The pion-photon transition form factor in QCD: Facts and fancy P. Kroll Fachbereich Physik, Univ. Wuppertal and Univ. Regensburg Dubna, September 2010 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 based on ongoing work in collaboration with V. Braun and M. Diehl PK 1
� � � e e � e � � � � � � + + e e + � Measuring the πγ form factor e space-like time-like Γ µν = − ie 2 F πγ ∗ ( Q 2 ) ǫ µναβ q α q ′ β γ ∗ γπ vertex: data from: BaBar(06) TPC/2 γ (90), CELLO(91) ηγ and η ′ γ at s = 112 GeV 2 CLEO(95,98) Q 2 < ∼ 8 GeV 2 ∼ Q 2 < ∼ 38 GeV 2 BaBar(09) 4 < also data on ηγ , η ′ γ , η c γ two-photon decay width of the mesons: normalization of FF at Q 2 = 0 L3(97), CLEO(95,98), BaBar(10) PK 2
� � � � Theory: collinear factorization P P ( � ) � ( � ) � √ 1 � 2 f π F πγ ( Q 2 ) dx Φ π ( x, µ F ) T H ( x, Q 2 , µ R ) for large Q 2 = 3 0 � �� ln Q 2 1 1 + C F α s ( µ R ) � 1 2(1 − x ) − 9 x ln x 2 + � 3 2 ln 2 x − T NLO 2 + ln x � = H µ 2 xQ 2 2 π R � γ n /β 0 � α s ( µ F ) � � � C 3 / 2 Φ π ( x, µ F ) = 6 x (1 − x ) 1 + a n ( µ 0 ) (2 x − 1) n α s ( µ 0 ) n =2 , 4 ,... 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
√ 1 + � a n ( µ F ) � � 2 f π Q 2 F πγ = � 1 /x � � 1 /x � = 3 LO: 3 due to evolution relative weights of the a n vary with ln Q 2 √ for ln Q 2 → ∞ Q 2 F πγ → Φ π → 6 x (1 − x ) = Φ AS 2 f π PK 4
Two virtual photons 2 = 1 2 ( Q 2 + Q ′ 2 ) ; ω = Q 2 − Q ′ 2 Q Q 2 + Q ′ 2 √ 1 � � 2 f π � Φ π ( x, µ F ) 1 + α s ( µ R ) 2 , ω ) = F πγ ∗ ( Q dx K ( ω, x ) 2 1 − (2 x − 1) 2 ω 2 π 3 Q 0 √ 2 f π 1 − α s 2 F πγ ∗ = � � + O ( ω 2 ) for ω → 0 : Q 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) α 2 s corrections Melic-M¨ uller-Passek (03) 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 − ) , but no data Bjorken sum rule, Ellis-Jaffe sum rule, .. PK 5
Situation before the advent of the BaBar data 0.25 Q 2 F πγ 0.20 0.15 Q ′ 2 = 0 CELLO 0.10 close to NLO result evaluated CLEO97 mHSA 0.05 from asymptotic distribution amplitude 0.00 0.0 2.0 4.0 6.0 8.0 10.0 2 [GeV 2 ] Q remaining ≃ 10% can be explained easily but differently: 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-K-Vogt (01), Bakulev et al (03), . . . PK 6
The new situation BaBar (09) 0.3 strong increase with Q 2 b a 2 , a 4 Q 2 | F ( Q 2 ) | [GeV] b √ a 2 � 0 . 25 b Q 2 / 10 GeV � b green: 2 f π 0.2 b b b b b b b c b b (to guide the eyes) b b b AS c b c b b c b b c b c b b c c b b c c b b c b c c b c b b c 0.1 fit AS: NLO corr. < 0 • BABAR ◦ CLEO a 2 (1 GeV) = 0 . 39 1 2 3 5 10 20 30 50 a 2 (1 GeV) = 0 . 39 , a 4 = 0 . 24 Q 2 [GeV 2 ] 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 4 ≃ − 3 . 5 in conflict with lattice QCD: a 2 (1 GeV) = 0 . 252 ± 0 . 143 Braun et al (06)) PK 7
Ways out flat DA Φ ≡ 1 : � 1 x + M 2 /Q 2 � − 1 = ln [ Q 2 /M 2 + 1] Polyakov(09) Q 2 F πγ ∼ � 0 dx Radyushkin(09) with Gaussian w.f. ∼ k 2 ⊥ /x (1 − x ) � 1 xQ 2 dx � �� Q 2 F πγ ∼ → ln [ Q 2 / 2 σ ] � 1 − exp − x 2(1 − x ) σ 0 � 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) corrections due to long-distance, hadron-like component of photon quark-transverse momenta and resummation of Sudakov-like effects: Braun-Diehl-K, Li-Mishima(09) PK 8
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) ~ exp [ � s ( � ; b ; Q )] l l Sudakov factor: higher order pQCD in NLL, resummed to all orders 1 √ S ∝ ln ln ( xQ/ 2Λ QCD ) exponentiation in b space ⇒ e − S + NLL + RG( µ F , µ R ) = ln (1 /b Λ QCD ) ( q − ¯ q separation) 0 with e − S = 0 for b > 1 / Λ QCD 1 0 � l ~ 0 1 b � � l QCD Φ π ( x, µ F ) exp [ − x (1 − x ) b 2 Ψ π ( x, b, µ F ) = 2 π f π ˆ ] √ 4 σ 2 6 π factorization scale µ F = 1 /b b plays role of IR cut-off: interface between soft gluons (in wave fct) and (semi-)hard gluons in Sudakov f. and T H � 1 � 1 / Λ QCD 3 π K 0 ( √ xQb ) 2 db 2 ˆ � � e − S √ F πγ = dx Ψ π 0 0 PK 9
A remarkable property with the Gegenbauer expansion √ � � � Q 2 F πγ = 2 f 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 0.8 Q 2 → ∞ : C 0 → 1 and C n → 0 n 0.6 due to evolution 2 low Q 2 : strong suppr. of higher terms 0.4 4 0.2 6 increasing Q 2 : higher n terms become 8 10 gradually more important 0 1 2 3 5 10 20 30 50 100 300 Q 2 [GeV 2 ] only lowest few Gegenbauer terms influence results on F πγ Φ AS suffices for low Q 2 (see fit to CLEO data) PK 10
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 ⊥ ) : √ xQ � √ xQ � Λ − 1 �� (coll. fact: suppression of bdbK 0 ( √ xQb ) = 1 QCD � 1 − Λ QCD K 1 xQ 2 Λ QCD large b only by pert. prop.) 0 √ xQ ≫ Λ QCD : K 1 term exponentially suppressed √ xQ ∼ Λ QCD : K 1 term of O (1) multiplication with distr. ampl. and integration over x � Λ 4 Λ 2 � F πγ ∼ 1 + a 2 + a 4 + . . . − 8 QCD Q 2 (1 + 6 a 2 + 15 a 4 + . . . ) + O 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 PK 11
Fit to BaBar data present data allow to fix only one Gegenbauer coefficient Braun-Diehl-K. fit to CLEO and Babar data (initial scale 1 GeV ): a 2 = 0 . 25 (fixed from lattice Braun(06)) σ = 0 . 42 ± 0 . 07 GeV − 1 (trans. size parameter) a 4 = 0 . 07 ± 0 . 10 dashed line: Φ AS K.-Raulfs(95) fit 0.3 b Q 2 | F ( Q 2 ) | [GeV] b b b 0.2 b b b b b b c b b b b b b b c b c b c b b c b b c b c b c c b c b b c b c c b c b c b 0.1 • BaBar ◦ CLEO 1 2 3 5 10 20 30 50 Q 2 [GeV 2 ] PK 12
Extension to ηγ and η ′ γ P = η, η ′ : F P γ = F 8 P γ + F 1 P γ F i P γ 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 n ( µ 0 ) ≃ 0 : evolution with γ (+) if intrinsic glue is small a g ≃ γ n n to NLO: also direct contribution from gg Fock state (K-Passek(03)) quark-flavor mixing scheme (Feldmann-K-Stech (98)) ⇒ 2 cos θ 8 F 8 − sin θ 1 F 1 Q 2 F ηγ = = 3 f 8 4 sin θ 8 F 8 + cos θ 1 F 1 Q 2 F η ′ γ √ = = ⇒ f 1 3 θ 8 = − 21 . 2 ◦ θ 1 = − 9 . 2 ◦ f 8 = 1 . 26 f π f 1 = 1 . 17 f π PK 13
ut ut Results for ηγ and η ′ γ 0.4 ηγ η ′ γ 0.3 0.3 Q 2 | F ( Q 2 ) | [GeV] Q 2 | F ( Q 2 ) | [GeV] c b 0.2 c b c b c b c b 0.2 c b b c c b b c b c b c 0.1 c b 0.1 ◦ CLEO ◦ CLEO △ L3 1 2 3 5 10 20 30 50 1 2 3 5 10 20 30 50 Q 2 [GeV 2 ] Q 2 [GeV 2 ] preliminary BaBar data; ICHEP 2010, Paris dashed: Φ AS Feldmann-K.(97) dotted: asymptotic behavior solid: σ 8 = σ 1 = 0 . 76 ± 0 . 06 GeV − 1 a 8 a 1 2 ( µ 0 ) = − 0 . 10 ± 0 . 09 2 ( µ 0 ) = − 0 . 20 ± 0 . 07 PK 14
Extension to η c γ 1.0 data BaBar(10) η c γ 0.8 √ b 6 e 2 2 | F ( Q 2 ) /F (0) | b c T H = b 0.6 xQ 2 + (1 + 4 x (1 − x )) m 2 c + k 2 b b ⊥ b 2nd scale, Sudakov unimportant 0.4 b b b ( x − 1 / 2) 2 b � � − σ 2 η c M 2 0.2 Φ η c = Nx (1 − x ) exp b η c x (1 − x ) • BaBar 1 2 3 5 10 20 30 50 Wirbel-Stech-Bauer(85) Q 2 [GeV 2 ] 1.0 solid (dashed, dotted) line: 0.8 m c = 1 . 35(1 . 49 , 1 . 21) GeV Q 2 | F ( Q 2 ) | [GeV] 0.6 η c PDG: m c = 1 . 25 ± 0 . 09 GeV 0.4 η ′ 0.2 in contrast to πγ form factor: η π behavior predicted Feldmann-K(97) 1 2 3 5 10 20 30 50 100 300 Q 2 [GeV 2 ] PK 15
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