Pion Transition Form Factor

First Prev Next Last Symposium on the 12% Rule and ρ π Puzzle in Vector Charmonium Decays Beijing, China, March 18-20, 2011. Pion Transition Form Factor ZE-KUN GUO In Collaboration With Qiang Zhao Institute of High Energy Physics, Chinese Academy of Sciences Beijing, China, 100049 Page 1 of 28 guozk@ihep.ac.cn

First Prev Next Last Outline Pion transition form factor before BaBar Effective Lagrangian and meson loops Conclusion and outlook Page 2 of 28

First Prev Next Last 1. Pion transition form factor before BaBar Photon-pion transition form factor (TFF) in the space-like momentum transfer region, an ideal subject to test the validity of pqcd in exclusive processes, and the most important non-perturbative parameter is the twist-2 pion light-cone distribution amplitude (DA) F γπ (Q 2 ) = 1 {[ 2f π du T 0 + α s(µ 2 R ) ] } 0 4π T 1 ϕ π (u, µ 2 F) +higher twist, (1) The conformal expansion in terms of Gegenbauer polynomials Cn 3/2 (ξ), ϕ π (u; µ 2 F) = Ω(u) an (µ 2 F) Cn 3/2 (ξ), n 0 Ω(u) 6u(1 u), a 0 = 1, a n>0 (µ 2 F ) 0 (2) Page 3 of 28

First Prev Next Last In the experimental accessible momentum transfer region, the perturbative and non-perturbative corrections are important, Page 4 of 28 Fig.1 QCD structure of γ π 0 transition form factor.

First Prev Next Last Operator product expansion (OPE) analysis of TFF Fig.1(a), the leading twist contribution Fig.1(b), the contribution from higher Fock-state or long-distance photon effect, gives another half of anomaly in light-cone QCD (LCQCD), T.Huang and X.G.Wu, Int. J. Mod. Phys. A 22, 3065 (2007);. Fig.1(c), the twist-4 quark-antiquark-gluon contribution, which is related to the quark transverse momentum through QCD equation of motion. Fig.1(d), the Feynman mechanism, corresponding to the overlap of initial photon and final pion wave functions. Page 5 of 28

First Prev Next Last Most importantly, the handbag diagram with massless quark propagator can not generate the twist-6 contribution because higher terms in the light-cone expansion of bilocal operator 0 ψ(0)γ 5 /zψ(z) p = ξ 2 (zp) z 2 =0 + z 2 ξ 4 (zp) z2 = 0 + (z 2 ) 2 ξ 6 (zp) z 2 =0 +... (3) cancel the singularity of the propagator /z(z 2 ) 2. Musatov and Radyushkin, Phys. Rev. D 56, 2713 (1997). The extra twist-6 part comes from the factorizable four quark pion DA, namely the product of quark condensate and a twist-3 chiral-odd DA. Page 6 of 28 Khodjamirian, Eur. Phys. J. C 6, 477 (1999).

First Prev Next Last The pqcd leading twist results of the rescaled TFF in the asymptotic region, Q 2 F γπ (Q 2 ) Q2 2f π 3 1 0 du Ω(u) u = 2f π, (4) The TFF at zero momentum transfer is related to the famous chiral anomaly, F γπ (0) = 1 4π 2 f π. (5) Then, we have the Brodsky-Lepage interpolation formula in the whole momentum transfer region F γp BL (Q2 ) = 1 4π 2 f P 1 1 + (Q 2 /8π 2 f 2 P ). (6) Brodsky and Lepage, Phys. Rev. D 24, 1808 (1981). Page 7 of 28

First Prev Next Last The analysis up to next-to-leading order (NLO) radiative and twist-4 of CELLO and CLEO space-like experiments (CELLO, Z. Phys. C 49, 401 (1991); CLEO, Phys. Rev. D 57, 33 (1998)) as well as other experiments show that the twist-2 DA of pion is nearly asymptotic or at least end-points suppressed Page 8 of 28 Fig.2 pion DA analysis taken from Ze-kun Guo and Jueping Liu, Phys. Rev. D 78, 076006 (2008).

First Prev Next Last Fig.3 The configurations of different pion DAs at µ 2 0 = 1GeV 2. The midpoint Page 9 of 28 values from above to below correspond to asymptotic, PR, BF (a2=0.44,a4=0.25), NLC01, NLC06, CZ, Agaev, BMS03, BMS06.

2. However, the celebrated confirmation of pqcd dominance at large momentum transfers was shortly overwhelmed by the surprising observation at BaBar in the space-like region up to Q 2 = 40 GeV 2 Page 10 of 28 Fig.4 Figure taken from BaBar, Phys. Rev. D 80, 052002 (2009). First Prev Next Last

First Prev Next Last The fit result of rescaled TFF presents an unexpected enhancement in Q 2 > 9 GeV 2 region and exceeds the asymptotic limit ( ) Q Q 2 F γπ (Q 2 2 β ) = A 10GeV 2, (7) A = 0.182 ± 0.002 GeV, and β = 0.25 ± 0.02, which favors endpoint concentrated DAs, e.g. CZ DA, Chernyak and Zhitnitsky, Phys. Rep. D 112, 173 (1984). Page 11 of 28 Fig.5 Figure taken from BaBar, Phys. Rev. D 80, 052002 (2009).

First Prev Next Last 3. 3.1. π 0 case The rising can not be explained by next-to-next-to-leading order (NNLO) perturbative correction, of which the correction is about -5% at Q 2 5.76GeV 2. Mikhailov and Stefanis, Nucl. Phys. B 821, 291 (2009). Radyushkin and M.V. Polyakov argue that, a flat pion DA (ϕ π (u) = f π ) combined with a massive propagator in the handbag diagram can explain the BaBar data, Q 2 F γπ (Q 2 ) = 2f π 3 and M 2 0.6 GeV 2. 1 0 Q 2 ) du (1 uq 2 + M 2 = log + Q2 M 2, (8) Radyushkin, Phys. Rev. D 80, 094009, 2009; M.V. Polyakov, JETP Lett. 90, 228 (2009). Page 12 of 28

First Prev Next Last But as discussed above this will lead to the extra (M 2 /uq 2 ) n power correction, which contradicts OPE. This is why the introduction of transverse momentum is more natural. So, taking the Gaussian ansatz for the transverse momentum dependence of the light-front wave function(lepage, Brodsky, Huang and Mackenzie, Hadronic Wave Functions In QCD, Report CLNS-82-522 (1982)), Radyushkin gives F γπ (Q 2 ) = 2f π 3 1 0 ϕ π (u) uq 2 [ 1 exp )] ( uq2 dx. (9) 2ūσ But the flat DA means the evolution of one-gluon-exchange switches off. The gluon line should be absorbed into the wave function, and the calculation of form factor is completely non-perturbative. Page 13 of 28 Radyushkin, Phys. Rev. D 80, 094009, (2009).

First Prev Next Last Taking the transverse momentum into account, several authors give their solutions to the, and the DAs are not unique: Page 14 of 28 Fig.6 Analyses taking transverse momentum into account.

First Prev Next Last The local or nonlocal chiral quark model (Dorokhov, arxiv:1003.4693 [hep-ph] ) and modified PCAC method (T. N. Pham and X. Y. Pham, arxiv:1101.3177 [hep-ph]) can also give a logarithmic or double logarithmic growth of the rescaled TFF through a massive quark loop, but the constituent quark masses are as low as about 130 MeV. The light-cone sum rules (LCSR) up to twist-6 with more Gegenbauer moments (Agaev, Braun, Offen and Porkert, arxiv:1012.4671 [hep-ph]) shows that the BaBar data needs endpoints concentrated (BF-like) or flat DA, and other processes such as pion electromagnetic form factor and weak decay B πlν l are insensitive to this new pion DA. The only exception is the πργ TFF (Mikhailov and Stefanis, Nucl. Phys. B 821, 291 (2009)). The endpoint concentrated DA like BF DA will enhance the endpoints contribution, so that the α s ln 2 x in collinear NLO hard kernel should be resumed. The difference between asymptotic DA and endpoints concentrated DA after resummation is small, and the correction is about -40% (F. Feng, J. P. Ma and Q. Wang, JHEP 0706, 039 (2007)). Page 15 of 28

First Prev Next Last 3.2. SU(3) F violation The new data of η and η TFF released by BaBar show that the rise of the rescaled TFF is three times weaker than the corresponding rise of π 0 rescaled TFF. Page 16 of 28 Fig.7 Figure taken from BaBar, arxiv:1101.1142 [hep-ex].

First Prev Next Last The weaker growth means that the u, d component in η and η is different from that in π 0, which implies the SU(3) F violation!!! (Kroll, arxiv:1012.3542 [hepph]) Page 17 of 28 Fig.8 Figure taken from BaBar, arxiv:1101.1142 [hep-ex].

First Prev Next Last 4. Effective Lagrangian and meson loops 4.1. Motivation The key problem is the origin of new mass scale in the logarithmic growth! The BaBar data below 9GeV 2 is consistent with CLEO data, and the abnormal rise begins at about 9GeV 2. This mass scale stands at about the J/Ψ threshold by analytical continuation into time-like region. It is reasonable to suspect the charm quark may join the game. If so, this is a pure non-perturbative flavor-changing effect, and the open charm meson loop mechanism may play an important role! Page 18 of 28 4.2. Method The effective Lagrangian for the D ( ) mesons (D ( )0, D ( )+, D s ( )+ ) couplings to light pseudoscalar mesons has the following expression:

L = ig D DP(D i µ P ij D j µ D i µ µ P ij D j ) + 1 2 g D D Pɛ µναβ D µ i ν P ij α D β j, (10) The corresponding Lagrangians for the photon and D ( ) couplings are L DDγ = iea µ D µ D + + A µ Ds µ D s +, (11) { e L D Dγ = 4 g D + D + γɛ µναβ F µν D + αβ D + e 4 g D + s D + s γ ɛµναβ F µν D + sαβ D s + e 4 g D 0 D 0 γɛ µναβ F µν D 0 αβ D 0 } + h.c., (12) L D D γ = iea µ { g αβ D α g µα β D α D + β µ D + β + gµβ Dα α D + β } { + iea µ g αβ Dsα } +g µβ Dsα α D + sβ gµα β Dsα D + sβ µ D + sβ. (13) Page 19 of 28 First Prev Next Last

First Prev Next Last The coupling constants g D + D + γ and g D 0 D 0 γ are extracted from D. Y. Chen, Y. B. Dong and X. Liu, Eur. Phys. J. C 70, 177 (2010): g D + D + γ = 0.5 GeV 1, g D 0 D 0 γ = 2.0 GeV 1, g D s D s γ = 0.3 GeV 1, (14) where the signs and relative size can also be deduced from constituent quark model, and will play an important role in the following analysis. Other couplings can be obtained from either experimental data, or the heavy quark effective theory (HQET) and SU(3) symmetry. Page 20 of 28

First Prev Next Last The corresponding D loops are drawn as follows: Page 21 of 28 Fig.9 D ( ) meson loop contributions to the transition form factor. The crossing diagrams can be obtained by exchanging the photon momenta. The contact terms give no contribution! Loops for Kaon are not drawn here.

First Prev Next Last The transition amplitude can be expressed as follows: d 4 p 3 T 3 T 4 T 5 M fi = (2π) 4 F (p 2 a 3 a 4 a 3), (15) 5 P olarization where T 3,4,5 are the vertex functions given by the effective Lagrangians, and a 3 p 2 3 m 2 3, a 4 p 2 4 m 2 4 (p p 3 ) 2 m 2 4, and a 5 p 2 5 m 2 5 (p 3 p + q 2 ) 2 m 2 5 are the denominators of propagators of the intermediate mesons, respectively. The form factor F (p 2 3) to take into account the off-shell effects of each vertex is introduced to suppress the ultraviolet divergence: F (p 2 3) = 5 i=3 (Λ 2 i m2 i ) (Λ 2 i p2 i ). (16) The cutoff Λ i can be parameterized as Λ i = m i + αλ QCD with Λ QCD = 220 MeV, and m i is the exchanged meson mass. Page 22 of 28

First Prev Next Last Constraints for α: I) The meson loop contributions to the anomaly are restricted to be at about 0.005 GeV 1 (absolute value). II) The form factor of Eq. (16) introduces additional singularities into the integrals empirically, which means that in order to reduce the model-dependence, parameter α should have a sufficiently large value, e.g. α > 1. Y. J. Zhang, G. Li and Q. Zhao, Phys. Rev. Lett. 102, 172001 (2009); X. H. Liu and Q. Zhao, Phys. Rev. D 81, 014017 (2010); F. K. Guo et al., Phys. Rev. D 83, 034013 (2011). Page 23 of 28 iii) We restrict that α D for D loops should not be larger than α K for Kaon loops, and neglect the B mesons contribution.

First Prev Next Last 4.3. π and η transition form factor Fig.10 (a) The rescaled pion form factor in comparison with the experimental data. The dashed line is the pqcd interpolation formula. The meson loops combining the pqcd results give the shadowed band with α D = 3 and α D = 4 for the lower and upper bound while α K = 4 fixed. (b) The α dependence of the meson loop contributions to the chiral anomaly. The dashed, dotted, solid lines are for the D loops, Kaon loops, and their sum, respectively. Page 24 of 28

First Prev Next Last Fig.11 (a) Charmed meson loop contributions to the pion transition form factor. The thin solid, dashed, dot-dashed, and dotted lines denote D D(D), D D(D ), D D (D ), D D (D) loops, and the thick solid line for the total. (b) Kaon loop contributions with the same notations as (a), but D ( ) replaced by Kaons. α D = 4 and α K = 4. Page 25 of 28

First Prev Next Last Fig.12 (a) The rescaled η form factor in comparison with the experimental data (BaBar, Phys. Rev. D 74, 012002 (2006); arxiv:1101.1142 [hep-ex]). The thick line denotes meson loops plus pqcd prediction which is the dashed line, while the dot-dashed, dotted and thin solid lines are for the nn, ss, and physical η from meson loops. α D = 3.5 and α K = 4. (b) The α dependence of the meson loop contributions to the chiral anomaly. The dashed, dotted and solid lines are from the D loops, Kaon plus φφ(η ( ) ) loops, and their sum, respectively. Page 26 of 28

First Prev Next Last 5. Conclusion and outlook I) The abnormal behavior of pion TFF in the large momentum transfer can be explained by the meson loop effect especially the charm loops. II) Meson loop effect can also explain the η TFF experimental data without violating SU(3) symmetry. III) Meson loop effect does not overlap with the pqcd analysis, and the conclusion before BaBar may survive. IV) If the meson loop effect is convinced to be the solution to the, it will be reasonable to expect meson loops important role in the time-like region, and may be useful to explain the famous ρ π puzzle. Page 27 of 28 V) The meson loop contributions to η TFF and other process need to be further studied.

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