The Pion Form Factor in the Covariant Spectator Theory

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1 The Pion Form Factor in the Covariant Sectator Theory Alfred Stadler University of Évora, and Centro de Física Teórica de Partículas (CFTP) Lisbon in collaboration with Teresa Peña, Elmar Biernat (CFTP Lisbon) Franz Gross (JLab) ublished in PRD89, (04) and PRD89, (04) Electron-Nucleus Scattering XIII, June 3-7, 04, Marciana Marina, Isola d Elba

2 New challenges in meson hysics Motivation Ucoming intense exerimental activity to exlore meson structure GlueX (Jlab), PANDA (GSI) Search for exotic mesons (hybrids, glueballs) Need to understand conventional q q -mesons in more detail Pion transition form factors Hadronic contributions to light-by-light scattering Source of uncertainty in rediction of anomalous magnetic moment of the muon Imortant in search for hysics beyond the Standard Model (a) [L.D.].

3 A unified model for all q q mesons Much imortant work was done on meson structure Cornell-tye static otential models (Isgur and Godfrey, Sence and Vary, etc.) But: nonrelativistic (or relativized ); structure of constituent quark and relation to existence of zero-mass ion in chiral limit not addressed Dyson-Schwinger aroach (C. Roberts et al.) But: Euclidean sace; only Lorentz vector confining interaction Lattice QCD (also Euclidean sace), EFT, Bethe-Saleter, Light-front, Point-form,... Our objectives Construct a model to describe all q q -tye mesons Covariant framework (CST) - light quarks require relativistic treatment Work in Minkowski sace (hysical momenta but have to confront singularities) Imrove on revious work by Gross, Milana, Savkli Quark self-energy from q q interaction kernel (consistent quark mass function) Chiral symmetry: massless ion in chiral limit of vanishing bare quark mass (NJL) Calculate meson sectrum, bound-state vertex functions and form factors Learn about confining interaction (scalar vs. vector, etc.)

4 Covariant two-body bound-state equation Start from the Bethe-Saleter (BS) equation for the bound-state vertex function Γ d 4 k BS(, P )i ( ) 4 V(, k; P ) S (k ) BS(k, P) S (k ) k P V k P P k k k (k k )/ (k, P) or (k,k ) V(, k) total momentum relative momentum vertex function kernel The dressed roagator satisfies the Dyson equation S() S S 0 S 0 Σ S m 0 / () i ( ) M( ) / i dressed quark roagator () A( )/B( ) ( ) V B( ) S 0 () m 0 / i bare roagator bare quark mass dressed mass satisfies m M(m ) M( ) m 0 A( ) B( ) self-energy wave function renormalization mass function

5 Kernel truncation The kernel contains all two-body irreducible diagrams V... In the BS equation the kernel is effectively iterated to all orders But the comlete kernel is a sum of an infinite number of irreducible diagrams has to be truncated (most often: ladder aroximation) Problems with the ladder truncation No one-body limit (missing crossed ladders) Not best suited to describe bound states (crossed-ladder contributions are significant) see Nieuwenhuis and Tjon, PRL77, 84 (996)...

6 From Bethe-Saleter to CST h i h i Covariant Sectator Theory (CST) Mini-review: A.S., F. Gross, Few-Body Syst. 49, 9 (00) E k µ E k µ Im k 0 E k µ E k µ Re k 0 Integration over relative energy k0: Kee only ole contributions from roagators Move kernel oles to higher-order kernels (reorganization of the BS series) Cancellations between ladder and crossed ladder diagrams can occur Reduction to 3D loo integrations, but covariant Works very well in few-nucleon systems If bound-state mass µ is small: both oles are close together (both imortant) Symmetrize ole contributions from both half lanes: resulting equation is symmetric under charge conjugation BS vertex (arox.) CST vertices { }

7 Four-channel CST equation Closed set of equations when external legs are systematically laced onshell Solutions: bound state masses μ and corresonding vertices Γ Aroximations for secial cases Smooth one-body limit: Dirac equation Smooth nonrelativistic limit: Schrödinger equation Heavy-light quark systems: channels Large bound-state mass: channel

8 Confining otential in momentum sace Phenomenological q q kernel Insired by Cornell otential: V (r) r C s r NR linear otential in momentum sace: Fourier transform of screened otential e r r r But simler: r lim e r!0 ṼA(r) Ṽ A (0) d V L (q) V A (q) ( ) 3 3 q 0 FT: (q) ( ) 3 V A(q 0 ) with hv L i() Static QCD otential from the lattice Allton et al, UKQCD Collab., PRD 65, (00) V A (q) 8 q 4 Gross, Milana, PRD 43, 40 (99) Savkli, Gross, PRC 63, (00) d 3 k ( ) 3 V L( k) (k) 8 d 3 k (k) () ( ) 3 ( k) 4 highly singular automatic subtraction only a Cauchy rincial value singularity remains

9 Covariant confining kernel in CST Covariant generalization: q! q This leads to a kernel that acts like d 3 k m hv L i() ( ) 3 V L (, E ˆk) (ˆk) k 8 ˆk d 3 k m (ˆk) (ˆ R ) ( ) 3 E k ( ˆk) 4 initial state: either quark or antiquark onshell any regular function ˆk (E k, k) ˆ R (E R, R ) Proerties: Subtraction regularizes kernel to Cauchy rincial value Nonrelativistic limit linear otential Satisfies the condition hv L i V L (, ˆk) 0 k R R ( 0, ) value of k at which kernel corresonds to becomes singular Shorthand d 3 k m ( ) 3! E k Ṽ nr L (r 0) 0 k But does it still confine? Yes: the vertex function vanishes if both quarks are onshell! 0 More details: Savkli, Gross, PRC 63, (00)

10 Sontaneous chiral symmetry breaking - ion equation Pion bound-state equation for finite mass μ { } For μ 0 this becomes (in the rest frame) lim µ!0 P (µ, 0) P 0 P 0 5 G( ) 5 G 0 5 G 0 For a kernel with Lorentz scalar and vector structure V(, k) V S (, k) 4 V V (, k)g µ µ one gets a condition for the existence of a massless ion solution 0 4m( B 0 ) k ale V V (ˆ, ˆk)V V (ˆ, ˆk) VS (ˆ, ˆk) B 0 B(m V S (ˆ, ˆk) ) 0 (m )

11 Sontaneous chiral symmetry breaking M ( )() /() the Sontaneous chiral symmetry breaking in the CST Sontaneous chiral symmetry breaking in CST ( ) in Dressed quark roagator S() Sontaneous chiral symmetry breaking () ) the CST m0 () i M ( i / (( )) M( ) ( ) M Dressed quark roagator is Dressed quark roagator is ( ) M ( ) S( ) S( ) 0 Dressed quark roagator is S( ) mσ( ) Σ(M )( ) M m0 ( )iε iε 0 m 0 Σ( ) M ( ) iε ( ) () A( ) /B( ) self-energy The q q kernel determines CST-Dyson equation: The dressed quark B( ) m A( ) M ( ) B( ) withfunction renormalization withwave mass function with m A m A A ) ) ) ( )M Σ( ) Σ(A() ) A(B( ) B( M ( 0 ) ; 0 (m0;) ( Σ( ) A( ) B( ) M ( ) ; ( ) B B B B the quark self-energy B the self-energy Σ is comuted from the kernel: the self-energy Σ is comuted from the kernel: () the self-energy Σ is comuted from the kernel: m m0 () / 0 / ) Σ( )m0 m0 m0 m 0Σ( m0 - m0- Σ( ) ( ) ( ) B Σ( ) Σ( ) Σ( ) ga Theequation ga equation gives the constraint The the constraint M (m )(m gives M (m ) m m The ga equation gives the(m constraint M ) m mass is the solution of the ga equation M )m In rest frame, (0, 0) : h i comare with the zero mass ion condition comare with the zero mass ion condition 0 A(0 ) VS (, k ) VS (, comare k ) VVwith (,the k ) zero VVmass (,ionk )condition 4 k 0 Ek B(0 ) VS (, k ) VS (, k ) VV (, k ) VV (, k ) 40 k m are the These These are the These are same if: same if: same if: 0m0 0 m0 A m0 ) 0 0 A(m V (, k ) ( 0, k ) 0 S B0 kvs B(m VS (), k ) k k 0 (m ) We can use M (m ) m m0 A0 mb0 in the equation for A at 0 m this yields another constraint

12 Sontaneous chiral symmetry breaking { The two equations become identical for m 0 0 rovided that Comare: k m 0/A 0 4m( B 0 ) 0 hv S (, ˆk)V S (, ˆk) i 0 4m( B 0 ) 0 k k constraint from ga equation for A hv V (ˆ, ˆk)V V (ˆ, ˆk)VS (ˆ, ˆk)V i S (ˆ, ˆk) ale V V (ˆ, ˆk)V V (ˆ, ˆk) VS (ˆ, ˆk) V S (ˆ, ˆk) zero-mass ion condition When in the chiral limit (m0 0), the Lorentz scalar interaction is zero if a finite quark mass m can be generated then a zero-mass ion solution exists or it satisfies the constraint (Goldstone boson) The linear confining kernel satisfies V L (, ˆk) 0 the confining kernel can k have a scalar comonent! A( The confining interaction does not contribute to { ) (m 0 0) ion equation Also: if m 0 > 0 the condition for A( )guarantees that no zero-mass ion solution exists the real ion must have a finite mass

13 NJL mechanism in diagrammatic form In the chiral limit: -body CST-Dyson equation scalar art multily by γ 5 and attach quark lines and a ion line with P0 -body ion equation toologically equivalent

14 Lorentz structure of the kernel We construct a simle exloratory model: Mixed scalar-vector linear confining kernel (becomes linear otential in nonrel. limit) V L (, ˆk) ( ) µ µ VL (, ˆk) Pure vector constant (in r-sace) interaction with V C (, ˆk) 4 g µ µ V C (, ˆk) V C (, ˆk; P )Ch( )h( )h(ˆk )h(k )( ) 3 E k m k 3 ( k) Quark form factors h rovide regularization (one factor for each quark line) Can be viewed as vertex corrections h( ) h( ) k In the chiral limit, VL does not contribute to A (or the ion vertex) For one gets also B L ( )0 L ( )0 k k A L ( )0 no contribution from the confining interaction to the self-energy

15 Quark mass function The constant vector interaction contributes only to A( ) M( )A( )m 0 Ch (m )h ( )m 0 C m c m 0 O(m 0) with m 0! 0 M ( )m h ( ) Lattice QCD data from Bowman et al PRD 7, 005 extraolated to chiral limit in chiral limit Fit to (Euclidean) LQCD data h( ) m fit for < 0 (to 50 oints with > -.94 GeV ).04 GeV m GeV 0.4! Lattice data don t converge to m0 for large - (finite lattice sacing effect) M GeV GeV Alfred Stadler, Elmar University Biernat of Évora (CFTP/IST) The Pion Quarks Form and Factor mesons in the Covariant in CST Sectator Theory May, 04 Elba 7 / XIII, 304

16 Quark mass function for finite m0 C m c m 0 O(m 0) c is fit (other arameters fixed in the chiral limit) M GeV m GeV, m GeV M GeV m GeV, m GeV GeV GeV M GeV m GeV, m GeV M GeV m GeV, m 0.46 GeV GeV GeV Elmar Biernat (CFTP/IST) Quarks and mesons in CST May, 04 8 / 3

17 Pion form factor The ion current in CST: triangle diagrams with 6 roagator oles (each diagram) RIA (Relativistic Imulse Aroximation): kee only the sectator oles q P P q P P j µ S( ) S( ) j µ S( ) S( ) P Γ Λ( ˆk) Γ P P Γ Λ(ˆk) Γ P Analysis of ole structure: RIA is a good aroximation for large Q (any μ) large μ (any Q ) For small Q and small μ, the remaining oles contribute significantly need to calculate the CIA (Comlete Imulse Aroximation) Needed ingredients: ion vertex function dressed quark current

18 A simle ion vertex function In the chiral limit, the self-energy function A( ) and the vertex Γ() satisfy the same equation we already have a ion vertex! General BS vertex structure for seudoscalar bound states BS(, )G (, ) 5 G (, )(/ 5 5 / ) G (, )(/ 5 5 / )G 3 (, )/ 5 / (, ) G( ) 5 in chiral limit (and rest frame) For real ion, assume the chiral limit structure dominates P 0 ˆ P P ˆ near the chiral limit (, ) G 0 h( ) 5

19 Dressed quark current Use the rescrition by Gross & Riska to derive a conserved quark current S() S( ) h( ) j µ h( ) h S() h S( ) h j µ h Vertex form factors are absorbed into modified roagators S() h ( )S() Imose Ward-Takahashi identity on reduced current q µ j Rµ ( 0,) e S () e S ( 0 ) Structure of the current: j µ R f ale j µ R (0,)h ( 0 )j µ ( 0,)h ( ) µ ale i µ q m 0 0 µ µ g 0 µ off-shell form factors determined by WTI in terms of h( ) M( ) / M( ) Differs from Ball-Chiu current (used by Tandy and Maris) by a transverse iece WTI cannot determine the current uniquely need a dynamical calculation

20 Results for the ion form factor The ion form factor is calculated with different ion masses Best fit with μ0.4 GeV (somewhat large, but allows us to use RIA at small Q!) Λ i F Π Q, Μ i Λ i F Π Q, Μ i µ 0.4 GeV, µ 0.8 GeV, µ 0.4 GeV Λ i F Π Q, Μ i Q GeVQ GeV Q GeV ρ ole Q FΠ Q Q GeV Fπ shows correct monoole fall-off F (Q ) Q µ with ' 0.63 GeV Q: Why does the model work well Q F π indeendent without a ρ-ole of h form (VMD)? factors this is indeendent of h( ) scaling Remarkable scaling relations hold at large Q Should relation calculate the dressed (used in the figure for μ0.4 and 0.8 GeV) F π (Q quark, λµ) current Q µ λdynamically F π (Q,µ) F (Q, µ) Q µ (to be done soon!) ' F (Q,µ) correct monoole behaviour F π (Q ) Q µ Q ν simle ρ-ole our model, μ0.4 GeV RIA fails for small ion masses µ

21 Summary Formalism Develoed the CST formalism for a dynamical q q model for all mesons CST equations are solved in Minkowski sace Consistency between bound-state and mass ga equations Describes confinement and sontaneous chiral symmetry breaking Can be alied in the time-like region Model calculations A very simle model can yield good agreement of the quark mass function with lattice calculations, and of the ion form factor with exerimental data Work in rogress or lanned for the near future Solution of the full q q CST equation and fit to the meson sectrum Study interaction kernels with different Lorentz structure; add gluon exchange Calculate quark-hoton vertex dynamically Study ππ scattering (constraints from axial-vector WTI)

22 But what does the ion look like? Most likely