Images of Dynamical Chiral Symmetry Breaking Ian Cloët Argonne National Laboratory Collaborators Lei Chang – Adelaide Craig Roberts – ANL Sebastian Schmidt – Jülich Jorge Segovia – ANL Peter Tandy – Kent State MESON 2014 Krakow, 29 May – 3 June 2014
The Challenge of QCD QCD is the only known example in nature of a fundamental quantum field theory that is innately non-perturbative a priori no idea what such a theory can produce Solving QCD will have profound implications for our understanding of the natural world e.g. it will explain how massless gluons and light quarks bind together to form hadrons, and thereby explain the origin of ∼ 98% of the mass in the visible universe given QCDs complexity, the best promise for progress is a strong interplay between experiment and theory QCD is characterized by two emergent phenomena: confinement & dynamical chiral symmetry breaking (DCSB) a world without DCSB would be profoundly different, e.g. m π ∼ m ρ Must discover the origin of confinement, its relationship to DCSB and understand how these phenomenon influence hadronic obserables table of contents MESON 29 May – 3 June 2 / 20
QCDs Dyson-Schwinger Equations The equations of motion of QCD ⇐ ⇒ QCDs Dyson–Schwinger equations an infinite tower of coupled integral equations must implement a symmetry preserving truncation The most important DSE is QCDs gap equation = ⇒ quark propagator − 1 − 1 = + ingredients – dressed gluon propagator & dressed quark-gluon vertex [M. S. Bhagwat et al. , Phys. Rev. C 68 , 015203 (2003)] Z ( p 2 ) S ( p ) = p + M ( p 2 ) i/ S ( p ) has correct perturbative limit mass function, M ( p 2 ) , exhibits dynamical mass generation complex conjugate poles no real mass shell = ⇒ confinement table of contents MESON 29 May – 3 June 3 / 20
Pion’s Parton Distribution Amplitude pion’s PDA – ϕ π ( x ) : is a probability amplitude that describes the momentum distribution of a quark and antiquark in the bound-state’s valence Fock state it’s a function of the lightcone momentum fraction x = k + p + and the scale Q 2 PDA PDA PDA P P D D A A GPDs GPDs PDAs enter numerous hard exclusive scattering processes table of contents MESON 29 May – 3 June 4 / 20
Pion’s Parton Distribution Amplitude pion’s PDA – ϕ π ( x ) : is a probability amplitude that describes the momentum distribution of a quark and antiquark in the bound-state’s valence Fock state it’s a function of the lightcone momentum fraction x = k + p + and the scale Q 2 The pion’s PDA is defined by d 4 k � k + − x p + � γ + γ 5 S ( k ) Γ π ( k, p ) S ( k − p ) � � � f π ϕ π ( x ) = Z 2 (2 π ) 2 δ Tr S ( k ) Γ π ( k, p ) S ( k − p ) is the pion’s Bethe-Salpeter wave function in the non-relativistic limit it corresponds to the Schrodinger wave function ϕ π ( x ) : is the axial-vector projection of the pion’s Bethe-Salpeter wave function onto the light-front [pseudo-scalar projection also non-zero] Pion PDA is interesting because it is calculable in perturbative QCD and, e.g., in this regime governs the Q 2 dependence of the pion form factor Q 2 →∞ Q 2 F π ( Q 2 ) → 16 π f 2 π α s ( Q 2 ) ϕ asy − ⇐ ⇒ π ( x ) = 6 x (1 − x ) table of contents MESON 29 May – 3 June 5 / 20
QCD Evolution & Asymptotic PDA ERBL ( Q 2 ) evolution for pion PDA [c.f. DGLAP equations for PDFs] � 1 µ d dµ ϕ ( x, µ ) = dy V ( x, y ) ϕ ( y, µ ) 0 This evolution equation has a solution of the form � � � ϕ π ( x, Q 2 ) = 6 x (1 − x ) n =2 , 4 ,... a 3 / 2 n ( Q 2 ) C 3 / 2 1 + (2 x − 1) n α = 3 / 2 because in Q 2 → ∞ limit QCD is invariant under the collinear conformal group SL (2; R ) Gegenbauer- α = 3 / 2 polynomials are irreducible representations SL (2; R ) The coefficients of the Gegenbauer polynomials, a 3 / 2 n ( Q 2 ) , evolve logarithmically to zero as Q 2 → ∞ : ϕ π ( x ) → ϕ asy π ( x ) = 6 x (1 − x ) At what scales is this a good approximation to the pion PDA E.g., AdS/QCD find ϕ π ( x ) ∼ x 1 / 2 (1 − x ) 1 / 2 at Q 2 = 1 GeV 2 expansion in terms of C 3 / 2 a 3 / 2 32 / a 3 / 2 (2 x − 1) convergences slowly: ∼ 10 % n 2 table of contents MESON 29 May – 3 June 6 / 20
Pion PDA from the DSEs [L. Chang, ICC, et al. , Phys. Rev. Lett. 110 , 132001 (2013)] [C.D. Roberts, Prog. Part. Nucl. Phys. 61 50 (2008)] asymptotic 1 . 4 1 . 2 1 . 0 ϕ ( x ) 0 . 8 rainbow-ladder 0 . 6 0 . 4 0 . 2 DCSB improved 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 x Both DSE results, each using a different Bethe-Salpeter kernel, exhibit a pronounced broadening compared with the asymptotic pion PDA scale of calculation is given by renormalization point ζ = 2 GeV Broading of the pion’s PDA is directly linked to DCSB As we shall see the dilation of pion’s PDA will influence the Q 2 evolution of the pion’s electromagnetic form factor table of contents MESON 29 May – 3 June 7 / 20
Pion PDA from lattice QCD asymptotic 1 . 4 Lattice QCD can only determine one 1 . 2 non-trivial moment 1 . 0 � 1 ϕ ( x ) 0 . 8 dx (2 x − 1) 2 ϕ π ( x ) = 0 . 27 ± 0 . 04 lattice QCD 0 . 6 0 0 . 4 [V. Braun et al. , Phys. Rev. D 74 , 074501 (2006)] DCSB improved 0 . 2 [ICC, et al. , Phys. Rev. Lett. 111 , 092001 (2013)] scale is Q 2 = 4 GeV 2 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 x Standard practice to fit first coefficient of “ asymptotic expansion ” to moment � � ϕ π ( x, Q 2 ) = 6 x (1 − x ) � n =2 , 4 ,... a 3 / 2 n ( Q 2 ) C 3 / 2 1 + (2 x − 1) n however this expansion is guaranteed to converge rapidly only when Q 2 → ∞ this procedure results in a double-humped pion PDA Advocate using a generalized expansion ϕ π ( x, Q 2 ) = N α x α − 1 / 2 (1 − x ) α − 1 / 2 � � � n =2 , 4 ,... a α n ( Q 2 ) C α 1 + n (2 x − 1) Find ϕ π ≃ x α (1 − x ) α , α = 0 . 35 +0 . 32 − 0 . 24 ; good agreement with DSE: α ≃ 0 . 30 table of contents MESON 29 May – 3 June 8 / 20
Pion PDA from lattice QCD Lattice QCD can only determine one non-trivial moment � 1 dx (2 x − 1) 2 ϕ π ( x ) = 0 . 27 ± 0 . 04 0 [V. Braun et al. , Phys. Rev. D 74 , 074501 (2006)] scale is Q 2 = 4 GeV 2 Standard practice to fit first coefficient of “ asymptotic expansion ” to moment � � ϕ π ( x, Q 2 ) = 6 x (1 − x ) � n =2 , 4 ,... a 3 / 2 n ( Q 2 ) C 3 / 2 1 + (2 x − 1) n however this expansion is guaranteed to converge rapidly only when Q 2 → ∞ this procedure results in a double-humped pion PDA Advocate using a generalized expansion ϕ π ( x, Q 2 ) = N α x α − 1 / 2 (1 − x ) α − 1 / 2 � � � n =2 , 4 ,... a α n ( Q 2 ) C α 1 + n (2 x − 1) Find ϕ π ≃ x α (1 − x ) α , α = 0 . 35 +0 . 32 − 0 . 24 ; good agreement with DSE: α ≃ 0 . 30 table of contents MESON 29 May – 3 June 9 / 20
When is the Pion’s PDA Asymptotic [I. C. Cloët, et al. , Phys. Rev. Lett. 111 , 092001 (2013)] [T. Nguyen, et al. , Phys. Rev. C 83 , 062201 (2011)] asymptotic 1 . 4 1 . 2 1 . 0 ϕ ( x ) Q 2 = 4 GeV 2 0 . 8 0 . 6 0 . 4 Q 2 = 100 GeV 2 0 . 2 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 x Under leading order Q 2 evolution the pion PDA remains broad to well above Q 2 > 100 GeV 2 , compared with ϕ asy π ( x ) = 6 x (1 − x ) Consequently, the asymptotic form of the pion PDA is a poor approximation at all energy scales that are either currently accessible or foreseeable in experiments on pion elastic and transition form factors Importantly, ϕ asy π ( x ) is only guaranteed be an accurate approximation to ϕ π ( x ) when pion valence quark PDF satisfies: q π v ( x ) ∼ δ ( x ) This is far from valid at forseeable energy scales table of contents MESON 29 May – 3 June 10 / 20
When is the Pion’s Valence PDF Asymptotic 0 . 7 � x q v ( x ) � � x g ( x ) � 0 . 6 � x sea ( x ) � momentum fractions 0 . 5 0 . 4 0 . 3 0 . 2 0 . 1 LHC 0 10 0 10 1 10 2 10 3 10 4 10 5 10 6 Q (GeV) LO QCD evolution of momentum fraction carried by valence quarks � γ (0)2 / (2 β 0 ) γ (0)2 � α s ( Q 2 ) qq qq � x q v ( x ) � ( Q 2 ) = � x q v ( x ) � ( Q 2 0 ) where > 0 α s ( Q 2 0 ) 2 β 0 therefore, as Q 2 → ∞ we have � x q v ( x ) � → 0 implies q v ( x ) = δ ( x ) At LHC energies valence quarks still carry 20% of pion momentum the gluon distribution saturates at � x g ( x ) � ∼ 55% Asymptotia is a long way away! table of contents MESON 29 May – 3 June 11 / 20
Pion Elastic Form Factor [L. Chang, I. C. Cloët, et al. , Phys. Rev. Lett. 111 , 141802 (2013)] Extended the pre-experiment DSE 0 . 5 prediction to Q 2 > 4 GeV 2 0 . 4 Q 2 F π ( Q 2 ) Predict max at Q 2 ≈ 6 GeV 2 ; within 0 . 3 using DSE pion PDA forthcoming JLab data domain accessible at JLab12 0 . 2 0 . 1 Magnitude directly related to DCSB using asymptotic pion PDA 0 0 5 10 15 20 Q 2 The QCD prediction can be expressed as � 1 Q 2 ≫ Λ 2 w π = 1 dx 1 Q 2 F π ( Q 2 ) 16 π f 2 π α s ( Q 2 ) w 2 QCD ∼ π ; x ϕ π ( x ) 3 0 Using ϕ asy π ( x ) significantly underestimates experiment Within DSEs there is consistency between the direct pion form factor calculation and that obtained using the DSE pion PDA 15% disagreement explained by higher order/higher-twist corrections We predict that QCD power law behaviour sets in at Q 2 ∼ 8 GeV 2 table of contents MESON 29 May – 3 June 12 / 20
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