The γ γ π 0 form factor in QCD

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1 The γ γ π 0 form factor in QCD Vladimir M. Braun Institut für Theoretische Physik Universität Regensburg based on S.S. Agaev, V.M. Braun, N. Offen, F.A. Porkert, Phys. Rev. D83 (2011)

2 Pion-photon transition form factor d 4 y e iq 1y π 0 (p) T{j em µ (y)j em ν (0)} 0 = ie 2 ε µναβ q α 1 q β 2 F γ γ π 0(q2 1, q 2 2) e e e e + e + e + Figure: The pion-photon transition form factor from e + e collisions. here: q 2 1 = Q 2 < 0, q V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

3 The BABAR puzzle P. del Amo Sanchez, et al. [BaBar collaboration] arxiv: BABAR, (3/5)Q 2 F n (Q 2 ) (GeV) BABAR (γ γ π 0 ) CLEO (γ γ η,η / ) CLEO (e + e - γη,γη / ) BABAR (γ γ η,η / ) BABAR (e + e - γη,γη / ) Q 2 (GeV 2 ) Strong scaling violation in γγ π up to Q 2 30 GeV 2?? No effect seen in γγ η(η )?? V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

4 The BABAR puzzle (2) Fixed-order NLO QCD calculation with µ = Q does not work: 0.3 a 2, a 4 Q 2 F(Q 2 ) [GeV] a 2 AS fit BABAR CLEO Input parameters at 1 GeV: magenta: a 0 = 1, blue: a 0 = 1, a 2 = 0.39, black: a 0 = 1, a 2 = 0.39, a 4 = Q 2 [GeV 2 ] Changing pion distribution amplitude does not help at all? Power-suppressed effects 1/Q p?? V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

5 ABC of QCD factorization: leading regions Ì À Ì ¼ À Ë µ µ µ Figure: Schematic structure of the QCD factorization for the F γ γ π 0(Q2 ) formfactor. A : B : C : hard subgraph that includes the both photon vertices real photon is emitted at large distances Feynman Mechanism: soft quark spectator 1 Q Q Q Q 4 Contributions of regions A, B, C are additive All other possibilities lead to exponentially small corrections exp[ Q 2 ] V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

6 Region A: Leading Twist Contribution 1/Q 2 related to OPE for T{jµ em (y)jν em (0)} to leading twist accuracy best studied object in QCD can be written in factorized form: T NLO H = F γ γ π 0(Q2 ) = { 1 xq 2 α s(µ) 1 + C F 2π 2fπ dx T H (x, Q 2, µ, α s(µ))φ π(x, µ) 3 NNLO coefficient function known in conformal scheme [ 1 2 ln2 x x ln x 2(1 x) ( ln x ) ln Q2 µ 2 ]} φ π(x, µ) = 6x(1 x) Pion Distribution Amplitude (DA) n=0 ( αs(µ) ) (0) γ n /2β 0 an(µ 0)Cn 3/2 (2x 1), a 0(µ) = 1 α s(µ 0) V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

7 my average for a 2: a 2(1 GeV) = 0.30 ± 0.15 a 2(2 GeV) = 0.20 ± 0.07 expect 10-15% error from new generation of lattice calculations; precision limited by discretisation errors in operators with derivatives weak constraints on a 4: QCD SRs with nonlocal condensates (model): a LCSRs for B-meson decays: a Lattice calculation not feasible no information on higher moments expect 1 = a 0 > a 2 > a 4 > a 6 >... models using Gegenbauer expansion truncated at some order V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

8 Scaling violation observed by BABAR cannot be explained by a fancy pion DA What are the options? 1 Resummation of higher order perturbative corrections (k t-factorization) 2 1/Q 4 corrections from region A (higher-twist) 3 1/Q 4 corrections from region B (photon emission from large distances) 4 1/Q 4 corrections from region C (soft overlap of wave functions) What methods are available? 1 Direct calculations/estimates/models 2 Dispersion relations and duality (LCSRs) V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

9 Region A: (Sudakov) resummation Basic idea: retain k t dependence in the hard kernel 1 xq 2 1 xq 2 + k 2 t Large corrections exponentiate in impact parameter space F γ γ π 0(Q2 ) = Sudakov form factor 2fπ dx 3 S = s(b, xq) + s(b, (1 x)q) + 2 s(b, Q) = 1 4 Q 2 b 2 0 /b2 dk 2 k 2 Botts, Sterman; Li, Sterman d 2 b 2π T H (x, Q 2, b, µ, α s(µ)) e S φ π(x, b 0/b) [ µ b 0 /b dµ µ γq(αs(µ )) ] ln Q2 k 2 Γ cusp(α s(k 2 )) + Γ(α s(k 2 )) Implementation involves subtleties which influence numerical outcome Usually used in combination with some model for soft effects V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

10 Region A: Twist-4 Contribution 1/Q 4 related to OPE for T{jµ em (y)jν em (0)} to twist-four accuracy well understood µ µ µ µ Figure: Twist-4 corrections to the pion transition form factor involve twist-4 quark-gluon pion distribution amplitudes ( ) 2fπ 1 dx 80 δ F γ γ π 0(Q2 π 2 ) = φπ(x) Q 2 3 x 27 Q 2 δ 2 π 0.2 GeV 2 A significant contribution at Q GeV 2 but unlikely to solve the puzzle V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

11 Region B: Photon emission from large distances Mainly overlap of twist-three photon and pion wave functions not well understood The LO pqcd calculation gives F (B) γ γ π 0 (Q 2 ) = which yields a correction F γ γ π 0(Q2 ) = 2fπ 3 16πα sχ qq 2 1 dx φp 3;π (x) 9fπ 2 Q 4 x 2fπ Q 2 ( dy φγ(y) ȳ 2 ) dx 0.2 GeV2 φπ(x) + ln 2 Q 2 x Q 2 µ 2 IR Infrared Divergence signals overlap with soft region C (may be) a significant contribution at Q GeV 2 but unlikely to solve the BABAR puzzle V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

12 Region C: Feynman (soft) contribution Overlap of soft wave functions truly nonperturbative Musatov Radyushkin Model Use Drell-Yan representation as convolution of light-cone WFs (Brodsky-Lepage) 1 (ε q )F qq f π γ γ π 0(Q2 ) = 4π 3 dx d 2 (ε (xq + k )) k 3 (xq 0 + k ) 2 iǫ Ψ qq(x, k ) with a model wave function ( ) Ψ qq(x, k ) = 4π2 σ φ π(x) exp k2 6 x x 2σx x to get 1 [ ( )] F MR 2fπ dx φ π(x) γ γ π 0(Q2 ) = 1 exp xq2 3 xq xσ using σ = 0.53 GeV 2 and flat pion DA φ π(x) = 1 can fit the BABAR data! caveat: dx d 2 k t Ψ qq(x, k ) 2 =,?! V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

13 MR presents a model for the soft contribution: 1 [ ( )] F MR 2fπ dx φ π(x) γ γ π 0(Q2 ) = 1 exp xq2 3 xq xσ The correction is exponentially suppressed for any finite value of x absent in OPE Upon integration one obtains F γ γ π 0(Q2 ) or F γ γ π 0(Q2 ) Q 2 = Q 2 = [ 2fπ Q 2 1 4σ ] Q 2, φ π = 6x(1 x) [ ] 2fπ Q ln Q2, φ π = 1 2σ The effect is roughly equivalent to the cutoff of the end-point region 1 2σ/Q 2 dx...? end-point contribution is not calculable in terms of L = 0 wave function, e.g. L = 1 is not suppressed Can one estimate soft corrections model-independently? Dispersion relations V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

14 Region C: Dispersion relations and duality (LCSRs) The QCD result satisfies an unsubtracted dispersion relation F QCD γ γ π 0(Q2, q 2 ) = 1 π ds 0 ImF QCD γ γ π 0(Q2, s) s + q 2. Khodjamirian An effect of soft terms is to correct the spectral density to look more like 2fρF F γ γ π 0(Q2, q 2 γ ) = ρ π 0(Q2 ) mρ ImF γ ds γ π 0(Q2, s) q2 π s + q 2. Duality: assume that above a certain threshold ImF γ γ π 0(Q2, s) = s 0 ImF QCD γ γ π 0(Q2, s) for s > s 0 Asymptotic freedom: QCD expression must be correct at q 2, therefore 2fρF γ ρ π 0(Q2 ) = 1 π s0 0 ds ImF QCD γ γ π 0(Q2, s). Duality sum rules: use this result to correct the QCD calculation V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

15 Leading order example QCD calculation LCSR F LCSR γ γ π 0 (Q 2 ) = F QCD γ γ π 0(Q2, q 2 2fπ ) = 3 { 1 2fπ The difference is a soft correction 3 x x0 dx φ π(x) xq dx φ π(x) xq 2 + xq 2. } dx φ π(x) xm ρ 2, x 0 = s 0 s 0 + Q 2 V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

16 Exercise: applying LCSR machinery to the MR model MR model extended to q 2 0: F QCD γ γ π 0(Q2, q 2 2fπ ) = 3 LCSR modified: F LCSR γ γ π 0 (Q 2 ) = + 1 2fπ 3 x 0 x0 2fπ [ ( )] dx φ π(x) xq 2 + xq 2 1 exp xq2 2 xσ [ ( dx φ π(x) xq 2 1 exp xq2 2 xσ [ ( dx φ π(x) xm 2 ρ 1 exp LCSR greatly reduce sensitivity to the soft region xq2 2 xσ )] )] x = 1 x º e s 0/σ º 1/Q 4 1/Q 6 V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

17 Literature LO A. Khodjamirian, Eur. Phys. J. C6, (1999) 477 NLO A. Schmedding, O.I. Yakovlev, Phys.Rev. D62 (2000) A.P. Bakulev, S.V. Mikhailov, N.G. Stefanis, Phys.Lett. B508 (2001) 279 A.P. Bakulev, S.V. Mikhailov, N.G. Stefanis, Phys.Rev. D67 (2003) S.S. Agaev, Phys.Rev. D72 (2005) S.V. Mikhailov, N.G. Stefanis, Nucl.Phys. B821 (2009) 291 revisited: S.S. Agaev, V.M. Braun, N. Offen, F.A. Porkert, Phys. Rev. D83 (2011) Higher orders in the Gegenbauer expansion of pion DA Twist-six operators in the OPE V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

18 Results (I) Simple ad hoc models do not work models of pion DA: φ as π (x) = 6x(1 x), φ hol π (x) = 8 π φ flat π (x) = 1 x(1 x), V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

19 Results (II) Two-component models a 2 < a 4 V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

20 Results (III) The same models describe pion form EM factor and B πlν l width NLO LCSRs in all cases V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

21 Conclusions There seems to be several separate issues: LCSR fits generally prefer a small value a 2(1 GeV) compared to a 2(1 GeV) 0.35 ± 0.15 from lattice calculations higher precision lattice data needed BABAR data in the Q 2 = GeV 2 range require large a 4(1 GeV) 0.25; older data/other reactions not sensitive because of lower effective Q 2 No natural explanation for the difference γ γ π and γ γ η more experimental data needed (KEK?) V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

22 Supplementary material V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

23 a 2 (µ 0 ) Method µ = 1 GeV µ = 2 GeV Reference LO QCDSR, CZ model [?,?] QCDSR [?] QCDSR 0.28 ± ± 0.05 [?] QCDSR, NLC 0.19 ± ± 0.04 [?,?,?] F πγγ, LCSR 0.19 ± ± 0.03 (µ = 2.4) [?] F πγγ, LCSR (µ = 2.4) [?] F πγγ, LCSR, R [?] F πγγ, LCSR, R [?] Fπ em,lcsr 0.24 ± 0.14 ± ± 0.09 ± 0.05 [?,?] Fπ em,lcsr, R 0.20 ± ± 0.02 [?] F B πlν, LCSR 0.19 ± ± 0.13 [?] F B πlν, LCSR [?] LQCD, N f = 2, CW ± ± QCDSF/UKQCD [?] LQCD, N f = 2+1, DWF ± ± RBS/UKQCD [?] Table: The Gegenbauer moment a 2 (µ 2 ). The CZ model involves a 2 = 2/3 at the low scale µ = 500 MeV; for the discussion of the extrapolation to higher scales, see Ref. [?]. The abbreviations stand for: QCDSR: QCD sum rules; NLC: non-local condensates; LCSR: light-cone sum rules; R: renormalon model for twist-4 corrections; LQCD: lattice calculation; CW: non-perturbatively O(a) improved Clover Wilson fermions; DWF: domain wall fermions. V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

24 Anatomy of the BMS model QCD sum rules with nonlocal condensates: φ π(x) C 1 6x x {1 + αs [5 π2 3π 3 + x ln2 x Partial resummation of higher-order terms in the OPE Bakulev, Mikhailov, Stefanis, 04 Mikhailov, Radyushkin, ]} + C 2 qq 2{ [11δ(x) + 2δ (x)] + (x x) } x Average virtuality of vacuum quarks λ 2 q = qd 2 q / qq 0.4 GeV 2 = λ 2 q/(2m 2 ) 0.2 Strong overlap with C 3/2 4 (2x 1) a 4 = 0.1, a 6,8,... 0 V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

25 Anatomy of the BMS model (2) Problems: Approximation theoretically inconsistent. Example: Chernyak hep-ph/ Model not tested in other applications and is known to fail in a few cases: parton (quark) distributions VB, Gornicki, Mankiewicz, PRD51 (1995) 6036 B ρl ν l form factors Ali, VB, Simma, ZPC63 (1994) 437 crucial point: The nonperturbative scale does not appear: The series in δ-functions and their derivatives gets smeared over the whole interval 0 < x < 1 Example: Photon wave function Ball, VB, Kivel, NPB649 (2003) 263 V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

26 How good is the Gegenbauer expansion? It has been argued: BABAR data indicate an unusual pion DA φ π(x) = 1 that does not vanish at the end points (Radyushkin) hence Gegenbauer expansion does not converge and cannot be applied (Polyakov) Is this true? φ flat π (x) = 1 = 6x(1 x) k=0,2,... ak flat C 3/2 k (2x 1), ak flat 2(2k + 3) = 3(k + 1)(k + 2) consider approximations to the flat DA as Gegenbauer expansion truncated at order n. n φπ flat,(n) (x) = 6x(1 x) ak flat C 3/2 k (2x 1) Question: k=0,2,... what happens with the predictions of the MR model if the DAs φπ flat,(n) (x) are used as an input? V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

27 Answer: BaBar flat n=4 n= n=2 n= Q x alternatively, check how much is contributed by each successive Gegenbauer polynomial: F MR flatda (Q2 = 20) = = n = 0 n = 2 n = 4 n = 6 The moral is The Gegenbauer expansion for the form factor calculated with flat DA converges very fast At Q 2 < GeV 2 using n = 4 truncation is sufficient End-point behavior of a true pion DA is irrelevant Exact analogy: partial wave expansion V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28

28 State-of-the-art calculations in k factorization k factorization fit ) [GeV] 2 (Q F πγ 2 Q Q [GeV ] Asymptotic DA, LO Asymptotic DA, NLO Flat DA, LO Flat DA, NLO Q 2 F(Q 2 ) [GeV] BaBar CLEO Q 2 [GeV 2 ] Li, Mishima, arxiv: P. Kroll, arxiv: flat: a 2 = 0.39, a 4 = 0.24 fit: a 2 = 0.25, a 4 = 0.07 V. M. Braun (Regensburg) The γ γ π 0 form factor in QCD Minneapolis, May / 28