-Sterman(90)) (probably leads to an oscillating phase) Gousset-Pire: take space-like Sudakov factor estimate: s = 112GeV 2 : s F ηγ 0.17GeV s F η γ 0.28GeV BaBar(06): s F ηγ = 0.229±0.031GeV s F η γ = 0.251±0.021GeV PK 17

18 Consequences for the pion elm. form factor F π 2 Q 2 Fπ(Q 2 ) [GeV 2 ] Φ AS fit (JK) Q 2 [GeV 2 ] perturbative contribution: with distr. amplitude from best fit to πγ form fator with Φ AS Jakob-K.(93) PK 18

19 Summary even this simple excl. observable, believed to be understood very well, is subject to strong power suppressed corrections visible even at Q 2 as large as 40GeV 2 casts severe doubts on any attempt to explain other excl. observables within coll. factorization frame work (e.g. pion or proton FF) quark-transverse momenta and Sudakov suppressions is one way to estimate power corrections; existing data on Pγ trans. form factor (P = π,η,η,η c ) can well be described within that approach. One Gegenbauer coeff. of DA can be determined from data preserves standard asymptotics Q 2 F πγ 2f π πγ form factor should be remeasured by BELLE PK 19

20 Comparison with Li-Mishima (09) Gaussian wf. with Φ π 1 accompanied by treshold factor resummed α s ln 2 (x) and α s ln 2 (1 x) arising from end-point singularities which occur for flat DA = effective DA Φ r = Γ(2+2r) [ ] r Γ 2 x(1 x) (1+r) limiting cases: Φ AS and Φ π 1 remains power-like under evolution with r r(µ) Fit to BaBar data: r(2gev) = 0.59±0.06 r(µ) (21) [9] comparison of evolution behavior LM claim: r 1 at low Q 2 (Φ Φ AS ) µ 2 [GeV 2 ] r small at Q 2 40GeV 2 (Φ 1) PK 20