Pion scattering and electro-production on nucleons in the resonance - PowerPoint PPT Presentation

Pion scattering and electro-production on nucleons in the resonance region in chiral quark models B. Golli, U of Ljubljana München | 16 June 2011 M. Fiolhais, P. Alberto, U of Coimbra S. Širca, U of Ljubljana P11: EPJA 38 (2008) 271 P11: EPJA 42 (2009) 185 S11: EPJA 47 (2011) 61 1

Motivation for study of P 11 and S 11 resonances • Large width of **** P 11 ( 1440 ) “Roper”; existence of *** P 11 ( 1710 ) unclear; difficult to identify directly (in cross-sections) • Atypical behaviour of Im T π N in J = I = 1 / 2 partial wave • Level ordering (parity inversion) of P 11 ( 1440 ) wrt. S 11 ( 1535 ) on Lattice • Many competing explanations of the Roper in models, e.g. q 3 g hybrid ⊲ Li, Burkert PRD 46 (1992) 70 qqqqq admixtures ⊲ Li, Riska PRC 74 (2006) 015202 dynamical generation by N + σ ⊲ Krehl++ PRC 62 (2000) 025207 ⊲ Döring++ NPA 829 (2009) 170 • Two negative-parity resonances: resonance M [MeV] Γ [MeV] decays S11(1535) 1535 150 πN 35–55 % , ηN 45–60 % 2 πN < 10 % , ( πR ) S11(1650) 1655 165 πN 60–95 % , ηN 3–10 % , K Λ 3–11 % 2 πN 10–20 % , ρ 4–12 % , ∆ 1–7 % 2

Present work • A coupled-channels approach that includes many-body states of quarks and mesons in the scattering formalism • Calculate scattering and electro-production amplitudes within the same framework • Investigate whether quark+meson description is sufficient i.e. no exotic degrees of freedom involved • Baryons treated as composite particles � → coupling constants and cut-offs of form-factors computed from the underlying model, not fitted � → smaller number of free parameters • Physical resonances appear as linear combinations of bare resonances • Bare quark-meson and quark-photon vertices are strongly modified by meson loops and mixing of resonances • K -matrix real & symmetric � → S -matrix unitary 3

Reminder: ∆ ( 1232 ) in quark models with pion cloud Helicity and electro-production amplitudes for γ ∗ N → ∆ ( 1232 ) → Nπ A 1 / 2 E 1 + /M 1 + S 1 + /M 1 + • M 1 + is ( ∼ 50 % pion cloud) + ( ∼ 50 % quarks) • E 1 + is ( ∼ 100 % pion cloud) Golli++ PLB 373 (1996) 229 4

Coupled-channel K -matrix formalism Our model Golli, Širca / EPJA 38 (2008) 271 Golli, Širca, Fiolhais / EPJA 42 (2009) 185 The meson field linearly couples to the quark core; no meson self-interaction � � � �� � V lmt (k)a lmt (k) + V lmt (k) † a † ω k a † d k H = H quark + lmt (k)a lmt (k) + lmt (k) lmt V lmt (k) induces also radial excitations of the quark core, e.g. 1 s → 2 s , 1 s → 1 p 1 / 2 , 1 s → 1 p 3 / 2 , . . . transitions For example: V(k) from Cloudy Bag Model 3 � k 2 1 ω s j 1 (kR bag ) V s → s σ i m τ i � (p − wave pions ) 1 mt (k) = t 2 f 12 π 2 ω k ω s − 1 kR bag i = 1 � 3 � k 2 1 j 1 (kR bag ) ω 2s ω s V s → 2 s m τ i � σ i (p − wave ) 1 mt (k) = t 2 f (ω 2s − 1 )(ω s − 1 ) 12 π 2 ω k kR bag i = 1 � 3 � k 2 1 ω p 1 / 2 ω s j 0 (kR bag ) s → p 1 / 2 τ i � (s − wave ) 1 = 0 ,t (k) = V t 2 f (ω p 1 / 2 + 1 )(ω s − 1 ) 4 π 2 ω k kR bag i = 1 � 3 � k 2 1 ω p 3 / 2 ω s j 2 (kR bag ) s → p 3 / 2 Σ i 2 m τ i � (d − wave ) (k) = V 2 mt t 2 f 2 π 2 ω k (ω p 3 / 2 − 2 )(ω s − 1 ) kR bag i = 1 5

Constructing the K -matrix Aim: include many-body states of quarks (and mesons) in the scattering formalism (Chew-Low type approach) Construct the K -matrix in spin-isospin (JI) basis: � ω M E B K JI k M W � Ψ MB M ′ B ′ MB = − π JI (W) || V M ′ (k) || Ψ B ′ � dressed states by using principal-value (PV) states � �� � � JI P ω M E B H − W [V(k M ) | Ψ B � ] JI a † (k M ) | Ψ B � | Ψ MB JI (W) � = − k M W normalized as � Ψ MB (W) | Ψ M ′ B ′ (W ′ ) � = δ(W − W ′ )δ MB,M ′ B ′ ( 1 + K 2 ) MB,MB 6

Ansatz for the channel PV states free meson bare (genuine)  � (defines the channel) baryons (3q)  � ω M E B  [a † (k M ) | � | Ψ MB Ψ B � ] JI c MB JI � = + R | Φ R � k M W R meson “clouds” with amplitudes χ  � d k χ M ′ B ′ MB (k, k M )  � ω k + E B ′ (k) − W [a † (k) | � Ψ B ′ � ] JI +  M ′ B ′ Above the meson-baryon ( MB ) threshold: � � ω M E B ω M ′ E B ′ χ M ′ B ′ MB (k, k M ) K M ′ B ′ MB (k, k M ) = π k M W k M ′ W 2 π decay through intermediate hadrons ( ∆ ( 1232 ) , N( 1440 ) ; σ , ρ , . . . ), e.g. πN → B ∗ → π ∆ → ππN , πN → B ∗ → σN → ( 2 π)N Solve Lippmann-Schwinger eqs for χ ; the solution has the form � χ M ′ B ′ MB (k, k M ) = − R V M ′ B ′ R (k) + D M ′ B ′ MB (k, k M ) c MB R dressed background vertex part 7

Solving the coupled equations The dressed vertices satisfy: � d k ′ K MB M ′ B ′ (k, k ′ ) V M ′ � B ′ R (k ′ ) V M B R (k) = V M B R (k) + ω ′ k + E B ′ (k ′ ) − W M ′ B ′ r MBR = V M B R (k) V M B R (k) Similarly, for the background part of the amplitude: D M ′ B ′ MB (k, k M ) = K M ′ B ′ MB (k, k M ) � d k ′ K M ′ B ′ M ′′ B ′′ (k, k ′ ) D M ′′ B ′′ MB (k ′ , k M ) � + ω ′ k + E B ′′ (k ′ ) − W M ′′ B ′′ The coefficients c MB R ′ of the quasi-bound states satisfy a set of equations: � A RR ′ (W) c MB R ′ (W) = V M B R (k M ) R ′ � d k V M ′ B ′ R (k) V M ′ � B ′ R ′ (k) A RR ′ = (W − M 0 R )δ RR ′ + ω k + E B ′ (k) − W B ′ 8

Calculating the K -matrix To solve the set of equations, diagonalize A to obtain U , along with the poles of the K - matrix, and wave-function normalization Z :   0 0 Z R (W)(W − M R )   UAU T =   0 0 Z R ′ (W)(W − M R ′ )   0 0 Z R ′′ (W)(W − M R ′′ ) As a consequence, Φ R mix: � � 1 � | � � Φ R � = U RR ′ | Φ R � V B R = U RR ′ V B R ′ Z R (W) R ′ R ′ Solution for the K -matrix:   � � V M ′ � � B R � V M ω M E B ω M ′ E B ′ B ′ R  K MB,M ′ B ′ = π (M R − W) + D MB,M ′ B ′ k M W k M ′ W R resonant background Solution for the T matrix: � T MB,M ′ B ′ = K MB,M ′ B ′ + i T MB,M ′′ B ′′ K M ′′ B ′′ ,M ′ B ′ M ′′ K ′′ 9

Pion electro-production: including the γN channel Only the strong T MB,M ′ B ′ appears on the RHS: � T MB,γN = K MB,γN + i T MB,M ′ B ′ K M ′ B ′ ,γN M ′ K ′ � � � � � ω γ E N Ψ M ′ B ′ � V γ � Ψ N K M ′ B ′ ,γN = − π JI k γ W Choosing a resonance, R = N ∗ , the principal-value state can be split into the resonant and background parts. Then � helicity amplitude � � �� � γ � ω γ E 1 ( bkg ) � Ψ ( res ) N ∗ (W) | ˜ N M MB γN = V γ | Ψ N � T MB MB + M MB γN π V BN ∗ ω M E B � �� � M ( res ) MB γN The resonant state takes the form: bare quark contrib meson cloud contrib   � d k   � � V M 1 BN ∗ (k) | Ψ ( res )  | � ω k + E B − W [a † (k) | Ψ B � ] JI � N ∗ (W) � = Φ N ∗ � −  Z N ∗ MB 10

Underlying quark model Cloudy Bag Model extended to pseudo-scalar SU(3) octet � � = − i L ( quark − meson ) a = 1 , 2 , . . . , 8 r − R bag 2 f qγ 5 λ a qφ a δ , CBM Parameters: R bag = 0 . 83 fm f π = 76 MeV f K = 1 . 2 f π or 1 . 2 f π f η = f π Similar results for 0 . 75 fm < R bag < 1 . 0 fm Free parameters: bare masses of the resonant states 11

πN → MB Pion scattering in P11 partial wave Channels : πN , π ∆ , σN , πR (preliminary: ηN , K Λ ) Parameters of the σN -channel: g σNR = 1 , m σ = 450 MeV , Γ σ = 550 MeV Thin lines: only N( 1440 ) included ( ( 1 s) 2 ( 2 s) 1 ) Thick lines : N( 1710 ) added ( ( 1 s) 1 ( 2 p) 2 ) with g πNN( 1710 ) = 0 g σNN( 1710 ) ≈ g σNN( 1440 ) 12

πN → MB Pion scattering in S11 partial wave Allow single-quark excitations (1 s → 1 p 1 / 2 and 1 s → 1 p 3 / 2 ) Φ ( 1535 ) = − sin ϑ s | 4 8 1 / 2 � + cos ϑ s | 2 8 1 / 2 � cos ϑ s | 4 8 1 / 2 � + sin ϑ s | 2 8 1 / 2 � Φ ( 1650 ) = ϑ s is a free parameter ( ≈ − 30 ◦ ) Myhrer, Wroldsen / Z. Phys. C 25 (1984) 281 13

Pion scattering in S11 partial wave Inelastic Channels P11, P13 contributions sizeable S11 contribution dominates P11, P13 negligible not yet included in our calculation PWA results on S11 : P11 : P13 uncertain 14

γp → N( 1440 ) [ 10 − 3 GeV − 1 / 2 ] Helicity amplitudes for 15

γN → Nπ 0 P11 transverse photo-production amplitudes [ 10 − 3 /m π ] [ 10 − 3 /m π ] 16

γp → S 11 ( 1535 ), S 11 ( 1650 ) Helicity amplitudes A 1 / 2 (Q 2 ) [ 10 − 3 GeV − 1 / 2 ] S 1 / 2 (Q 2 ) [ 10 − 3 GeV − 1 / 2 ] 17

γp → pπ 0 S11 transverse amplitudes [ 10 − 3 /m π ] [ 10 − 3 /m π ] 18

Eta, kaon photoproduction work in progress 19

Summary • Using a single set of parameters we reproduce the main features of pion- and photon-induced production of π , η , and K mesons in P11 and S11 partial waves. • Importance of the meson cloud: – it enhances the bare baryon-meson couplings; – it improves the behaviour of the helicity amplitudes at low Q 2 . • Enhancement of couplings stronger for P11 and P33 than in the case of S11 resonances which are dominated by quark-core contributions. 20

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