Abelian Anomaly Neutral Pion Production Craig Roberts Physics

Abelian Anomaly & Neutral Pion Production Craig Roberts Physics Division

Why γ* γ → π0 ? Ø The process γ* γ → π0 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! Ø The case for attempting this has received a significant boost with the publication of data from the Ba. Bar Collaboration (Phys. Rev. D 80 (2009) 052002) because: – They agree with earlier experiments on their common domain of squaredmomentum transfer (CELLO: Z. Phys. C 49 (1991) 401 -410; CLEO: Phys. Rev. D 57 (1998) 33 -54) – But the Ba. Bar 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

Why γ* γ → π0 ? Ø The process γ* γ → π0 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! Ø The case for attempting this has received a significant boost with the p. QCD publication of data from the Ba. Bar Collaboration (Phys. Rev. D 80 (2009) 052002) because: – They agree with earlier experiments on their common domain of squaredmomentum transfer (CELLO: Z. Phys. C 49 (1991) 401 -410; CLEO: Phys. Rev. D 57 (1998) 33 -54) – But the Ba. Bar 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

Γν (l 12, l 2) – dressed Transition Form Factor γ*(k 1)γ(k 2) → π0 quark-photon vertex S(l 2) γν*(k 1) Γπ(l 1, l 2) – pion Bethe-Salpeter amplitude S(l 12) Γμ (l 1, l 12) π0 S(l 1) – dressed-quark propagator γμ(k 1) Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production • All computable quantities • UV behaviour fixed by p. QCD • IR Behaviour informed by DSE- and lattice-QCD 4

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

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

Frontiers of Nuclear Science: Theoretical Advances 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. Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 7

Frontiers of Nuclear Science: Theoretical Advances Ø Dynamical Chiral Symmetry Breaking = Mass from Nothing In QCD a quark's mass depends on its Critical for understanding pion 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 2<9 Ge. V 2 Jlab 12 Ge. V: Scanned by 2<Q low energies. Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production elastic & transition form factors. 8

Γν (l 12, l 2) – dressed Transition Form Factor γ*(k 1)γ(k 2) → π0 quark-photon vertex S(l 2) γν*(k 1) Γπ(l 1, l 2) – pion Bethe-Salpeter amplitude S(l 12) Γμ (l 1, l 12) π0 S(l 1) – dressed-quark propagator γμ(k 1) Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production • All computable quantities • UV behaviour fixed by p. QCD • IR Behaviour informed by DSE- and lattice-QCD 9

Maris, Roberts and Tandy nucl-th/9707003 π0: Goldstone Mode & bound-state of strongly-dressed quarks Ø Pion’s Bethe-Salpeter amplitude Critically! Pseudovector components are necessarily nonzero. Cannot be ignored! Ø Dressed-quark propagator Ø Axial-vector Ward-Takahashi identity entails Exact in Chiral QCD Goldstones’ theorem: Solution of one-body problem solves the two-body problem Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 10

Γν (l 12, l 2) – dressed Transition Form Factor γ*(k 1)γ(k 2) → π0 quark-photon vertex S(l 2) γν*(k 1) Γπ(l 1, l 2) – pion Bethe-Salpeter amplitude S(l 12) Γμ (l 1, l 12) π0 S(l 1) – dressed-quark propagator γμ(k 1) Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production • All computable quantities • UV behaviour fixed by p. QCD • IR Behaviour informed by DSE- and lattice-QCD 11

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

Transition Form Factor γν*(k 1) S(l 12) γμ(k 2) S(l 1) π0 γ*(k 1)γ(k 2) → π0 • Calculation now straightforward • However, before proceeding, consider slight modification Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 13

Transition Form Factor γν*(k 1) S(l 12) γμ*(k 2) S(l 1) π0 γ*(k 1)γ*(k 2) → π0 • Calculation now straightforward • However, before proceeding, consider slight modification Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production Only changes cf. γ*(k 1)γ(k 2) → π0 14

Maris & Roberts, Phys. Rev. C 58 (1998) 3659 Transition Form Factor Maris & Tandy, Phys. Rev. C 65 (2002) 045211 γ*(k 1)γ*(k 2) → π0 Ø Anomalous Ward-Takahashi Identity chiral-limit: G(0, 0, 0) = Ø Inviolable prediction ½ – No computation believable if it fails this test – No computation believable if it doesn’t confront this test. Ø DSE prediction, model-independent: Q 2=0, G(0, 0, 0)=1/2 Corrections from mπ2 ≠ 0, just 0. 4% Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 15

Maris & Tandy, Phys. Rev. C 65 (2002) 045211 Transition Form Factor Ø p. QCD prediction γ*(k 1)γ*(k 2) → π0 Obtained if, and only if, asymptotically, Γπ(k 2) ~ 1/k 2 Ø Moreover, absolutely no sensitivity to φπ(x); viz. , pion distribution amplitude Ø Q 2=1 Ge. V 2: VMD broken Ø Q 2=10 Ge. V 2: GDSE(Q 2)/Gp. QCD(Q 2)=0. 8 Ø p. QCD approached from below p. QCD Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 16

π(P+Q) γ(Q) Pion Form Factor Fπ(Q 2) Maris & Tandy, Phys. Rev. C 62 (2000) 055204 π(P) Ø DSE computation appeared before data; viz. , a prediction Ø p. QCD-scale Q 2 Fπ(Q 2)→ 16πα(Q 2)fπ2 Ø VMD-scale: mρ2 Ø Q 2=10 Ge. V 2 p. QCD-scale/VMD-scale = 0. 08 Internally consistent calculation CAN & DOES overshoot p. QCD limit, and approach it from above; viz, at ≈ 12 Ge. V 2 Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 17

Single-parameter, Internally-consistent Framework Ø Dyson-Schwinger Equations – applied extensively to spectrum & interactions of mesons with masses less than 1 Ge. V; & nucleon & Δ. Ø 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. , Ø Ø Prediction of elastic pion and kaon form factors: nucl-th/0005015 Pion and kaon valence-quark distribution functions: 1102. 2448 [nucl-th] Unification of these and other observables – ππ scattering: hep-ph/0112015 Nucleon form factors: ar. Xiv: 0810. 1222 [nucl-th] Ø Readily extended to explain properties of the light neutral pseudoscalar mesons (η cf. ή): 0708. 1118 [nucl-th] Ø One parameter: gluon mass-scale = m. G = 0. 8 Ge. V Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 18

Transition Form Factor γν*(k 1) S(l 12) γμ(k 2) S(l 2) π0 S(l 1) γ*(k 1)γ(k 2) → π0 Maris & Tandy, Phys. Rev. C 65 (2002) 045211 Ø DSE result Ø no parameters varied; Ø exhibits ρ-pole; Ø perfect agreement with CELLO & CLEO Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 19

Transition Form Factor γν*(k 1) S(l 12) γμ(k 2) S(l 2) π0 S(l 1) Ø Three , internally-consistent calculations – Maris & Tandy γ*(k 1)γ(k 2) → π0 H. L. L. Roberts et al. , Phys. Rev. C 82 (2010) 065202 Hallmark of internally-consistent computations • Dash-dot: γ*(k 1)γ(k 2) → π0 • Dashed: γ*(k 1)γ*(k 2) → π0 – H. L. L Roberts et al. • Solid: γ*(k 1)γ(k 2) → π0 contact-interaction, omitting pion’s pseudovector component Ø All approach UV limit from below Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 20

Transition Form Factor γν*(k 1) S(l 12) γμ(k 2) S(l 2) π0 S(l 1) Ø All approach UV limit from below Ø UV scale in this case is 10 -times larger than for Fπ(Q 2): γ*(k 1)γ(k 2) → π0 H. L. L. Roberts et al. , Phys. Rev. C 82 (2010) 065202 Hallmark of internally-consistent computations – 8 π2 fπ2 = ( 0. 82 Ge. V )2 – cf. mρ2 = ( 0. 78 Ge. V )2 Ø 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

Transition Form Factor γν*(k 1) S(l 12) γμ(k 2) S(l 2) π0 S(l 1) Ø UV-behaviour: light-cone OPE Ø Integrand sensitive to endpoint: x=1 – Perhaps φπ(x) ≠ 6 x(1 -x) ? – Instead, φπ(x) ≈ constant? γ*(k 1)γ(k 2) → π0 H. L. L. Roberts et al. , Phys. Rev. C 82 (2010) 065202 Ø There is one-to-one correspondence between behaviour of φπ(x) and shortrange interaction between quarks Ø φπ(x) = constant is achieved if, and only if, the interaction between quarks is momentum-independent; namely, of the Nambu – Jona. Lasinio form Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 22

Guttiérez, Bashir, Cloët, Roberts ar. Xiv: 1002. 1968 [nucl-th] Ø Pion’s Bethe-Salpeter amplitude Pion’s GT relation Contact interaction Remains! Ø Dressed-quark propagator 1 MQ Ø Bethe-Salpeter amplitude can’t depend on relative momentum; propagator can’t be momentum-dependent Ø Solved gap and Bethe-Salpeter equations P 2=0: MQ=0. 4 Ge. V, Eπ=0. 098, Fπ=0. 5 MQ Nonzero and significant Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 23

Transition Form Factor γν*(k 1) S(l 12) γμ(k 2) S(l 2) π0 S(l 1) γ*(k 1)γ(k 2) → π0 H. L. L. Roberts et al. , Phys. Rev. C 82 (2010) 065202 Ø Comparison between Internally-consistent calculations: Ø φπ(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 Ø φπ(x) = 6 x(1 -x) yields p. QCD limit, approaches from below Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 24

Transition Form Factor γν*(k 1) S(l 12) γμ(k 2) S(l 2) π0 S(l 1) γ*(k 1)γ(k 2) → π0 H. L. L. Roberts et al. , Phys. Rev. C 82 (2010) 065202 Ø 2σ shift of any one of the last three high-points – one has quite a different picture Ba. Bar Ø η production CLEO Ø η' production Ø Both η & η’ production in perfect agreement with p. QCD Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 25

Epilogue Ø 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. Ø 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

π0(p) Epilogue π0(p’) Ø The large-Q 2 Ba. Bar data is inconsistent with – p. QCD – All extant, fully-self-consistent studies Ø Conclusion: the large-Q 2 data reported by Ba. Bar is not a true representation of the γ∗ γ → π0 transition form factor Ø Explanation? – Possible erroneous way to extract pion transition form factor from the data is problem of π0 π0 subtraction. – This channel – γ∗ γ → π0 π0 • scales in the same way (Diehl et al. , Phys. Rev. D 62 (2000) 073014) • Misinterpretation of some events, where 2 nd π0 is not seen, may be larger at large-Q 2. Craig Roberts, Physics Division, APS April Meeting 2011: Abelian Anomaly & Neutral Pion Production 27