Andreas Nyffeler. Chiral Dynamics 2012, August 6-10, 2012 Jefferson Laboratory, Newport News, VA, USA

Transcription

1 Hadronic light-by-light scattering in the muon g : impact of proposed measurements of the π 0 γγ decay width and the γ γ π 0 transition form factor with the KLOE- experiment Andreas Nyffeler Regional Centre for Accelerator-based Particle Physics (RECAPP) Harish-Chandra Research Institute, Allahabad, India nyffeler@hri.res.in Chiral Dynamics 0, August 6-0, 0 Jefferson Laboratory, Newport News, VA, USA

2 Outline Monte-Carlo simulation of planned KLOE- measurements Hadronic light-by-light scattering in the muon g Pion-exchange versus pion-pole: off-shell versus on-shell form factors Impact of KLOE- measurements Conclusions

3 Monte-Carlo simulation of planned KLOE- measurements On the possibility to measure the π 0 γγ decay width and the γ γ π 0 transition form factor with the KLOE- experiment D. Babusci, H. Czyż, F. Gonnella, S. Ivashyn, M. Mascolo, R. Messi, D. Moricciani, A. Nyffeler, G. Venanzoni and the KLOE- Collaboration Eur. Phys. J. C7, 97 (0) [arxiv:09.46 [hep-ph]] Within year of data taking, collecting 5 fb, KLOE- will be able to measure: Γ π 0 γγ to % statistical precision. γ γ π 0 transition form factor F (Q ) in the region of very low, space-like momenta 0.0 GeV Q 0. GeV with a statistical precision of less than 6% in each bin. KLOE- can (almost) directly measure slope of form factor at origin. ) F(Q γ*γ π Q [GeV] Simulation of KLOE- measurement of F (Q ) (red triangles). MC program EKHARA.0 (Czyż, Ivashyn ) and detailed detector simulation. Solid line: F (0) given by chiral anomaly (WZW). Dashed line: form factor according to on-shell LMD+V model (Knecht, Nyffeler, EPJC 0). CELLO (black crosses) and CLEO (blue stars) data at higher Q.

4 The lepton taggers Important for γγ physics: installment of taggers for the leptons which leave interaction region inside KLOE detector at small polar angles θ < θ max : (Graphics courtesy of Matteo Mascolo) LET = low-energy taggers, HET = high-energy taggers From this one can infer virtualities of photons emitted from leptons in process e + e e + e γ γ e + e π 0 (double tag experiment).

5 Current status of the muon g Discrepancy: a exp µ a SM µ (50 300) 0, corresponding to σ (Jegerlehner + Nyffeler 09; Davier et al. 0; Jegerlehner + Szafron ; Hagiwara et al. ; Aoyama et al. ) Largest source of error in SM prediction: hadronic contributions Different types: (a) (b) (c) γ Z Light quark loop not well defined Hadronic blob (a) Hadronic vacuum polarization O(α ), O(α 3 ) (b) Hadronic light-by-light (LbyL) scattering O(α 3 ) (c) -loop electroweak contributions O(αG F m µ) Had. VP: can be related via dispersion relation to σ(e + e hadrons) can be improved systematically with more precise data. Had. LbyL: not directly related to experimental data need hadronic (resonance) model (or lattice QCD). Constrain models using experimental data (form factors of hadrons with photons) or theory (short-distance constraints, matching with pqcd).

6 Hadronic light-by-light scattering: Summary of selected results µ (p ) k k = p p µ (p) = ρ π + π 0,, η, η Exchange of other resonances + (f 0, a,...) Chiral counting: p 4 p 6 p 8 p 8 N C -counting: N C N C N C Relevant scales in VVVV (off-shell!): 0 GeV, i.e. much larger than m µ! + ρ Q

7 Hadronic light-by-light scattering: Summary of selected results µ (p ) k k = p p µ (p) = ρ π + π 0,, η, η Exchange of other resonances + (f 0, a,...) Chiral counting: p 4 p 6 p 8 p 8 N C -counting: N C N C N C Relevant scales in VVVV (off-shell!): 0 GeV, i.e. much larger than m µ! Contribution to a µ 0 : BPP: +83 (3) HKS: +90 (5) KN: +80 (40) MV: +36 (5) 007: +0 (40) PdRV:+05 (6) N,JN: +6 (40) GFW: +7 (9) GdR: +50 (3) -9 (3) -5 (8) 0 (0) ud.: (9) -9 (3) +85 (3) +83 (6) +83 () +4 (0) +4 (3) +99 (6) +8 () +68 (3) ud.: + ud. = undressed, i.e. point vertices without form factors -4 (3) [f 0, a ] +.7 (.7) [a ] + (5) [a ] +8 () [f 0, a ] +5 (7) [f 0, a ] + ρ Q + (3) +0 () [c-quark] + (3) +36 (59) (*) +8 (6) ud.: +60 BPP = Bijnens, Pallante, Prades 96, 0; HKS = Hayakawa, Kinoshita, Sanda 96, 98, 0; KN = Knecht, Nyffeler 0; MV = Melnikov, Vainshtein 04; 007 = Bijnens, Prades; Miller, de Rafael, Roberts; PdRV = Prades, de Rafael, Vainshtein 09; N,JN = Nyffeler 09; Jegerlehner, Nyffeler 09; GFW = Goecke, Fischer, Williams, (*) Value corrected to 64 (3) (total = 45(3)) (preliminary; error only from numerics) by Fischer, Cracow, May 0; GdR = Greynat, de Rafael (error only reflects variation M Q = 40 ± 0 MeV, 0%-30% systematic error) Recall (in units of 0 ): δa µ(had. VP) 45; δa µ(exp [BNL]) = 63; δa µ(future exp) = 5

8 Pion-pole in VVVV versus pion-exchange in had. LbyL in a µ To uniquely identify contribution of exchanged neutral pion π 0 in Green s function VVVV, we need to pick out pion-pole: q q π crossed diagrams q q 3 lim (q +q ) m π ((q + q ) m π ) VVVV Residue of pole: on-shell vertex function 0 VV π on-shell form factor F π 0 γ γ (q, q )

9 Pion-pole in VVVV versus pion-exchange in had. LbyL in a µ To uniquely identify contribution of exchanged neutral pion π 0 in Green s function VVVV, we need to pick out pion-pole: q q π crossed diagrams q q 3 lim ((q + q ) m (q +q ) mπ π ) VVVV Residue of pole: on-shell vertex function 0 VV π on-shell form factor F π 0 γ γ (q, q ) But in contribution to the muon g, we have to evaluate Feynman diagrams, integrating over the photon momenta with exchanged off-shell pions. For all pseudoscalars: π 0, η, η,... Shaded blobs represent off-shell form factor F PS γ γ where PS = π0, η, η, π 0,... Off-shell form factors are either inserted by hand starting from constant, pointlike Wess-Zumino-Witten (WZW) form factor or using e.g. some resonance Lagrangian. Similar statements apply for exchanges (or loops) of other resonances.

10 Off-shell pion form factor F π 0 γ γ from VVP Following Bijnens, Pallante, Prades 95, 96; Hayakawa, Kinoshita, Sanda 95, 96; Hayakawa, Kinoshita 98, we can define off-shell form factor for π 0 : Z d 4 x d 4 y e i(q x+q y) 0 T {j µ(x)j ν(y)p 3 (0)} 0 = ε µναβ q α qβ i ψψ F π i (q + q ) mπ F π 0 γ γ ((q + q ), q, q ) +... Up to small mixing effects of P 3 with η and η and neglecting exchanges of heavier states like π 0, π 0,... j µ = light quark part of the electromagnetic current: j µ(x) = (ψ ˆQγ µψ)(x) 0 u d s A, ˆQ = diag(,, )/3 P 3 λ = ψiγ 3 5 uiγ ψ = 5 u diγ 5 d /, ψψ = single flavor quark condensate Bose symmetry: F π 0 γ γ ((q + q ), q, q ) = F π 0 γ γ ((q + q ), q, q ) Note: for off-shell pions, instead of P 3 (x), we could use any other suitable interpolating field, like ( µ A 3 µ )(x) or even an elementary pion field π3 (x)!

11 On-shell form factor F π 0 γ γ and transition form factor F (Q ) On-shell π 0 γ γ form factor between an on-shell pion and two off-shell photons: Z i d 4 x e iq x 0 T {j µ(x)j ν(0)} π 0 (q + q ) = ε µνρσq ρ qσ F π 0 γ γ (q, q) Relation to off-shell form factor: F π 0 γ γ (q, q ) F π 0 γ γ (m π, q, q ) Form factor for real photons is related to π 0 γγ decay width: F π 0 γ γ (q = 0, q 4 = 0) = πα mπ 3 Often normalization with chiral anomaly is used: Pion-photon transition form factor: F π 0 γ γ (0, 0) = 4π F π Γ π 0 γγ F (Q ) F π 0 γ γ ( Q, q = 0), Q q Note that q = 0, but q 0 for on-shell photon!

12 Pion-exchange versus pion-pole contribution to a LbyL;π0 µ Off-shell form factors have been used to evaluate the pion-exchange contribution in Bijnens et al 96, Hayakawa et al 96, 98. Rediscovered by Jegerlehner in 07, 08. Consider diagram: q q + q π 0 q q = 0 4 q = q + q 3 F π 0 γ γ ((q + q ), q, q ) F π 0 γ γ ((q + q ), (q + q ), 0)

13 Pion-exchange versus pion-pole contribution to a LbyL;π0 µ Off-shell form factors have been used to evaluate the pion-exchange contribution in Bijnens et al 96, Hayakawa et al 96, 98. Rediscovered by Jegerlehner in 07, 08. Consider diagram: q q + q π 0 q q = 0 4 q = q + q 3 F π 0 γ γ ((q + q ), q, q ) F π 0 γ γ ((q + q ), (q + q ), 0) On the other hand, Knecht, Nyffeler 0 used on-shell form factors: F π 0 γ γ (m π, q, q ) F π 0 γ γ (m π, (q + q ), 0) But form factor at external vertex F π 0 γ γ (m π, (q + q ), 0) for (q + q ) m π violates momentum conservation, since momentum of external soft photon vanishes! Often the following misleading notation was used F π 0 γ γ ((q + q ), 0) F π 0 γ γ (m π, (q + q ), 0) At external vertex identification with transition form factor was made (wrongly!).

14 Pion-exchange versus pion-pole contribution to a LbyL;π0 µ Off-shell form factors have been used to evaluate the pion-exchange contribution in Bijnens et al 96, Hayakawa et al 96, 98. Rediscovered by Jegerlehner in 07, 08. Consider diagram: q q + q π 0 q q = 0 4 q = q + q 3 F π 0 γ γ ((q + q ), q, q ) F π 0 γ γ ((q + q ), (q + q ), 0) On the other hand, Knecht, Nyffeler 0 used on-shell form factors: F π 0 γ γ (m π, q, q ) F π 0 γ γ (m π, (q + q ), 0) But form factor at external vertex F π 0 γ γ (m π, (q + q ), 0) for (q + q ) m π violates momentum conservation, since momentum of external soft photon vanishes! Often the following misleading notation was used F π 0 γ γ ((q + q ), 0) F π 0 γ γ (m π, (q + q ), 0) At external vertex identification with transition form factor was made (wrongly!). Melnikov, Vainshtein 04 had already observed this inconsistency and proposed to use F π 0 γ γ (m π, q, q ) F π 0 γ γ (m π, m π, 0) i.e. a constant form factor at the external vertex given by the WZW term.

15 Pion-exchange versus pion-pole contribution to a LbyL;π0 µ (continued) However, this prescription will only yield the so-called pion-pole contribution and not the full pion-exchange contribution! In general, any evaluation e.g. using some resonance Lagrangian, will lead to off-shell form factors at both the vertices in the Feynman integral. Strictly speaking, the identification of the pion-exchange contribution is only possible, if the pion is on-shell. In the numerical results later, we will denote by (JN): pion-exchange contribution with off-shell pion form factors F π 0 γ γ at both vertices. (MV): pion-pole contribution with on-shell pion form factor F π 0 γ γ one vertex and constant form factor (WZW) at external vertex. at

16 KLOE- impact on a LbyL;π0 µ Value of a LbyL;π0 µ is currently obtained using various hadronic models. Any experimental information on the relevant form factors can therefore help to check the consistency of models and reduce the error. As stressed before, what enters in a LbyL;π0 µ is the fully off-shell form factor F π 0 γ γ ((q + q), q, q ) (vertex function). A measurement of the transition form factor F π 0 γ γ (m π, q, 0) can, in general, only be sensitive to a subset of the model parameters and, in general, does not allow to reconstruct the full off-shell form factor. Good description for transition form factor is only necessary, not sufficient, in order to uniquely determine a LbyL;π0 µ. From one model to another, uncertainty of aµ LbyL;π0 related to the off-shell pion can be very different. Complete error on aµ LbyL;π0 should take into account model dependence.

17 KLOE- impact on a LbyL;π0 µ (continued) For illustration, but not to present some new realistic estimate, we will study the impact of the KLOE- measurements on two models: VMD (off-shell): has only two parameters. Other models with very few parameters are constituent quark models or holographic models (AdS/QCD). LMD+V (off-shell) (Knecht, Nyffeler, EPJC 0): has many poorly constrained parameters. Including the uncertainties related to the off-shellness of the pion, which dominate the final error, one obtains the estimate: a LbyL;π0 µ;lmd+v = (7 ± ) 0 (Nyffeler 09; Jegerlehner, Nyffeler 09).

18 The VMD form factor Vector Meson Dominance: F VMD π 0 γ γ ((q + q), q, q ) = N C M V M V π F π q M V q M V on-shell = off-shell form factor! Only two model parameters even for off-shell form factor: F π and M V Transition form factor: F VMD (Q ) = N C MV π F π Q + MV

19 The LMD+V form factor (off-shell) Knecht, Nyffeler, EPJC 0; Nyffeler 09 Ansatz for VVP and thus F π 0 γ γ in large-n C QCD in chiral limit with multiplet of lightest pseudoscalars (Goldstone bosons) and multiplets of vector resonances, ρ, ρ (lowest meson dominance (LMD) + V) F π 0 γ γ fulfills all leading (and some subleading) QCD short-distance constraint from Operator Product Expansion (OPE) Reproduces Brodsky-Lepage (BL): lim F π Q 0 γ γ (m π, Q, 0) /Q (OPE and BL cannot be fulfilled simultaneously with only one vector resonance) Normalized to decay width Γ π 0 γγ Off-shell LMD+V form factor: F LMD+V π 0 γ γ (q 3, q, q ) = Fπ 3 q q (q + q + q 3) + P V H (q, q, q 3) (q M V ) (q M V ) (q M V ) (q M V ) P V H (q, q, q 3) = h (q + q ) + h q q + h 3 (q + q ) q 3 + h 4 q 4 3 +h 5 (q + q ) + h 6 q 3 + h 7 q 3 = (q + q ) F π = 9.4 MeV, M V = M ρ = MeV, M V = M ρ =.465 GeV Free parameters: h i

20 The LMD+V form factor (on-shell) On-shell LMD+V form factor: F LMD+V π 0 γ γ (q, q ) = Fπ 3 q q (q + q ) + h (q + q ) + h q q + h 5 (q + q ) + h 7 (q M V ) (q M V ) (q M V ) (q M V ) h = h + m π h 5 = h 5 + h 3m π h 7 = h 7 + h 6m π + h 4m 4 π Transition form factor: F LMD+V (Q ) = Fπ 3 M V M V h Q 4 h 5Q + h 7 (Q + M V )(Q + M V ) h = 0 in order to reproduce Brodsky-Lepage behavior. Can treat h as free parameter to fit the BABAR data, but the form factor does then not vanish for Q, if h 0.

21 Form factor F (Q ): data sets and normalization Data sets used for fits: A0 : A : A : B0 : B : B : CELLO, CLEO, PDG CELLO, CLEO, PrimEx CELLO, CLEO, PrimEx, KLOE- CELLO, CLEO, BABAR, PDG CELLO, CLEO, BABAR, PrimEx CELLO, CLEO, BABAR, PrimEx, KLOE- Normalization for F (0): Γ PDG π 0 γγ = 7.74 ± 0.48 ev (6.% precision) for PDG 00 value Γ PrimEx π 0 γγ = 7.8 ± 0. ev (.8% precision) from PrimEx experiment Γ KLOE π 0 γγ = 7.73 ± 0.08 ev (% precision) for the KLOE- simulation As noted in Nyffeler, PoS 09, the uncertainty in the normalization of the form factor was not taken into account in most evaluations of a LbyL;π0 µ. In most papers, simply F π = 9.4 MeV is used without any error attached to it. Value close to F π = (9.0 ± 0.4) MeV obtained from π + µ + ν µ(γ).

22 Fitting the models Model Data χ /d.o.f. Parameters VMD A0 6.6/9 M V = 0.778(8) GeV F π = 0.094(8) GeV VMD A 6.6/9 M V = 0.776(3) GeV F π = 0.099(3) GeV VMD A 7.5/7 M V = 0.778() GeV F π = 0.093(4) GeV VMD B0 77/36 M V = 0.89(6) GeV F π = (9) GeV VMD B 78/36 M V = 0.83(8) GeV F π = 0.095(3) GeV VMD B 79/44 M V = 0.83(5) GeV F π = 0.095(4) GeV LMD+V, h = 0 A0 6.5/9 h5 = 6.99(3) GeV 4 h7 = 4.8(45) GeV 6 LMD+V, h = 0 A 6.6/9 h 5 = 6.96(9) GeV 4 h 7 = 4.90() GeV 6 LMD+V, h = 0 A 7.5/7 h 5 = 6.99(8) GeV 4 h 7 = 4.83(7) GeV 6 LMD+V, h = 0 B0 65/36 h 5 = 7.94(3) GeV 4 h 7 = 3.95(4) GeV 6 LMD+V, h = 0 B 69/36 h 5 = 7.8() GeV 4 h 7 = 4.70(0) GeV 6 LMD+V, h = 0 B 70/44 h5 = 7.79(0) GeV 4 h7 = 4.8(7) GeV 6 LMD+V, h 0 A0 6.5/8 h5 = 6.90(7) GeV 4 h7 = 4.83(46) GeV 6 h = 0.03(8) GeV LMD+V, h 0 A 6.5/8 h 5 = 6.85(67) GeV 4 h 7 = 4.9() GeV 6 h = 0.03(7) GeV LMD+V, h 0 A 7.5/6 h5 = 6.90(64) GeV 4 h7 = 4.84(7) GeV 6 h = 0.0(7) GeV LMD+V, h 0 B0 8/35 h 5 = 6.46(4) GeV 4 h 7 = 4.86(44) GeV 6 h = 0.7() GeV LMD+V, h 0 B 8/35 h5 = 6.44() GeV 4 h7 = 4.9() GeV 6 h = 0.7() GeV LMD+V, h 0 B 9/43 h 5 = 6.47() GeV 4 h 7 = 4.84(7) GeV 6 h = 0.7() GeV Main improvement in normalization parameter, F π for VMD and h 7 for LMD+V. But more data also better determine the other parameters M V or h 5.

23 Results for a LbyL;π0 µ Model Data aµ LbyL;π0 0 VMD A0 (57. ± 4.0) JN VMD A (57.7 ±.) JN VMD A (57.3 ±.) JN LMD+V, h = 0 A0 (7.3 ± 3.5) JN (79.8 ± 4.) MV LMD+V, h = 0 A (73.0 ±.7) JN (80.5 ±.0) MV LMD+V, h = 0 A (7.5 ± 0.8) JN (80.0 ± 0.8) MV LMD+V, h 0 A0 (7.4 ± 3.8) JN LMD+V, h 0 A (7.9 ±.) JN LMD+V, h 0 A (7.4 ±.5) JN LMD+V, h 0 B0 (7.9 ± 3.4) JN LMD+V, h 0 B (7.4 ±.6) JN LMD+V, h 0 B (7.8 ± 0.7) JN error does not include uncertainty due to off-shellness of pion Error in aµ LbyL;π0 related to model parameters determined by Γ π 0 γγ (normalization; not taken into account before) and F (Q ) is reduced as follows: Sets A0, B0: δaµ LbyL;π0 4 0 Sets A, B: δa LbyL;π0 µ 0 (+ Γ PrimEx π 0 γγ ) Sets A, B: δa LbyL;π0 µ (0.7.) 0 (+ KLOE- data)

24 VMD versus LMD+V with h = 0 Both VMD and LMD+V with h = 0 can fit the data sets A0, A and A very well with essentially the same χ /d.o.f. Nevertheless, the results for a LbyL;π0 µ differ by about 0%: a LbyL;π0 µ;vmd a LbyL;π0 µ;lmd+v (JN) [a LbyL;π0 µ;lmd+v 80 0 (MV)] Due to the different behavior in these models of the fully off-shell form factor F π 0 γ γ ((q + q), q, q ) on all momentum variables. VMD model is known to have a wrong high-energy behavior F π 0 γ γ (m π, Q, Q ) /Q 4 instead of /Q according to the OPE. The small final error of ±. 0 for the VMD model with only two parameters, F π and M V, which are both fixed by the width and form factor measurements, might therefore be very deceptive.

25 Conclusions Planned measurements at KLOE- can help to reduce some of the uncertainty in the (presumably!) numerically dominant pion exchange contribution to had. LbyL scattering. Error in a LbyL;π0 µ related to the model parameters determined by Γ π 0 γγ and F (Q ) will be reduced as follows: δa LbyL;π0 µ 4 0 (with current data for F (Q ) + Γ PDG π 0 γγ ) δa LbyL;π0 µ 0 (+ Γ PrimEx π 0 γγ ) δa LbyL;π0 µ (0.7.) 0 (+ KLOE- data) Note that this error does not account for other potential uncertainties in aµ LbyL;π0, e.g. related to the off-shellness of the pion or the choice of model. Recall (in units of 0 ): a LbyL;π0 µ;lmd+v (N,JN) = 7 ± δa LbyL µ (N,JN) = 40; δa LbyL µ (PdRV) = 6 δa µ(had. VP) 45; δa µ(exp [BNL]) = 63; δa µ(future exp) = 5

26 Backup slides

27 HET + HET coincidence Γ π 0 γγ Entries 4 0 Pion Energy Entries Entries Decay photon θ-angle Entries Energy [GeV] π 0 energy distribution in lab frame with (dark) and without (light-gray) HET-HET coincidence. HET-HET coincidence therefore selects π 0 almost at rest θ [deg] Polar angle distribution of decay photons in lab frame (w.r.t. beam axis) with (dark) and without (light-gray) the HET-HET coincidence. Photons from pion decay at rest emitted at large angle, about 95% above 5 (and below 55 ), resulting in large acceptance for photons reaching Electromagnetic Calorimeter (EMC) of KLOE.

28 Form factor measurement: HET + KLOE F (Q ) Event selection: One lepton inside the KLOE detector: 0 < θ < 60, corresponding to 0.0 GeV < q < 0. GeV The other lepton in the HET detector, corresponding to q 0 4 GeV for most of the events Can measure the differential cross section (dσ/dq ) data, where Q q Detection efficiency is about 0% The form factor F (Q ) can then be evaluated through the relation: F (Q ) F (Q ) MC = ( dσ dq ) data ( dσ dq ) MC ( dσ dq ) MC is the differential cross section obtained from the MC with the form factor F (Q ) MC

29 Slope of the transition form factor at the origin An important quantity is the slope of the form factor at the origin: a mπ d F γ (q, 0) π0γ F π 0 γ γ (0, 0) d q q =0 Within ChPT, the slope is related to low-energy constants of the chiral Lagrangian of order p 6 in the odd intrinsic-parity sector. A precise measurement could help to distinguish between estimates of the low-energy constants, which have been made using various models: e.g. resonance Lagrangians (VMD, LMD, LMD+V), constituent quark models, holographic models (AdS/QCD),... For time-like photon virtualities (q > 0), the slope can be measured directly in the rare decay π 0 e + e γ, but the current experimental uncertainty is big. The PDG quotes since many years a = 0.03 ± This value is essentially the result obtained by the CELLO collaboration for space-like momenta q = Q < 0. CELLO fitted their data, collected for Q 0.5 GeV, with a simple VMD parametrization for the form factor and then used the analytical expression to obtain the slope. The potential model dependence of this extrapolation from rather large values of Q to the origin is not accounted for by the PDG in the central value and the error for the slope parameter. Also contributions from loops in ChPT at order p 6, a loops (µ = M ρ) = (Bijnens et al. 90), are not taken into account.

30 Relevant momentum regions in a LbyL;π0 µ In Knecht, Nyffeler 0, a -dimensional integral representation was derived for a certain class (VMD-like) of on-shell form factors (schematically): Z Z X aµ LbyL;π0 = dq dq w i (Q, Q ) f i (Q, Q ) 0 0 i with universal weight functions w i. Dependence on form factors resides in the f i. Expressions with on-shell form factors are not valid as they stand. One needs to set form factor at external vertex to a constant to obtain pion-pole contribution. Expressions are valid for WZW and off-shell VMD form factors. w f (Q,Q ) w g (M V,Q,Q ) Q [GeV] 0.5 Q [GeV] 0 0 w g (M π,q,q ) 0 0 Q [GeV] Q [GeV] Q [GeV] Q [GeV] w g (M V,Q,Q ) Q [GeV] Q [GeV] w f (Q, Q ) enters for WZW form factor. Tail leads to ln Λ divergence for momentum cutoff Λ. w g (M V, Q, Q ) enters for VMD form factor. Relevant momentum regions are therefore around GeV. As long as form factors in different models lead to damping, we expect comparable results for a LbyL;π0 µ, at the level of 0%. Similarly for η, η. Jegerlehner, Nyffeler 09 derived 3-dimensional integral representation for general form factors. Integration over Q, Q, cos θ, where Q Q = Q Q cos θ.

31 Relevant momentum regions in a LbyL;PS µ Result for pseudoscalar exchange contribution a LbyL;PS µ 0 for off-shell LMD+V and VMD form factors obtained with momentum cutoff Λ in 3-dimensional integral representation of JN 09 (integration over square). In brackets, relative contribution of the total obtained with Λ = 0 GeV. Λ π 0 η η [GeV] LMD+V (h 3 = 0) LMD+V (h 4 = 0) VMD VMD VMD (0.6%) 4.8 (0.3%) 4.4 (5.%).76 (.%) 0.99 (7.9%) (53.8%) 38.8 (53.%) 36.6 (64.%) 6.90 (47.5%) 4.5 (36.%) (7.%) 5. (7.7%) 47.7 (83.8%) 0.7 (73.4%) 7.83 (6.5%) (8.7%) 59. (8.4%) 5.6 (9.3%).6 (86.6%) 9.90 (79.%) (90.%) 65.6 (90.%) 55.8 (97.8%) 4.0 (96.%).7 (93.%) (93.9%) 68.3 (93.8%) 56.5 (99.%) 4.3 (98.6%). (97.4%) (98.8%) 7.9 (98.8%) 56.9 (99.9%) 4.5 (99.9%).5 (99.9%) (00%) 7.8 (00%) 57.0 (00%) 4.5 (00%).5 (00%) π 0 : Although weight functions plotted earlier are not applicable to off-shell LMD+V form factor, region below Λ = GeV gives the bulk of the result: 8% for LMD+V, 9% for VMD. No damping from off-shell LMD+V form factor at external vertex since χ 0 (new short-distance constraint). Note: VMD falls off too fast, compared to OPE. η, η : Mass of intermediate pseudoscalar is higher than pion mass expect a stronger suppression from propagator. Peak of relevant weight functions shifted to higher values of Q i. For η, vector meson mass is also higher M V = 859 MeV. Saturation effect and the suppression from the VMD form factor only fully set in around Λ =.5 GeV: 96% of total for η, 93% for η.