Photon-to-pion transition FF and endpoint behavior of pion DA v 1 Alexander Pimik o v 1 Stefanis 2 in ollab oration with S. Mikhailo , N. Russia) 1 Bogoliub o v Lab. Theor. Ph ys., JINR (Dubna, y) 2 ITP-I I, Ruhr-Univ ersit� at (Bo h um, German "Bogoliub o v Readings", Dubna 2010 p. 1
Outline: Introduction to “factorization” and its components Transition γγ ∗ → π 0 form factor (FF) Pion distribution amplitudes (DA) ϕ π ( x ) Data on the pion to photon transition form factor (FF) QCD sum rules (SR) approach Nonlocal scalar quark condensate QCD SR for pion DA QCD SR for slope of pion DA at the origin: derivative ϕ ′ π (0) and “integral derivatives”. Comparison of our results with other models Conclusions "Bogoliub o v Readings", Dubna 2010 p. 2
“Factorization” γ ∗ ( q 1 ) γ ∗ ( q 2 ) → π 0 ( P ) in pQCD � � � d 4 xe − iq 1 · z � π 0 ( P ) | T { j µ ( z ) j ν (0) }| 0 � = iǫ µναβ q α 1 q β 2 · F γ ∗ γ ∗ π ( Q 2 , q 2 ) , ∗ γ 1 = Q 2 > 0 , − q 2 2 = q 2 ≥ 0 where − q 2 π γ∗ Collinear factorization at Q 2 , q 2 ≫ (hadron scale ∼ m 2 ρ ) F ) + O ( 1 F γ ∗ γ ∗ π ( Q 2 , q 2 ) = T ( Q 2 , q 2 , µ 2 F ; x ) ⊗ ϕ π ( x ; µ 2 Q 4 ) , where µ 2 F – boundary between large scale and hadronic one. ( q ) � � 1 xP � √ ( P ) � � � � 1 2 1 F γ ∗ γ ∗ π ( Q 2 , q 2 ) = 3 f π dx x ϕ π ( x ) � Q 2 x + q 2 ¯ � xP ( q ) � � 0 2 √ √ � � � 1 2 dx 2 Q 2 F γ ∗ γπ ( Q 2 , q 2 → 0) = 3 f π � x − 1 � π 3 f π x ϕ π ( x ) ≡ 0 "Bogoliub o v Readings", Dubna 2010 p. 3
Feynman diagram for e + e − → e + e − π 0 One of the most accurate data on exclusive reactions is data on transition FF 2 ) provided by series of experiments e + e − → e + e − π 0 with q 2 F γ ∗ γ ∗ π 0 ( q 2 1 , q 2 2 ≈ 0 . e ± (p) e ± tag (p / ) CELLO (1991) 0 . 7 − 2 . 2 GeV 2 , CLEO (1998) 1 . 6 − 8 . 0 GeV 2 , q 1 π 0 BaBar (2009) 4 − 40 GeV 2 . q 2 − − + + e e "Bogoliub o v Readings", Dubna 2010 p. 4
Pion distribution amplitude ϕ π ( x, µ 2 ) The pion DA parameterizes this matrix element: � � � 1 � � dx e ix ( zP ) ϕ π ( x, µ 2 ) . � 0 | ¯ d ( z ) γ ν γ 5 [ z, 0] u (0) | π ( P ) � z 2 =0 = if π P ν � 0 where the path-ordered exponential � � z � t a A a µ ( y ) dy µ , [ z, 0]= P exp ig 0 i.e., the light-like gauge link, ensures the gauge invariance. Pion DA describes the transition of a physical pion into two valence quarks, separated at light cone. "Bogoliub o v Readings", Dubna 2010 p. 5
Pion distribution amplitude ϕ π ( x, µ 2 ) The pion DA parameterizes this matrix element: � � � 1 � � dx e ix ( zP ) ϕ π ( x, µ 2 ) . � 0 | ¯ d ( z ) γ ν γ 5 [ z, 0] u (0) | π ( P ) � z 2 =0 = if π P ν � 0 Distribution amplitudes are nonperturbative quantities to be derived from QCD SR [CZ 1984] , NLC QCD SR [M&Radyushkin 1988-91,B&Mikhailov&S 1998,2001–04] instanton-vacuum approaches, e.g. [Dorokhov et al. 2000; Polyakov et al. 1998, 2009] Lattice QCD, [Braun et al. 2006; Donnellan et al. 2007] from experimental data [Schmedding&Yakovlev 2000, BMS 2003–2006] DA evolves with µ 2 F according to ERBL equation in pQCD . "Bogoliub o v Readings", Dubna 2010 p. 5
Pion distribution amplitude ϕ π ( x, µ 2 ) The pion DA parameterizes this matrix element: � � � 1 � � dx e ix ( zP ) ϕ π ( x, µ 2 ) . � 0 | ¯ d ( z ) γ ν γ 5 [ z, 0] u (0) | π ( P ) � z 2 =0 = if π P ν � 0 ϕ π ( x ) Curve Approach 1.5 Asymptotic 1.0 BMS from NLC QCD SR CZ from QCD SR 0.5 AdS/QCD result x 0.0 0.0 0.2 0.4 0.6 0.8 1.0 There are numbers of models for pion DA on a market. We could qualitatively collect them in two groups by their behavior at the end-point region x = 0 : end-point suppressed and end-point enhanced pion DAs. "Bogoliub o v Readings", Dubna 2010 p. 5
CELLO and CLEO data on the transition FF 0.25 0.25 Q 2 F ( Q 2 ) [ GeV 2 ] Curve Approach 0.2 0.2 Asy CZ 84 0.15 0.15 NLO BMS 01-09 � CELLO 91 0.1 0.1 ▲ CLEO 98 0.05 0.05 Q 2 [ GeV 2 ] 2 2 4 4 6 6 8 8 10 10 CLEO and CELLO data favor endpoint-suppressed π DA; BMS “bunch” within 1 σ level. Endpoint suppression controlled by vacuum-quark nonlocality: λ 2 q = 0 . 4 GeV 2 . "Bogoliub o v Readings", Dubna 2010 p. 6
BaBar data on the γγ ∗ → π transition FF 0.25 0.25 Curve Approach Q 2 F ( Q 2 ) [ GeV 2 ] Asy 0.2 0.2 CZ 84 NLO 0.15 0.15 BMS 01-09 � CELLO 91 0.1 0.1 ▲ CLEO 98 � BaBar 09 0.05 0.05 Q 2 [ GeV 2 ] 2 2 4 4 6 6 8 8 10 10 For momentum transfer up to 9 GeV 2 , new BaBar data agree well with the previous CLEO data and prefer the DA with endpoints strongly suppressed. (NLO in LCSR) "Bogoliub o v Readings", Dubna 2010 p. 7
BaBar data on the γγ ∗ → π transition FF Curve Approach Q 2 F ( Q 2 ) [ GeV 2 ] 0.3 0.3 Asy 0.25 0.25 CZ 84 NLO BMS 01-09 0.2 0.2 � CELLO 91 0.15 0.15 ▲ CLEO 98 0.1 0.1 � BaBar 09 0.05 0.05 Q 2 [ GeV 2 ] Radyushkin 09 5 5 10 10 15 15 20 20 25 25 30 30 35 35 40 40 BaBar( Q 2 > 10 GeV 2 ) χ 2 CLEO&CELLO CLEO&BaBar BaBar ¯ Asy 3 . 9 11 . 5 19 . 2 19 . 8 BMS 0 . 56 4 . 4 7 . 8 11 . 9 CZ 5 . 2 20 . 9 36 . 0 6 . 0 Table from [M.&Stefanis Nucl.Phys.B821, 291-326, 2009] "Bogoliub o v Readings", Dubna 2010 p. 7
BaBar data—the facts 9 BaBar data points conform with the QCD paradigm and approach the √ 2 f π (horizontal dashed line) of Q 2 F ( Q 2 ) from below pQCD boundary 8 BaBar data points deviate from the “orthodox” scaling behavior of Q 2 F ( Q 2 ) ; they lie above the pQCD boundary and move with Q 2 farther away from it up to the highest measured point at 40 GeV 2 Moreover, they contradict the collinear factorization formula in QCD per se. They contradict the “counting rules” – the most reliable method up to now. As a corollary, CZ DA does not conform with the BaBar data: In the CLEO region it is off by 4 σ ; above 10 GeV 2 it starts to scale— no growth � Opposite statements by BaBar, PRD80(2009)052002 and Druzhinin in arXiv:0909.3148 [hep-ex] are unfounded . � "Bogoliub o v Readings", Dubna 2010 p. 8
QCD correction to transition FF Mikhailov&Stefanis Mod.Phys.Lett.A24:2858-2867, 2009. Higher radiative corrections up to NNLO β provide to Q 2 F ( Q 2 ) suppression Twist-4 contribution to Q 2 F ( Q 2 ) also provides suppression Hadronic content of real photon, parameterized via a Breit-Wigner resonance , gives some enhancement at low Q 2 Stefanis, 2008, Nucl.Phys.Proc.Suppl.199 "Bogoliub o v Readings", Dubna 2010 p. 9
Contextual explanations of the “BaBar effect” Dorokhov , arXiv:1003.4693 Quark-loop (triangle) diagram : Q 2 F γ ∗ γπ 0 ( Q 2 ) ∼ ln ( Q 2 /M 2 q ) with typical values of M 2 q = 0 . 2 − 0 . 3 GeV 2 Radyushkin , PRD80 (2009) 094009: Flattop pion DA – no radiative corrections, no evolution Polyakov , JETP Lett. 90 (2009) 228: π DA close to unity with φ ′ π (0) / 6 ≫ 1 at µ = 0 . 6 ÷ 0 . 8 GeV—convex DA obtained from χ quark model. Evolution included Li, Mishima , PRD80 (2009) 074024: k T -dependent hard kernel convoluted with flat π DA and resumming terms ∼ α s ln 2 x at low- Q 2 —Sudakov resummation Klopot, Oganesian, Teryaev , arXiv:1009.1120: Uses Axial anomaly SR to show importance of higher-state contributions to transition FF Kuraev et al. , arXiv:0912.3668: Sudakov suppression of quark-photon vertex in triangle πγγ ∗ diagram Kochelev, Vento , PRD81 (2010) 034009: Includes gluonic components to F γ ∗ γπ 0 stemming from nonperturbative QCD vacuum in the instanton liquid model Broniowski, Ruiz-Arriola , arXiv:0910.0869, – S pectral Q uark M odel, arXiv:1008.2317 – Regge approach. Chernyak , arXiv:0912.0623: Explains BaBar data by denying Q 2 growth Lih , arXiv:0912.2147: L ight- F ront Q uark M odel Noguera, Vento , arXiv:1001.3075: Match low- Q 2 description with high- Q 2 QCD-based calculation involving ϕ π ( x ) = 1 and evolving from Q 0 to Q ; twist-3 effects also included "Bogoliub o v Readings", Dubna 2010 p. 10
Motivations and aims 1.0 ϕ π ( x ) 1 � � � ϕ π ( x ) ϕ ′ Model dx π (0) x 0.8 0 0.6 Asy 3 6 0.4 � BMS 3 . 15 � � 6 0.2 CZ 4 . 5 26 x ∼ x 0 . 1 Rad. 0.0 ≫ 3 ≫ 6 0.00 0.02 0.04 0.06 0.08 0.10 High- Q 2 BaBar data call for endpoints enhanced π -DAs. But, CLEO and low- Q 2 BaBar data prefer endpoint-suppressed π -DAs. Observable pion FF is mainly defined by unobservable pion DA slope at the origin through inverse moment in collinear factorization. BMS bunch is based on NLC QCD SR and has large errors in endpoint region. Our purpose is the analysis of pion DA endpoint behavior using NLC QCD SR Mikhailov, et.al. PRD82 (2010) 054020 "Bogoliub o v Readings", Dubna 2010 p. 11
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