1. Abelian Anomaly & Neutral Pion Production Craig Roberts Physics Division

2. Why γ * γ → π 0 ?  The process γ* γ → π 0 is fascinating –To explain this transition form factor within the standard model on the full domain of momentum transfer, one must combine an explanation of the essentially nonperturbative Abelian anomaly with the features of perturbative QCD. –Using a single internally-consistent framework!  The case for attempting this has received a significant boost with the publication of data from the BaBar Collaboration (Phys.Rev. D80 (2009) 052002) because:Phys.Rev. D80 (2009) 052002 –They agree with earlier experiments on their common domain of squared- momentum transfer (CELLO: Z.Phys. C49 (1991) 401-410; CLEO: Phys.Rev. D57 (1998) 33-54)CELLO: Z.Phys. C49 (1991) 401-410CLEO: Phys.Rev. D57 (1998) 33-54) –But the BaBar data are unexpectedly far above the prediction of perturbative QCD at larger values of Q 2. Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 2

3. Why γ * γ → π 0 ?  The process γ* γ → π 0 is fascinating –To explain this transition form factor within the standard model on the full domain of momentum transfer, one must combine an explanation of the essentially nonperturbative Abelian anomaly with the features of perturbative QCD. –Using a single internally-consistent framework!  The case for attempting this has received a significant boost with the publication of data from the BaBar Collaboration (Phys.Rev. D80 (2009) 052002) because:Phys.Rev. D80 (2009) 052002 –They agree with earlier experiments on their common domain of squared- momentum transfer (CELLO: Z.Phys. C49 (1991) 401-410; CLEO: Phys.Rev. D57 (1998) 33-54)CELLO: Z.Phys. C49 (1991) 401-410CLEO: Phys.Rev. D57 (1998) 33-54) –But the BaBar data are unexpectedly far above the prediction of perturbative QCD at larger values of Q 2. Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 3 pQCD

4. Transition Form Factor γ * (k 1 )γ(k 2 ) → π 0 Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 4 S(l 1 ) – dressed-quark propagator Γ π (l 1,l 2 ) – pion Bethe-Salpeter amplitude γ ν * (k 1 ) γ μ (k 1 ) Γ ν (l 12,l 2 ) – dressed quark-photon vertex S(l 2 ) S(l 12 ) Γ μ (l 1,l 12 ) π0π0 All computable quantities UV behaviour fixed by pQCD IR Behaviour informed by DSE- and lattice-QCD

5. S(p) … Dressed-quark propagator - nominally, a 1-body problem  Gap equation  D μν (k) – dressed-gluon propagator  Γ ν (q,p) – dressed-quark-gluon vertex Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 5

6. S(p) :Dressed-quark propagator - nominally, a 1-body problem  D μν (k) – dressed-gluon propagator ~ 1/(k 2 + m(k 2 ) 2 )  Γ ν (q,p) – dressed-quark-gluon vertex ~ numerous tensor structures Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 6 DSE- and Lattice-QCD results

7. Frontiers of Nuclear Science: Theoretical Advances Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 7 In QCD a quark's mass depends on its momentum. The function describing this can be calculated and is depicted here. Numerical simulations of lattice QCD (data, at two different bare masses) have confirmed model predictions (solid curves) that the vast bulk of the constituent mass of a light quark comes from a cloud of gluons that are dragged along by the quark as it propagates. In this way, a quark that appears to be absolutely massless at high energies (m =0, red curve) acquires a large constituent mass at low energies.

8.  Dynamical Chiral Symmetry Breaking = Mass from Nothing Critical for understanding pion Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 8 In QCD a quark's mass depends on its momentum. The function describing this can be calculated and is depicted here. Numerical simulations of lattice QCD (data, at two different bare masses) have confirmed model predictions (solid curves) that the vast bulk of the constituent mass of a light quark comes from a cloud of gluons that are dragged along by the quark as it propagates. In this way, a quark that appears to be absolutely massless at high energies (m =0, red curve) acquires a large constituent mass at low energies. Jlab 12GeV: Scanned by 2<Q 2 <9 GeV 2 elastic & transition form factors. Frontiers of Nuclear Science: Theoretical Advances

9. Transition Form Factor γ * (k 1 )γ(k 2 ) → π 0 Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 9 S(l 1 ) – dressed-quark propagator Γ π (l 1,l 2 ) – pion Bethe-Salpeter amplitude γ ν * (k 1 ) γ μ (k 1 ) Γ ν (l 12,l 2 ) – dressed quark-photon vertex S(l 2 ) S(l 12 ) Γ μ (l 1,l 12 ) π0π0 All computable quantities UV behaviour fixed by pQCD IR Behaviour informed by DSE- and lattice-QCD

10. π 0 : Goldstone Mode & bound-state of strongly-dressed quarks  Pion’s Bethe-Salpeter amplitude  Dressed-quark propagator  Axial-vector Ward-Takahashi identity entails Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 10 Maris, Roberts and Tandy nucl-th/9707003 Exact in Chiral QCD Critically! Pseudovector components are necessarily nonzero. Cannot be ignored! Goldstones’ theorem: Solution of one-body problem solves the two-body problem

11. Transition Form Factor γ * (k 1 )γ(k 2 ) → π 0 Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 11 S(l 1 ) – dressed-quark propagator Γ π (l 1,l 2 ) – pion Bethe-Salpeter amplitude γ ν * (k 1 ) γ μ (k 1 ) Γ ν (l 12,l 2 ) – dressed quark-photon vertex S(l 2 ) S(l 12 ) Γ μ (l 1,l 12 ) π0π0 All computable quantities UV behaviour fixed by pQCD IR Behaviour informed by DSE- and lattice-QCD

12. Dressed-quark-photon vertex  Linear integral equation –Eight independent amplitudes  Readily solved  Leading amplitude Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 12 ρ-meson pole generated dynamically - Foundation for VMD Asymptotic freedom Dressed-vertex → bare at large spacelike Q 2 Ward-Takahashi identity

13. Transition Form Factor γ * (k 1 )γ(k 2 ) → π 0 Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 13 γ ν * (k 1 ) γ μ (k 2 ) π0π0 S(l 1 ) S(l 12 ) S(l 2 ) Calculation now straightforward However, before proceeding, consider slight modification

14. Transition Form Factor γ * (k 1 )γ * (k 2 ) → π 0 Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 14 γ ν * (k 1 ) γ μ * (k 2 ) π0π0 S(l 1 ) S(l 12 ) S(l 2 ) Only changes cf. γ * (k 1 )γ(k 2 ) → π 0 Calculation now straightforward However, before proceeding, consider slight modification

15.  Anomalous Ward-Takahashi Identity chiral-limit: G(0,0,0) = ½  Inviolable prediction –No computation believable if it fails this test –No computation believable if it doesn’t confront this test.  DSE prediction, model-independent: Q 2 =0, G(0,0,0)=1/2 Corrections from m π 2 ≠ 0, just 0.4% Transition Form Factor γ * (k 1 )γ * (k 2 ) → π 0 Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 15 Maris & Tandy, Phys.Rev. C65 (2002) 045211 Maris & Roberts, Phys.Rev. C58 (1998) 3659

16. Transition Form Factor γ * (k 1 )γ * (k 2 ) → π 0 Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 16  pQCD prediction Obtained if, and only if, asymptotically, Γ π (k 2 ) ~ 1/k 2  Moreover, absolutely no sensitivity to φ π (x); viz., pion distribution amplitude  Q 2 =1GeV 2 : VMD broken  Q 2 =10GeV 2 : G DSE (Q 2 )/G pQCD (Q 2 )=0.8  pQCD approached from below Maris & Tandy, Phys.Rev. C65 (2002) 045211 pQCD

17. Pion Form Factor F π (Q 2 ) Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 17 γ(Q) π(P) π(P+Q) Maris & Tandy, Phys.Rev. C62 (2000) 055204  DSE computation appeared before data; viz., a prediction  pQCD-scale Q 2 F π (Q 2 ) → 16πα(Q 2 )f π 2  VMD-scale: m ρ 2  Q 2 =10GeV 2 pQCD-scale/VMD-scale = 0.08 Internally consistent calculation CAN & DOES overshoot pQCD limit, and approach it from above; viz, at ≈ 12 GeV 2

18. Single-parameter, Internally-consistent Framework  Dyson-Schwinger Equations – applied extensively to spectrum & interactions of mesons with masses less than 1 GeV; & nucleon & Δ.  On this domain the rainbow-ladder approximation – leading-order in systematic, symmetry-preserving truncation scheme, nucl-th/9602012 – is accurate, well-understood tool: e.g.,nucl-th/9602012  Prediction of elastic pion and kaon form factors: nucl-th/0005015nucl-th/0005015  Pion and kaon valence-quark distribution functions: 1102.2448 [nucl-th]1102.2448 [nucl-th]  Unification of these and other observables – ππ scattering: hep-ph/0112015hep-ph/0112015  Nucleon form factors: arXiv:0810.1222 [nucl-th]arXiv:0810.1222 [nucl-th]  Readily extended to explain properties of the light neutral pseudoscalar mesons (η cf. ή): 0708.1118 [nucl-th]0708.1118 [nucl-th]  One parameter: gluon mass-scale = m G = 0.8 GeV Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 18

19. Transition Form Factor γ * (k 1 )γ(k 2 ) → π 0 Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 19 γ ν * (k 1 ) γ μ (k 2 ) π0π0 S(l 1 ) S(l 12 ) S(l 2 ) Maris & Tandy, Phys.Rev. C65 (2002) 045211  DSE result  no parameters varied;  exhibits ρ-pole;  perfect agreement with CELLO & CLEO

20. Transition Form Factor γ * (k 1 )γ(k 2 ) → π 0  Three, internally-consistent calculations –Maris & Tandy Dash-dot: γ * (k 1 )γ(k 2 ) → π 0 Dashed: γ * (k 1 )γ * (k 2 ) → π 0 –H.L.L Roberts et al. Solid: γ * (k 1 )γ(k 2 ) → π 0 contact-interaction, omitting pion’s pseudovector component  All approach UV limit from below Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 20 Hallmark of internally-consistent computations γ ν * (k 1 ) S(l 12 ) γ μ (k 2 ) S(l 2 ) S(l 1 ) π0π0 H.L.L. Roberts et al., Phys.Rev. C82 (2010) 065202

21. Transition Form Factor γ * (k 1 )γ(k 2 ) → π 0  All approach UV limit from below  UV scale in this case is 10-times larger than for F π (Q 2 ): –8 π 2 f π 2 = ( 0.82 GeV ) 2 –cf. m ρ 2 = ( 0.78 GeV ) 2  Hence, internally-consistent computations can and do approach the UV-limit from below. Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 21 Hallmark of internally-consistent computations γ ν * (k 1 ) S(l 12 ) γ μ (k 2 ) S(l 2 ) S(l 1 ) π0π0 H.L.L. Roberts et al., Phys.Rev. C82 (2010) 065202

22. Transition Form Factor γ * (k 1 )γ(k 2 ) → π 0  UV-behaviour: light-cone OPE  Integrand sensitive to endpoint: x=1 –Perhaps φ π (x) ≠ 6x(1-x) ? –Instead, φ π (x) ≈ constant? Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 22 γ ν * (k 1 ) S(l 12 ) γ μ (k 2 ) S(l 2 ) S(l 1 ) π0π0 H.L.L. Roberts et al., Phys.Rev. C82 (2010) 065202  There is one-to-one correspondence between behaviour of φ π (x) and short- range interaction between quarks  φ π (x) = constant is achieved if, and only if, the interaction between quarks is momentum-independent; namely, of the Nambu – Jona- Lasinio form

23. Pion’s GT relation Contact interaction Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 23 Guttiérez, Bashir, Cloët, Roberts arXiv:1002.1968 [nucl-th]  Pion’s Bethe-Salpeter amplitude  Dressed-quark propagator  Bethe-Salpeter amplitude can’t depend on relative momentum; propagator can’t be momentum-dependent  Solved gap and Bethe-Salpeter equations P 2 =0: M Q =0.4GeV, E π =0.098, F π =0.5M Q 1 M Q Nonzero and significant Remains!

24. Transition Form Factor γ * (k 1 )γ(k 2 ) → π 0  Comparison between Internally-consistent calculations:  φ π (x) ≈ constant, in conflict with large-Q 2 data here, as it is in all cases –Contact interaction cannot describe scattering of quarks at large-Q 2  φ π (x) = 6x(1-x) yields pQCD limit, approaches from below Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 24 γ ν * (k 1 ) S(l 12 ) γ μ (k 2 ) S(l 2 ) S(l 1 ) π0π0 H.L.L. Roberts et al., Phys.Rev. C82 (2010) 065202

25. Transition Form Factor γ * (k 1 )γ(k 2 ) → π 0  2σ shift of any one of the last three high-points –one has quite a different picture  η production  η' production  Both η & η’ production in perfect agreement with pQCD Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 25 γ ν * (k 1 ) S(l 12 ) γ μ (k 2 ) S(l 2 ) S(l 1 ) π0π0 H.L.L. Roberts et al., Phys.Rev. C82 (2010) 065202 CLEO BaBar

26. Epilogue  In fully-self-consistent treatments of pion: static properties; and elastic and transition form factors, the asymptotic limit of the product Q 2 G(Q 2 ) which is determined a priori by the interaction employed, is not exceeded at any finite value of spacelike momentum transfer: –The product is a monotonically-increasing concave function.  A consistent approach is one in which: –a given quark-quark scattering kernel is specified and solved in a well-defined, symmetry-preserving truncation scheme; –the interaction’s parameter(s) are fixed by requiring a uniformly good description of the pion’s static properties; –and relationships between computed quantities are faithfully maintained. Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 26

27. Epilogue  The large-Q 2 BaBar data is inconsistent with –pQCD –All extant, fully-self-consistent studies  Conclusion: the large-Q 2 data reported by BaBar is not a true representation of the γ ∗ γ → π 0 transition form factor  Explanation? –Possible erroneous way to extract pion transition form factor from the data is problem of π 0 π 0 subtraction. –This channel – γ ∗ γ → π 0 π 0 scales in the same way (Diehl et al., Phys.Rev. D62 (2000) 073014)Diehl et al., Phys.Rev. D62 (2000) 073014 Misinterpretation of some events, where 2 nd π 0 is not seen, may be larger at large-Q 2. Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 27 π 0 (p) π 0 (p’)